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Partial Differential Equations in Physics
The Method of Characteristics
Separation of Variables Fourier Method
12 other sections not shown
arbitrary Bessel functions boundary conditions bounded Cauchy Cauchy's problem closed domain coefficients condition of type conductor Consequently Consider const constant contour converges uniformly coordinates cylindrical functions d'Alembert's formula defined density differential equation eigenfunctions eigenvalues equal Example fact find the solution Fn(x following theorem formula Fredholm's function p(x fundamental sequence generalised function given Green's function Hankel functions harmonic heat Hilbert-Schmidt theorem homogeneous improper integral infinite inhomogeneous initial conditions initial functions initial temperature integral equations kernel K(x Legendre polynomials lemma linear linearly independent maintained at zero membrane Method of Characteristics obtain orthogonal plane pn(x point M0 positive roots potential due pp(x Proof Let Property prove radius range respect satisfies the boundary Section seek the solution segment separation of variables solution of Cauchy's solve sphere spherical string Substituting Suppose surface symmetric temperature distribution velocity vibrations volume potential written zero temperature