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Linear equations with constant coefficients in
The heat equation and related equations
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absolutely integrable arbitrary assume auxiliary conditions Bessel's equation boundary conditions boundary value problem bounded called coefficients consider constant continuous function convergence theorem corresponding cosine series defined disc discussion domain dx dy eigenfunction expansion eigenfunctions eigenfunctions belonging eigenvalue problem equa equilibrium temperature example EXERCISES finite interval fixed Fourier cosine Fourier series Fourier sine series Fourier transform given graph Green's formula heat equation heat flow hence infinite inhomogeneous initial value problem initial-boundary value problem interval a,b Jo(r Laplace's equation linearly independent maximum membrane method nirx nonnegative nx dx obtain ordinary differential equations partial derivatives partial differential equation piecewise smooth pn(x proof prove real-valued satisfies the D.E. second derivatives Section Show sinh solu solution of problem Solve statement strict solution string Substituting temperature distribution temperature zero tion uniform convergence variables verify vibrations wave equation