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Differential Equations

Differential equations

In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. Differential equations have many applications in physics and chemistry, and are widespread in mathematical models explaining biological, social, and economic phenomena. Differential equations are divided into two types:
- An ordinary differential equation (ODE) only contains functions of one independent variable, and derivatives in that variable.
- A partial differential equation (PDE) contains functions of multiple independent variables and their partial derivatives. The order of a differential equation is that of the highest derivative that it contains. For instance, a first-order differential equation contains only first derivatives. A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process. The study of differential equations is a wide field in both pure and applied mathematics. Pure mathematicians study the types and properties of differential equations, such as whether or not solutions exist, and should they exist, whether they are unique. Applied mathematicians, physicists and engineers are usually more interested in how to compute solutions to differential equations. These solutions are then used to simulate celestial motions, design bridges, automobiles, aircraft, sewers, etc. Often, these equations do not have closed form solutions and are solved using numerical methods. Famous differential equations include:
- Maxwell's equations in electromagnetism
- Einstein's field equation in general relativity
- The Schrödinger equation in quantum mechanics
- The heat equation in thermodynamics
- The wave equation
- Laplace's equation, which defines harmonic functions
- Poisson's equation
- The Navier-Stokes equations in fluid dynamics
- The Lotka-Volterra equation in population dynamics
- The Black-Scholes equation in finance
- The Cauchy-Riemann equations in complex analysis

References

- D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.
- A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition)", Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2.
- W. Johnson, [http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=abv5010.0001.001 A Treatise on Ordinary and Partial Differential Equations], John Wiley and Sons, 1913, in [http://hti.umich.edu/u/umhistmath/ University of Michigan Historical Math Collection]
- ko:미분방정식 ja:微分方程式 th:สมการเชิงอนุพันธ์

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.
History
:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.
Inspiration, pure and applied mathematics, and aesthetics
Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.
Notation, language, and rigor
Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.
Is mathematics a science?
Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.
Overview of fields of mathematics
As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.
Major themes in mathematics
An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.
Quantity
This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements. : :Number – Natural number – Integers – Rational numbers – Real numbers – Complex numbers – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – p-adic numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base
Structure
:Pinning down ideas of size, symmetry, and mathematical structure. : :Abstract algebra – Number theory – Algebraic geometry – Group theory – Monoids – Analysis – Topology – Linear algebra – Graph theory – Universal algebra – Category theory – Order theory – Measure theory
Space
:A more visual approach to mathematics. : :Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry
Change
:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions
Foundations and methods
:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols
Discrete mathematics
:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory
Applied mathematics
:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory
Important theorems
:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.
Important conjectures
See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.
History and the world of mathematicians
See also list of mathematics history topics :History of mathematics – Timeline of mathematics – Mathematicians – Fields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematical abilities and gender issues
Mathematics and other fields
:Mathematics and architecture – Mathematics and education – Mathematics of musical scales
Common misconceptions
Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.
See also

- Mathematical game
- Mathematical problem
- Mathematical puzzle
- Puzzle
Bibliography

- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).
External links

- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

Derivative

In mathematics, the derivative is one of the two central concepts of calculus. (The other is the integral; the two are related via the fundamental theorem of calculus.) The simplest type of derivative is the derivative of a real-valued function of a single real variable. It has several interpretations:
- The derivative gives the slope of a tangent to the graph of the function at a point. In this way, derivatives can be used to determine many geometrical properties of the graph, such as concavity or convexity.
- The derivative provides a mathematical formulation of rate of change; it measures the rate at which the function's value changes as the function's argument changes. This derivative is the kind usually encountered in a first course on calculus, and historically was the first to be discovered. However, there are also many generalizations of the derivative. The remainder of this article discusses only the simplest case (real-valued functions of real numbers).

Differentiation and differentiability

In physical terms, differentiation expresses the rate at which a quantity, y, changes with respect to the change in another quantity, x, on which it has a functional relationship. Using the symbol Δ to refer to change in a quantity, this rate is defined as a limit of difference quotients :

\frac

as Δx approaches 0. In Leibniz's notation for derivatives, the derivative of y with respect to x is written :

\frac

suggesting the ratio of two infinitesimal quantities. The above expression is pronounced in various ways such as "dy by dx" or "dy over dx". The form "dy dx" is also used conversationally, although it may be confused with the notation for element of area. Modern mathematicians do not bother with "dependent quantities", but simply state that differentiation is a mathematical operation on functions. The precise definition of this operation (which therefore need not deal with infinitesimal quantities) is given as: :

\lim_\frac.

A function is differentiable at a point x if its derivative exists at that point; a function is differentiable on an interval if it is differentiable at every x within the interval. If a function is not continuous at x, then there is no tangent line and the function is therefore not differentiable at x; however, even if a function is continuous at x, it may not be differentiable there. In other words, differentiability implies continuity, but not vice versa. One famous example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function. The derivative of a differentiable function can itself be differentiable. The derivative of a derivative is called a second derivative. Similarly, the derivative of a second derivative is a third derivative, and so on.

Newton's difference quotient

The derivative of a function f at x is geometrically the slope of the tangent line to the graph of f at x. Without the concept which we are about to define, it is impossible to directly find the slope of the tangent line to a given function, because we only know one point on the tangent line, namely (x, f(x)). Instead, we will approximate the tangent line with multiple secant lines that have progressively shorter distances between the two intersecting points. When we take the limit of the slopes of the nearby secant lines in this progression, we will get the slope of the tangent line. The derivative is then defined by taking the limit of the slope of secant lines as they approach the tangent line. tangent tangent To find the slopes of the nearby secant lines, choose a small number h. h represents a small change in x, and it can be either positive or negative. The slope of the line through the points (x,f(x)) and (x+h,f(x+h)) is :

.

