Transform Methods Summary

PDEs · Axiom Academy

SUMMARY Transform Methods for PDEs Key concepts and techniques for solving partial differential equations using integral transforms. Definition: ( ) = _ - ^ u(x)e^ -i x \,dx Inverse Transform: u(x) = 2 _ - ^ ( )e^ i x \,d Derivative Property: \ u_x\ = i , \ u_ xx \ = - ^2 Best For: Infinite domains, problems on - < x < Definition: \ u(t)\ = U(s) = _ 0 ^ u(t)e^ -st \,dt Inverse Transform: u(t) = 2 i _ -i ^ +i U(s)e^ st \,ds Derivative Property: \ u_t\ = sU(s) - u(0) , \ u_ tt \ = s^2U(s) - su(0) - u_t(0) Best For: Initial value problems, time-dependent equations with t > 0 Fourier Transform: Transforms to _t = -k ^2 Solution: ( ,t) = ( ,0)e^ -k ^2 t Fundamental Solution: u(x,t) = e^ -x^2/(4kt) (Gaussian spreading) Equation: u_ tt = c^2 u_ xx , - 0 Fourier Transform: Transforms to _ tt = -c^2 ^2 Solution: ( ,t) = A( ) (c t) + B( ) (c t) D'Alembert Form: u(x,t) = 2 [f(x+ct) + f(x-ct)] + 2c _ x-ct ^ x+ct g(s)\,ds Convolution and Superposition Principles Convolution Theorem: \ u * v\ = where (u*v)(x) = _ - ^ u(x-y)v(y)\,dy Application: Solutions with source terms can be expressed as convolutions with the fundamental solution Superposition: For linear PDEs, sum of solutions is a solution. Use to build general solutions from particular ones Green's Function: Represents response to point source; general solution is convolution with Green's function Infinite Domain: Use Fourier transform on - < x < Semi-Infinite: Use Fourier sine/cosine transforms on 0 < x <

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