Theory of differential equations
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Qualitative theory of differential equations
Year SJR 0. Citations per document. Year International Collaboration Documents Year Value Uncited documents 0 Uncited documents 36 Uncited documents 39 Uncited documents 54 Uncited documents 39 Uncited documents 43 Uncited documents 50 Uncited documents 43 Uncited documents 47 Uncited documents 58 Uncited documents Uncited documents Uncited documents Uncited documents Uncited documents Uncited documents Uncited documents Uncited documents Uncited documents Cited documents 0 Cited documents 4 Cited documents 8 Cited documents 13 Cited documents 9 Cited documents 17 Cited documents 16 Cited documents 20 Cited documents 27 Cited documents 29 Cited documents 65 Cited documents Cited documents Cited documents Cited documents Cited documents Cited documents Cited documents Cited documents In terms of differential equations, if coefficients of a differential operator are piece-wise continuous then we make use of distributions how and why it works?
I am more interested in their relation with Green's function. Please help me understand, how can I use distribution theory for solving differential equations.
The most basic application is the use of the fundamental solution also known as the Green's function to solve inhomogeneous linear problems. Duhamel's principle lets us extend this to time-dependent problems, provided the spatial differential operator is constant and again linear. I'll try to present the role that distributions play when we're looking for solutions of differential equations. It is in general much easier to show that a particular differential equation has a weak solution than a solution in the usual sense.
Such a distribution will often also be called a weak solution. While it is not clear whether or not this will always occur, this is the setting in with the machinery of functional analysis is best suited to tackle the problem.
There, show we can find a solution, and show that this solution in fact was a member of the original function space to start with of course, this last sentence will not always work out, but this is the philosophy and the hope! Below is the minimal mathematics - or rather the minimal Linear Systems Theory - needed to understand the role of distributions and Green's functions with differential equations.
As extracted from old notes about "Signals and Systems".
Theory of Differential Equations in Engineering and Mechanics - CRC Press Book
Suppose that a linear and homogeneous system is excited with a Dirac-delta as its input. I hope it nevertheless serves a purpose and that the reader is capable of thinking how to generalize this material to more than one dimension. Sign up to join this community. The best answers are voted up and rise to the top.
Conduction of heat, the theory of which was developed by Joseph Fourier , is governed by another second-order partial differential equation, the heat equation. It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black—Scholes equation in finance is, for instance, related to the heat equation. The number of differential equations that have received a name, in various scientific areas is a witness of the importance of the topic.
See List of named differential equations. From Wikipedia, the free encyclopedia. Not to be confused with Difference equation.
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Tables of Contents
Main articles: Ordinary differential equation and Linear differential equation. Main article: Partial differential equation. Main article: Non-linear differential equations. See also: Time scale calculus.
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Studies in the History of Mathematics and Physical Sciences. Bibcode : AmJPh..
Herman HJ Lynge and Son. Pierce, Acoustical Soc of America, ; page Discovering the Principles of Mechanics , p. A First Course in Differential Equations 5th ed. Areas of mathematics. Category theory Information theory Mathematical logic Philosophy of mathematics Set theory. Abstract Elementary Linear Multilinear. Calculus Real analysis Complex analysis Differential equations Functional analysis. Combinatorics Graph theory Order theory Game theory. Arithmetic Algebraic number theory Analytic number theory Diophantine geometry. Algebraic Differential Geometric.
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History of mathematics Recreational mathematics Mathematics and art Mathematics education. Category Portal Commons WikiProject. Differential equations. Difference discrete analogue Stochastic Stochastic partial Delay. Inspection Separation of variables Method of undetermined coefficients Variation of parameters Integrating factor Integral transforms Euler method Finite difference method Crank—Nicolson method Runge—Kutta methods Finite element method Finite volume method Galerkin method Perturbation theory.
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