This expression is Newton's difference quotient. The derivative of f at x is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line: :

f'(x)=\lim_.

difference quotient If the derivative of f exists at every point x in the domain, we can define the derivative of f to be the function whose value at a point x is the derivative of f at x. Since immediately substituting 0 for h results in division by zero, calculating the derivative directly can be unintuitive. One technique is to simplify the numerator so that the h in the denominator can be cancelled. This happens easily for polynomials; see calculus with polynomials. For almost all functions however, the result is a mess. Fortunately, many guidelines exist.

Notations for differentiation

Lagrange's notation

The simplest notation for differentiation that is in current use is due to Joseph Louis Lagrange and uses the prime mark:

Leibniz's notation

The other common notation is Leibniz's notation for differentiation which is named after Leibniz. For the function whose value at x is the derivative of f at x, we write: : $\frac.$ We can write the derivative of f at the point a in two different ways: : $\frac\left.\right|_ = \left(\frac\right)(a).$ If the output of f(x) is another variable, for example, if y=f(x), we can write the derivative as: : $\frac.$ Higher derivatives are expressed as : $\frac$ or $\frac$ for the n-th derivative of f(x) or y respectively. Historically, this came from the fact that, for example, the 3rd derivative is: : $\frac$ which we can loosely write as: : $\left(\frac\right)^3 \left(f(x)\right) =
\frac \left(f(x)\right).$ Dropping brackets gives the notation above. Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially relevant for partial differentiation. It also makes the chain rule easy to remember, because the "du" terms appear symbolically to cancel: : $\frac = \frac \cdot \frac.$ (In the popular formulation of calculus in terms of limits, the "du" terms cannot literally cancel, because on their own they are undefined; they are only defined when used together to express a derivative. In nonstandard analysis, however, they can be viewed as infinitesimal numbers that cancel.)
Newton's notation
Newton's notation for differentiation (also called the dot notation for differentiation) requires placing a dot over the function name: : $\dot = \frac = x'(t)$ : $\ddot = x$ (t) and so on. Newton's notation is mainly used in mechanics, normally for time derivatives such as velocity and acceleration, and in ODE theory. It is usually only used for first and second derivatives.

Euler's notation

Euler's notation uses a differential operator, denoted as D, which is prefixed to the function with the variable as a subscript of the operator: This notation can also be abbreviated when taking derivatives of expressions that contain a single variable. The subscript to the operator is dropped and is assumed to be the only variable present in the expression. In the following examples, u represents any expression of a single variable: Euler's notation is useful for stating and solving linear differential equations.

Critical points

Points on the graph of a function where the derivative is undefined or equals zero are called critical points or sometimes stationary points (in the case where the derivative equals zero). If the second derivative is positive at a critical point, that point is a local minimum; if negative, it is a local maximum; if zero, it may or may not be a local minimum or local maximum. Taking derivatives and solving for critical points is often a simple way to find local minima or maxima, which can be useful in optimization. In fact, local minima and maxima can only occur at critical points. This is related to the extreme value theorem.

Physics

Arguably the most important application of calculus to physics is the concept of the "time derivative"—the rate of change over time—which is required for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in Newtonian physics:
- Velocity (instantaneous velocity; the concept of average velocity predates calculus) is the derivative (with respect to time) of an object's position.
- Acceleration is the derivative (with respect to time) of an object's velocity.
- Jerk is the derivative (with respect to time) of an object's acceleration. For example, if an object's position

p(t) = -16t^2 + 16t + 32

; then, the object's velocity is

\dot p(t) = p'(t) = -32t + 16

; the object's acceleration is

\ddot p(t) = p

(t) = -32; and the object's jerk is $p(t) = 0.$ If the velocity of a car is given, as a function of time, then, the derivative of said function with respect to time describes the acceleration of said car, as a function of time.
Algebraic manipulation
Messy limit calculations can be avoided, in certain cases, because of differentiation rules which allow one to find derivatives via algebraic manipulation; rather than by direct application of Newton's difference quotient. One should not infer that the definition of derivatives, in terms of limits, is unnecessary. Rather, that definition is the means of proving the following "powerful differentiation rules"; these rules are derived from the difference quotient.
- Constant rule: The derivative of any constant is zero.
- Constant multiple rule: If c is some real number; then, the derivative of $cf(x)$ equals c multiplied by the derivative of f(x) (a consequence of linearity below).
- Linearity: $(af + bg)' = af' + bg'$ for all functions f and g and all real numbers a and b.
- Power rule: If $f(x) = x^r$ , for some real number r; $f'(x) = rx^$ .
- Product rule: $(fg)' = f'g + fg'$ for all functions f and g.
- Quotient rule: $(f/g)' = (f'g - fg')/(g^2)$ unless g is zero.
- Chain rule: If $f(x) = h(g(x))$ , then $f'(x) = h'(g(x)) g'(x)$ .
- Inverse function: If $g(x) = f^(x)$ , and $f(x)$ is injective, then $g'(x) = 1/f'(f^(x))$ .
- Derivative of one variable with respect to another when both are functions of a third variable: Let $x = f(t)$ and $y = g(t)$ . Now $d y/d x = (d y/d t)/(d x/d t)$ . This is the chain rule in the Leibniz notation.
- Implicit differentiation: If $f(x,y) = 0$ is an implicit function, we have: dy/dx = - (∂f / ∂x) / (∂f / ∂y). In addition, the derivatives of some common functions are useful to know. See the table of derivatives. As an example, the derivative of : $f(x) = 2x^4 + \sin (x^2) - \ln (x)\;e^x + 7$ is : $f'(x) = 8x^3 + 2x\cos (x^2) - \frac\;e^x - \ln (x)\;e^x.$
Using derivatives to graph functions
Derivatives are a useful tool for examining the graphs of functions. In particular, the points in the interior of the domain of a real-valued function which take that function to local extrema will all have a first derivative of zero. However, not all critical points are local extrema; for example, f(x)=x³ has a critical point at x=0, but it has neither a maximum nor a minimum there. The first derivative test and the second derivative test provide ways to determine if the critical points are maxima, minima or neither. In the case of multidimensional domains, the function will have a partial derivative of zero with respect to each dimension at local extrema. In this case, the Second Derivative Test can still be used to characterize critical points, by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is a saddle point, and if none of these cases hold then the test is inconclusive (e.g., eigenvalues of 0 and 3). Once the local extrema have been found, it is usually rather easy to get a rough idea of the general graph of the function, since (in the single-dimensional domain case) it will be uniformly increasing or decreasing except at critical points, and hence (assuming it is continuous) will have values in between its values at the critical points on either side.
Generalizations
Where a function depends on more than one variable, the concept of a partial derivative is used. Partial derivatives can be thought of informally as taking the derivative of the function with all but one variable held temporarily constant near a point. Partial derivatives are represented as ∂/∂x (where ∂ is a rounded 'd' known as the 'partial derivative symbol'). Some people pronounce the partial derivative symbol as 'der' rather than the 'dee' used for the standard derivative symbol, 'd'. The concept of derivative can be extended to more general settings. The common thread is that the derivative at a point serves as a linear approximation of the function at that point. Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles. In order to differentiate all continuous functions and much more, one defines the concept of distribution. For complex functions of a complex variable differentiability is a much stronger condition than that the real and imaginary part of the function are differentiable with respect to the real and imaginary part of the argument. For example, the function f(x + iy) = x + 2iy satisfies the latter, but not the first. See also the article on holomorphic functions.
See also

