PDE - Mathematics 421 Essay 4 Partial Differential...

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Mathematics 421 Essay 4 Partial Differential Equations Spring 2008 0. Introduction The Method of Separation of Variables is used to construct special solutions of some linear partial differential equations. Other solutions can then be found as linear combinations of these special solutions. Our applications will use eigenfunction expansions of initial conditions to produce an expression for the solution of equations with certain boundary conditions. 1. Some partial differential equations Equations involving partial derivatives arise in certain physical models. The model gives rise to the equation and certain initial conditions (in time) and boundary conditions (in space). The combined problem is interesting because of its relation to the physical model. The mathematical theory of these equations guarantees a unique solution only under conditions more stringent than those used in the solution that we will find, so there is a question of whether this solution is the physically correct solution. In addition to describing the method, we shall give an outline of a method for verifying its validity. The form of the solution involves a series whose convergence is not assured, so more work must be done to even show that it satisfies the equation. The equations for u.x;t/ that we study, together with additional conditions are: k @ 2 u @x 2 D @u @t .k > 0/; .H/ the heat equation ; and a 2 @ 2 u @x 2 D @ 2 u @t 2 ; .W / the wave equation . In both, there will be boundary conditions for x and initial conditions for t . The equations of second order in x , so boundary conditions at two points are suitable according to the theory developed in the notes on Boundary Value Problems (Essay 3). Since .H/ is a first order equation with respect to t , the value of u.x;0/ should suffice to determine a solution; but .W / is a second order equation with respect to t , so the value of u and @[email protected] at t D 0 should be specified. Further justification of these assumptions will be given in later sections. In all examples, we will take the boundary conditions to be u.0;t/ D u.L;t/ D 0 . Once you are familiar with the method, the changes necessary to deal with other boundary conditions will be easy to implement. Different boundary conditions lead to different eigenfunctions, so the special features of the selected boundary value problems must be considered before starting to compute the solution. Some boundary conditions (including the one we have chosen) lead to something resembling the half- range Fourier series . Although the functions are determined by their properties on the interval Œ0;LŁ , they have extensions that are periodic with period 2L . If the series can be identified with a Fourier series, then known Fourier series and operational properties of Fourier series can be used to obtain the eigenfunction expansion of the initial values that are used to give an expression for the solution of the given equation.
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PDE - Mathematics 421 Essay 4 Partial Differential...

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