- Derivative (examples)
- Derivative (generalizations)
- Partial derivative
- Total derivative
- Table of derivatives
- Smooth function
- Differintegral
- Automatic differentiation
External links

- [http://wims.unice.fr/wims/wims.cgi?module=tool/analysis/function.en WIMS Function Calculator] makes online calculation of derivatives; this software enables also interactive exercises.
References

- Spivak, Michael; Calculus (3rd edition, 1994) Publish or Perish Press. ISBN 0914098896. Explains why all this works.
- Thompson, Silvanus Phillips, Calculus made easy : being a very-simplest introduction to those beautiful methods of reckoning which are generally called by the terrifying names of the differential calculus and the integral calculus New York : St. Martin's Press, 1998 ISBN 0312185480. Introduced by Martin Gardner. "What one fool can do, another can."
- Larson, Ron; Hostetler, Robert P.; and Edwards, Bruce H. (2003). Calculus of a Single Variable: Early Transcendental Functions (3rd edition). Houghton Mifflin Company. ISBN 061822307X.
- Anton, Howard (1980). Calculus with analytical geometry.. New York:John Wiley and Sons. ISBN 0-471-03248-4.
- ko:미분 ja:微分 simple:Derivative th:อนุพันธ์

Mathematical model

:Note: The term model has a different meaning in model theory, a branch of mathematical logic. A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. Mathematical models are used particularly in the natural sciences and engineering disciplines (such as physics, biology, and electrical engineering) but also in the social sciences (such as economics, sociology and political science); physicists, engineers, computer scientists, and economists use mathematical models most extensively.

Examples of mathematical models

- Model of a particle in a potential field. In this model we consider a particle as being a point of mass m which describes a trajectory which is modeled by a function x: R → R³ given its coordinates in space as a function of time. The potential field is given by a function V:R³ → R and the trajectory is a solution of the differential equation ::

m \frac x(t)  = - \operatorname V(x(t)).

:Note this model assumes the particle is a point mass, which is certainly known to be false in many cases we use this model, for example, as a model of planetary motion.
- Model of rational behavior for a consumer. In this model we assume a consumer faces a choice of n commodities labelled 1,2,...,n each with a market price p₁, p₂,..., p_n. The consumer is assumed to have a cardinal utility function U (cardinal in the sense that it assigns numerical values to utilities), depending on the amounts of commodities x₁, x₂,..., x_n consumed. The model further assumes that the consumer has a budget M which she uses to purchase a vector x₁, x₂,..., x_n in such a way as to maximize U(x₁, x₂,..., x_n). The problem of rational behavior in this model then becomes one of constrained maximization, that is maximize ::

U(x_1,x_2,\ldots, x_n)

:subject to ::

\sum_^n p_i x_i = M.

: This model has been used in models of general equilibrium theory, particularly to show existence and Pareto optimality of economic equilibria. However, the fact that this particular formulation assigns numerical values to levels of satisfaction is the source of criticism (and even ridicule). However, it is not an essential ingredient of the theory and again this is an idealization.

Background

Often when engineers analyze a system to be controlled or optimized, they use a mathematical model. In analysis, engineers can build a descriptive model of the system as a hypothesis of how the system could work, or try to estimate how an unforeseeable event could affect the system. Similarly, in control of a system, engineers can try out different control approaches in simulations. A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. The values of the variables can be practically anything; real or integer numbers, boolean values or strings, for example. The variables represent some properties of the system, for example, measured system outputs often in the form of signals, timing data, counters, event occurrence (yes/no). The actual model is the set of functions that describe the relations between the different variables.

Building blocks

There are six basic groups of variables: decision variables, input variables, state variables, exogenous variables, random variables, and output variables. Since there can be many variables of each type, the variables are generally represented by vectors. Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants. The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system (represented by the state variables). Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of the model's user. Depending on the context, an objective function is also known as an index of performance, as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved (computationally).

Classifying mathematical models

Mathematical models can be classified in several ways, some of which are described below. #Linear vs. nonlinear: If the objective functions and constraints are represented entirely by linear equations, then the model is known as a linear model. If one or more of the objective functions or constraints are represented with a nonlinear equation, then the model is known as a nonlinear model. #Deterministic vs. probabilistic (stochastic): A deterministic model performs the same way for a given set of initial conditions, while in a stochastic model, randomness is present, even when given an identical set of initial conditions. #Static vs. dynamic: A static model does not account for the element of time, while a dynamic model does. Dynamic models typically are represented with difference equations or differential equations. #Lumped parameters vs. distributed parameters: If the model is homogeneous (consistent state throughout the entire system) the parameters are lumped. If the model is heterogeneous (varying state within the system), then the parameters are distributed. Distributed parameters are typically represented with partial differential equations.

A priori information

Mathematical modelling problems are often classified into black box or white box models, according to how much a priori information is available of the system. A black-box model is a system of which there is no a priori information available. A white-box model (also called glass box or clear box) is a system where all necessary information is available. Practically all systems are somewhere between the black-box and white-box models, so this concept only works as an intuitive guide for approach. Usually it is preferable to use as much a priori information as possible to make the model more accurate. Therefore the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly. Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in a human system, we know that usually the amount of medicine in the blood is an exponentially decaying function. But we are still left with several unknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine in blood? This example is therefore not a completely white-box model. These parameters have to be estimated through some means before one can use the model. In black-box models one tries to estimate both the functional form of relations between variables and the numerical parameters in those functions. Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately. If there is no a priori information we would try to use functions as general as possible to cover all different models. An often used approach for black-box models are neural networks which usually do not assume almost anything about the incoming data. The problem with using a large set of functions to describe a system is that estimating the parameters becomes increasingly difficult when the amount of parameters (and different types of functions) increases.

Complexity

Another basic issue is the complexity of a model. If we were, for example, modelling the flight of an airplane, we could embed each mechanical part of the airplane into our model and would thus acquire an almost white-box model of the system. However, the computational cost of adding such a huge amount of detail would effectively inhibit the usage of such a model. Additionally, the uncertainty would increase due to an overly complex system, because each separate part induces some amount of variance into the model. It is therefore usually appropriate to make some approximations to reduce the model to a sensible size. Engineers often can accept some approximations in order to get a more robust and simple model. For example Newton's classical mechanics is an approximated model of the real world. Still, Newton's model is quite sufficient for most ordinary-life situations, that is, as long as particle speeds are well below the speed of light, and we study macro-particles only.

Training

Any model which is not pure white-box contains some parameters that can be used to fit the model to the system it shall describe. If the modelling is done by a neural network, the optimization of parameters is called training. In more conventional modelling through explicitly given mathematical functions, parameters are determined by curve fitting.

Model evaluation

An important part of the modelling process is the evaluation of an acquired model. How do we know if a mathematical model describes the system well? This is not an easy question to answer. Usually the engineer has a set of measurements from the system which are used in creating the model. Then, if the model was built well, the model will adequately show the relations between system variables for the measurements at hand. The question then becomes: How do we know that the measurement data is a representative set of possible values? Does the model describe well the properties of the system between the measurement data (interpolation)? Does the model describe well events outside the measurement data (extrapolation)? A common approach is to split the measured data into two parts; training data and verification data. The training data is used to train the model, that is, to estimate the model parameters (see above). The verification data is used to evaluate model performance. Assuming that the training data and verification data are not the same, we can assume that if the model describes the verification data well, then the model describes the real system well. However, this still leaves the extrapolation question open. How well does this model describe events outside the measured data? Consider again Newtonian classical mechanics-model. Newton made his measurements without advanced equipment, so he could not measure properties of particles travelling at speeds close to the speed of light. Likewise, he did not measure the movements of molecules and other small particles, but macro particles only. It is then not surprising that his model does not extrapolate well into these domains, even though his model is quite sufficient for ordinary life physics.

Ordinary differential equation

:ODE redirects here. For the real-time physics engine, see open dynamics engine. In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is an equation that involves the derivatives of an unknown function of one variable. A simple example of an ordinary differential equation is :

f' = f \,

, where

f \,

is an unknown function, and

f'\,

is its derivative. See differential calculus and integral calculus for basic calculus background. An ODE may be thought of as an equation depending on a single spatial variable. An ODE is called autonomous if it is time-independent, and nonautonomous otherwise. Important theorems in the field of ODEs include broad existence and uniqueness theorems and for ODEs in the plane, the Poincaré-Bendixson theorem.

Definition

Let y represent an unknown function of x, and let :

y', y

,\ \dots,\ y^ denote the derivatives : $\frac,\ \frac,\ \dots,\ \frac.$ An ordinary differential equation (ODE) is an equation involving : $x,\ y,\ y',\ y$ ,\ \dots . The order of a differential equation is the order n of the highest derivative that appears. If the highest derivative appears only in integer powers, then the degree of the equation is the highest power of the highest derivative. A solution of an ODE is a function y(x) whose derivatives satisfy the equation. Such a function is not guaranteed to exist and, if it does exist, is usually not unique. A general solution of an nth-order equation is a solution containing

n

arbitrary variables, corresponding to n constants of integration. A particular solution is derived from the general solution by setting the constants to particular values. A singular solution is a solution that can't be derived from the general solution. When a differential equation of order n has the form :

F\left(x, y', y

,\ \dots,\ y^\right) = 0 it is called an implicit differential equation whereas the form : $F\left(x, y', y$ ,\ \dots,\ y^\right) = y^ is called an explicit differential equation. A differential equation not depending on x is called autonomous, and one with no terms depending only on x is called homogeneous.

General application

An important special case is when the equations do not involve

x

. These differential equations may be represented as vector fields. This type of differential equation has the property that space can be divided into equivalence classes based on whether two points lie on the same solution curve. Since the laws of physics are believed not to change with time, the physical world is governed by such differential equations. (See also symplectic topology for abstract discussion.) In the case where the equations are linear, the original equation can be solved by breaking it down into smaller equations, solving those, and then adding the results back together. Unfortunately, many of the interesting differential equations are non-linear, which means that they cannot be broken down in this way. There are also a number of techniques for solving differential equations using a computer (see numerical ordinary differential equations). Ordinary differential equations are to be distinguished from partial differential equations where

y

is a function of several variables, and the differential equation involves partial derivatives.

Existence and nature of solutions

The problem of solving a differential equation is to find the function

y

whose derivatives satisfy the equation. For example, the differential equation :

y

+ y = 0 \, has the general solution : $y = A \cos + B \sin \,$ , where A, B are constants determined from boundary conditions. In general, an n-th order equation allows both $x$ and $y$ to be fixed, as well as all the $n-1$ lower order derivatives of $y$ ; the remaining equation can be solved (at least conceptionally) for $y^$ . If the equation has finite degree $d$ , then we now have a polynomial equation in $y^$ with at most $d$ roots. Therefore there can be as many as $d$ possible values for $y^$ at any given point and for any possible values of the lower order derivatives, though there may be ranges of these points and values where there are fewer solutions (or none at all). A Lipschitz condition must also be satisfied for a solution to exist. Thus, in the previous example, a second-order, first-degree equation, any point on the plane and any slope through that point can be selected and yield a unique solution (since the single root of $y$ exists for any value of

y

). Note in particular that there are an infinity of solutions through any given point; this is a general characteristic of equations of order higher than one. Lipschitz condition Consider now :

(y')^2 + xy' - y = 0 \,

with general solution :

y = Ax + A^2 \,

This is a first-order, second-degree equation, thus any point can have at most two solutions passing through it, corresponding to the two roots of

y'

in the quadratic equation that would result after fixing

x

and

y

. Studying the quadratic equation's discriminant (

x^2 + 4y

) leads to the conclusion that only a single solution exists along the parabola

y = - \frac x^2

(where the discriminant is zero) and that no solution exists below this parabola (where both roots are complex). The parabola in this problem is an example of a cusp locus; a curve along which two or more roots of the differential equation are identical. Along such a locus it is possible to move from one general solution to another while still obeying the differential equation; thus the presence of cusp loci introduce the possibility of singular solutions. In this example, the parabola

y = - \frac x^2

is such a singular solution; it satisfies the original differential equation, and a full set of solutions must include such possibilities as the hybrid solution:

y = \begin x + 1, & \mbox x < -2 \\ - \frac x^2, & \mbox -2 <= x < 2 \\ -x + 1, & \mbox x >= 2 \end

where the cusp locus has been used to connect two particular solutions; note that the first derivative (the only derivative to appear in the differential equation) is continuous at the transitions. ([http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=abv5010.0001.001 Johnson], Chapter 5)

Types of differential equations with some history

The influence of geometry, physics, and astronomy, starting with Newton and Leibniz, and further manifested through the Bernoullis, Riccati, and Clairaut, but chiefly through d'Alembert and Euler, has been very marked, and especially on the theory of linear partial differential equations with constant coefficients.

Homogeneous linear ODEs with constant coefficients

The first method of integrating linear ordinary differential equations with constant coefficients is due to Euler, who realized that solutions have the form

e^

, for possibly-complex values of

z

. Thus :

\frac   + A_\frac   + \cdots + A_y = 0

has the form :

z^n e^ + A_1 z^ e^ + \cdots + A_n e^ = 0

so dividing by

e^

gives the nth-order polynomial :

F(z) = z^ + A_z^ + \cdots + A_n = 0

In short the terms :

\frac  \quad\quad(k = 1, 2, \cdots, n).

of the original differential equation are replaced by z^k. Solving the polynomial gives n values of z,

z_1, \dots,z_n

. Plugging those values into

e^

gives a basis for the solution; any linear combination of these

e^

will satisfy the differential equation. This equation F(z) = 0, is the "characteristic" equation considered later by Monge and Cauchy. If z is a (possibly not real) zero of F(z) of multiplicity m and $k\in\ \,$ then $y=x^ke^ \,$ is a solution of the ODE. These functions make up a basis of the ODE's solutions. If the A_i are real then real-valued solutions are preferable. Since the non-real z values will come in conjugate pairs, so will their corresponding ys; replace each pair with their linear combinations Re(y) and Im(y). A case that involves complex roots can be solved with the aid of Euler's formula.
- Example: Given $y$ -4y'+5y=0 \,. The characteristic equation is

z^2-4z+5=0 \,

which has zeroes 2+i and 2−i. Thus the solution basis

\

\ \,

. Now y is a solution iff

y=c_1y_1+c_2y_2 \,

for

c_1,c_2\in\mathbb C

. Because the coefficients are real,
- we are likely not interested in the complex solutions
- our basis elements are mutual conjugates The linear combinations :

u_1=\mbox(y_1)=\frac=e^\cos(x) \,

and :

u_2=\mbox(y_1)=\frac=e^\sin(x) \,

will give us a real basis in

\

Linear ODEs with constant coefficients

Suppose instead we face :

\frac   + A_\frac   + \cdots + A_y = f(x)

For later convenience, define the characteristic polynomial :

P(v)=v^n+A_1v^+\cdots+A_n

We find the solution basis

\

as in the homogeneous (f=0) case. We now seek a particular solution y_p by the variation of parameters method. Let the coefficients of the linear combination be functions of x: :

y_p=u_1y_1+u_2y_2+\cdots+u_ny_n

Using the "operator" notation

D=d/dx

and a broad-minded use of notation, the ODE in question is

P(D)y=f

; so :

f=P(D)y_p=P(D)(u_1y_1)+P(D)(u_2y_2)+\cdots+P(D)(u_ny_n)

With the constraints :

0=u'_1y_1+u'_2y_2+\cdots+u'_ny_n

0=u'_1y'_1+u'_2y'_2+\cdots+u'_ny'_n

:… :

0=u'_1y^_1+u'_2y^_2+\cdots+u'_ny^_n

the parameters commute out, with a little "dirt": :

f=u_1P(D)y_1+u_2P(D)y_2+\cdots+u_nP(D)y_n+u'_1y^_1+u'_2y^_2+\cdots+u'_ny^_n

But

P(D)y_j=0

, therefore :

f=u'_1y^_1+u'_2y^_2+\cdots+u'_ny^_n

This, with the constraints, gives a linear system in the

u'_j

. This much can always be solved; in fact, combining Cramer's rule with the Wronskian, :

u'_j=(-1)^f\frac

The rest is a matter of integrating

u'_j

. The particular solution is not unique;

y_p+c_1y_1+\cdots+c_ny_n

also satisfies the ODE for any set of constants c_j. See also variation of parameters. Example: Suppose

y

-4y'+5=sin(kx). We take the solution basis found above $\$ . : : : Using the list of integrals of exponential functions : : And so : (Notice that u₁ and u₂ had factors that canceled y₁ and y₂; that is typical.) For interest's sake, this ODE has a physical interpretation as a driven damped harmonic oscillator; y_p represents the steady state, and $c_1y_1+c_2y_2$ is the transient.
Linear ODEs with variable coefficient

Method of undetermined coefficients
The method of undetermined coefficients (MoUC), is useful in finding solution for $y_p$ . Given the ODE $P(D)y = f(x)$ , find another differential operator $A(D)$ such that $A(D)f(x) = 0$ . This operator is called the annihilator, and thus the method of undetermined coefficients is also known as the annihilator method. Applying $A(D)$ to both sides of the ODE gives an homogeneous ODE $\big(A(D)P(D)\big)y = 0$ for which we find a solution basis $\$ as before. Then the original nonhomogeneous ODE is used to construct a system of equations restricting the coefficients of the linear combinations to satisfy the ODE. Undetermined coefficients is not as general as variation of parameters in the sense that an annihilator does not always exist. Example: Given $y$ -4y'+5=\sin(kx),

P(D)=D^2-4D+5

. The simplest annihilator of

\sin(kx)

A(D)=D^2+k^2

. The zeros of

A(z)P(z)

are

\

, so the solution basis of

A(D)P(D)

\=\

. Setting

y=c_1y_1+c_2y_2+c_3y_3+c_4y_4

we find : giving the system :

i=(k^2+4ik-5)c_3+(-k^2+4ik+5)c_4

0=(k^2+4ik-5)c_3+(k^2-4ik-5)c_4

which has solutions :

c_3=\frac i

c_4=\frac i

giving the solution set :

Method of variation of parameters

As explained above, the general solution to a non-homogeneous, linear differential equation

y

(x) + p(x) y'(x) + q(x) y(x) = g(x) can be expressed as the sum of the general solution $y_h(x)$ to the corresponding homogenous, linear differential equation $y$ (x) + p(x) y'(x) + q(x) y(x) = 0 and any one solution

y_p(x)

y

(x) + p(x) y'(x) + q(x) y(x) = g(x). Like the method of undetermined coefficients, described above, the method of variation of parameters is a method for finding one solution to $y$ (x) + p(x) y'(x) + q(x) y(x) = g(x), having already found the general solution to

y

(x) + p(x) y'(x) + q(x) y(x) = 0. Unlike the method of undetermined coefficients, which fails except with certain specific forms of g(x), the method of variation of parameters will always work; however, it is significantly more difficult to use. For a second-order equation, the method of variation of parameters makes use of the following fact:
Fact
Let p(x), q(x), and g(x) be functions, and let $y_1(x)$ and $y_2(x)$ be solutions to the homogeneous, linear differential equation $y$ (x) + p(x) y'(x) + q(x) y(x) = 0. Further, let u(x) and v(x) be functions such that

u'(x) y_1(x) + v'(x) y_2(x) = 0

and

u'(x) y_1'(x) + v'(x) y_2'(x) = g(x)

for all x, and define

y_p(x) = u(x) y_1(x) + v(x) y_2(x)

. Then

y_p(x)

is a solution to the non-homogeneous, linear differential equation

y

(x) + p(x) y'(x) + q(x) y(x) = g(x).
Proof
$y_p(x) = u(x) y_1(x) + v(x) y_2(x)$ $y_p$ (x) + p(x) y'_p(x) + q(x) y_p(x) = g(x) + u(x) y_1(x) + v(x) y_2(x) + p(x) u(x) y_1'(x) + p(x) v(x) y_2'(x) + q(x) u(x) y_1(x) + q(x) v(x) y_2(x) $= g(x) + u(x) (y_1$ (x) + p(x) y_1'(x) + q(x) y_1(x)) + v(x) (y_2(x) + p(x) y_2'(x) + q(x) y_2(x)) = g(x) + 0 + 0 = g(x)
Usage
To solve the second-order, non-homogeneous, linear differential equation $y$ (x) + p(x) y'(x) + q(x) y(x) = g(x) using the method of variation of parameters, use the following steps: #Find the general solution to the corresponding homogeneous equation $y$ (x) + p(x) y'(x) + q(x) y(x) = 0. Specifically, find two linearly independent solutions $y_1(x)$ and $y_2(x)$ . #Since $y_1(x)$ and $y_2(x)$ are linearly independent solutions, their Wronskian $y_1(x) y_2'(x) - y_1'(x) y_2(x)$ is nonzero, so we can compute $-(g(x) y_2(x))/()$ and $()/()$ . If the former is equal to u(x) and the latter to v(x), then u and v satisfy the two constraints given above: that $u'(x) y_1(x) + v'(x) y_2(x) = 0$ and that $u'(x) y_1'(x) + v'(x) y_2'(x) = g(x)$ . We can tell this after multiplying by the denominator and comparing coefficients. #Integrate $-(g(x) y_2(x))/()$ and $()/()$ to obtain u(x) and v(x), respectively. (Note that we only need one choice of u and v, so there is no need for constants of integration.) #Compute $y_p(x) = u(x) y_1(x) + v(x) y_2(x)$ . The function $y_p$ is one solution of $y$ (x) + p(x) y'(x) + q(x) y(x) = g(x). #The general solution is $c_1 y_1(x) + c_2 y_2(x) + y_p(x)$ , where $c_1$ and $c_2$ are arbitrary constants.
Higher-order equations
The method of variation of parameters can also be used with higher-order equations. For example, if $y_1(x)$ , $y_2(x)$ , and $y_3(x)$ are linearly independent solutions to $y(x) + p(x) y$ (x) + q(x) y'(x) + r(x) y(x) = 0, then there exist functions u(x), v(x), and w(x) such that $u'(x) y_1(x) + v'(x) y_2(x) + w'(x) y_3(x) = 0$ , $u'(x) y_1'(x) + v'(x) y_2'(x) + w'(x) y_3'(x) = 0$ , and $u'(x) y_1$ (x) + v'(x) y_2(x) + w'(x) y_3(x) = g(x). Having found such functions (by solving algebraically for u(x), v(x), and w(x), then integrating each), we have $y_p(x) = u(x) y_1(x) + v(x) y_2(x) + w(x) y_3(x)$ , one solution to the equation $y(x) + p(x) y$ (x) + q(x) y'(x) + r(x) y(x) = g(x).
Example
Solve the previous example, $y$ + y = \sec x Recall $\sec x = \frac = f$ . From technique learned from 3.1, LHS has root of $r = \pm i$ that yield $y_c = C_1 \cos x + C_2 \sin x$ , (so $y_1 = \cos x$ , $y_2 = \sin x$ ) and its derivatives : $\left\$
Partial differential equation
In mathematics, a partial differential equation (PDE) is an equation relating the partial derivatives of an unknown function of several variables. A solution of the equation is a function satisfying this relation. The idea is to try to deduce information about an unknown function by first discovering a relationship between itself and its partial derivatives in the form of a PDE. The PDE can then be used to uncover information about the unknown function, and sometimes an explicit formula for the unknown function can be discovered. A PDE usually has many (possibly infinitely many) solutions; a particular problem often requires additional boundary conditions which constrain the solution set. Where ordinary differential equations have solutions that are families with each solution characterized by the values of some parameters, for a PDE the solutions often are parametrized by functions (informally put, this means that the set of solutions is much larger). Partial differential equations are ubiquitous in science and especially in physics, as physical laws can usually be written in form of PDEs. They describe phenomena such as fluid flow, the growth of crystals, diffusion, gravitation, and the behavior of electromagnetic fields. They are important in fields such as aircraft simulation, computer graphics, and weather prediction. The central equations of general relativity and quantum mechanics are also partial differential equations.
Notation and examples
In PDEs, it is common to write the unknown function as u and its partial derivative with respect to the variable x as u_x, that is: : $u_x =$ : $u_ =$ Especially in (mathematical) physics, one often prefers use of the nabla operator $\nabla=(\part_x,\part_y,\part_z)$ for spatial derivatives and a dot ( $\dot u$ ) for time derivatives, e.g. to write the wave equation (see below) as $\ddot u=c^2\nabla^2u$ .
Laplace's equation
A very important and basic PDE is Laplace's equation: : $u_ + u_ + u_ = 0$ for the unknown function u(x,y,z). Solutions to this equation, known as harmonic functions, serve as the potentials of vector fields in physics, such as the gravitational or electrostatic fields. A generalization of Laplace's equation is Poisson's equation: : $u_ + u_ + u_ = f$ where f(x,y,z) is a given function. The solutions to this equation describe potentials of gravitational and electrostatic fields in the presence of masses or electrical charges, respectively.
Wave equation
The wave equation is an equation for an unknown function u(x,y,z,t) (where we think of t as a time variable) which reads: : $u_ = c^2( u_ + u_ + u_ )$ Its solutions describe waves such as sound or light waves; c is a number which represents the speed of the wave. In lower dimensions, this equation describes the vibration of a string or drum. Solutions will typically be combinations of oscillating sine waves.
Heat equation
The heat equation describes the temperature in a given region over time. It is: : $u_t = k ( u_ + u_ + u_ )$ Solutions will typically "even out" over time. The number k describes the thermal diffusivity of the material.
Euler-Tricomi equation
The Euler-Tricomi equation is used in the investigation of transonic flow. It is : $u_=xu_$
Advection equation
The advection equation describes the transport of a conserved scalar $\psi$ in a velocity field $=(u,v,w)$ . It is: : $\psi_t+(u\psi)_x+(v\psi)_y+(w\psi)_z=0.$ If the velocity field is solenoidal (that is, $\nabla\cdot=0$ ), then the equation may be simplified to : $\psi_t+u\psi_x+v\psi_y+w\psi_z=0.$ The one dimensional steady flow advection equation $\psi_t+u.\psi_x=0$ (where $u$ is constant) is commonly referred to as the pigpen problem. If $u$ is not constant and equal to $\psi$ the equation is referred to as Burgers' equation.
Ginzburg-Landau equation
The Ginzburg-Landau equation is used in modelling superconductivity. It is : $iu_t+pu_ +q|u|^2u=i\gamma u$ where $p,q\in\mathbb$ and $\gamma\in\mathbb$ are constants and $i$ is the imaginary unit.
The Dym equation
The Dym equation is named for Harry Dym and occurs in the study of solitons. It is : $u_t = u^3u_.$
Other examples
The Schrödinger equation is a PDE at the heart of non-relativistic quantum mechanics. In the WKB approximation it is the Hamilton-Jacobi equation. Except for the Dym equation and the Ginzburg-Landau equation, the above equations are linear in the sense that they can be written in the form Au = f for a given linear operator A and a given function f. Other important non-linear equations include the Navier-Stokes equations describing the flow of fluids, and Einstein's field equations of general relativity.
Methods to solve PDEs
Linear PDEs are generally solved, when possible, by decomposing the equation according to a set of basis functions, solving those individually and using superposition to find the solution corresponding to the boundary conditions. The method of separation of variables has many important particular applications. There are no generally applicable methods to solve non-linear PDEs. Still, existence and uniqueness results (such as the Cauchy-Kovalevskaya theorem) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of analysis). Nevertheless, some techniques can be used for several types of equations. The h-principle is the most powerful method to solve underdetermined equations. The Riquier-Janet theory is an effective method for obtaining information about many analytic overdetermined systems. The method of characteristics can be used in some very special cases to solve partial differential equations. In some cases, a PDE can be solved via perturbation analysis in which the solution is considered to be a correction to an equation with a known solution. Alternatives are numerical analysis techniques from simple finite difference schemes to the more mature multigrid and finite element methods. Many interesting problems in science and engineering are solved in this way using computers, sometimes high performance supercomputers.
Classification
Second-order partial differential equations, and systems of second-order PDEs, can usually be classified as parabolic, hyperbolic or elliptic. This classification gives an intuitive insight into the behavior of the system itself. Assuming $u_=u_,$ the general second-order PDE is of the form : $Au_ + Bu_ + Cu_ + \cdots = 0,$ which looks remarkably similar to the equation for a conic section: : $Ax^2 + Bxy + Cy^2 + \cdots = 0.$ Just as one classifies conic sections into parabolic, hyperbolic, and elliptic based on the discriminant $B^2 - 4AC$ , the same can be done for a second-order PDE. # $B^2 - 4AC < 0$ : elliptic equations tend to smooth out any disturbances. A typical example is Laplace's equation. The motion of a fluid at sub-sonic speeds can be approximated with elliptic PDEs. # $B^2 - 4AC = 0$ : parabolic equations tend to smooth out any pre-existing disturbances in the data. A typical example is the heat equation. # $B^2 - 4AC > 0$ : hyperbolic equations tend to amplify any disturbances in the data. A typical example is the wave equation. The motion of a fluid at super-sonic speeds can be approximated with hyperbolic PDEs. This method of classification can easily be extended to equations with more than two independent variables by examining the eigenvalues of the coefficient matrix. In this situation, the classification scheme becomes: # Elliptic: The eigenvalues are all positive or all negative. # Parabolic : The eigenvalues are all positive or all negative, save one which is zero. # Hyperbolic : There is at least one negative and at least one positive eigenvalue, and none of the eigenvalues are zero. This matches with positive-definite and negative-definite matrix analysis, of the sort that comes up during a discussion of maxima and minima.
Equations of mixed type
If a PDE has coefficients which are not constant, it is possible that it will not belong to any of these categories but rather be of mixed type. A simple but important example is the Euler-Tricomi equation : $u_=xu_$ which is called elliptic-hyperbolic because it is elliptic in the region x < 0, hyperbolic in the region x > 0, and degenerate parabolic on the line x = 0.
External links

- [http://eqworld.ipmnet.ru/en/pde-en.htm Partial Differential Equations: Exact Solutions] at EqWorld: The World of Mathematical Equations.
- [http://eqworld.ipmnet.ru/en/solutions/eqindex/eqindex-pde.htm Partial Differential Equations: Index] at EqWorld: The World of Mathematical Equations.
- [http://eqworld.ipmnet.ru/en/methods/meth-pde.htm Partial Differential Equations: Methods] at EqWorld: The World of Mathematical Equations.
- [http://www.exampleproblems.com/wiki/index.php?title=Partial_Differential_Equations Example problems with solutions] at exampleproblems.com
References

- L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2
- A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9
- A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2004. ISBN 1-58488-355-3
- A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002. ISBN 0-415-27267-X
- D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997. Category:Multivariate calculus
- ja:偏微分方程式

Stochastic differential equation

: For the video display issue known as SDE, see Screen door effect. A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process.

Use in physics

For example a general, coupled set of first-order SDEs (note that there are standard techniques for transforming a higher-order equation into several coupled first-order equations by introducing new unknowns) is often written in the form: :

\dot_i = \frac = f_i(\mathbf) + \sum_^ng_i^m(\mathbf)\eta_m(t),\,

where

\mathbf=\

is the set of unknowns, the

f_i

and

g_i

are arbitrary functions and the

\eta_m

are random functions of time, often referred to as "noise terms". If the

g_i

are constants, the system is said to be subject to additive noise, otherwise it is said to be subject to multiplicative noise. The main method of solution is by use of the Fokker-Planck equation, which provides a deterministic equation satisfied by the time dependant probability density. Alternatively numerical solutions can be obtained by Monte Carlo simulation. Other techniques, such as path integration have also been used, drawing on the analogy between statistical physics and