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METHOD OF SEPARATION OF VARIABLES
VIBRATING STRINGS AND MEMBRANES
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analyze approximation assume boundary value problem Consider converges density derivative determine Dirac delta function dt dx dx dx dx dy eigenfunction expansion example Exercise exponentially finite Fourier coefficients Fourier cosine series Fourier series Fourier sine series Fourier transform frequency Green's formula Green's function heat energy heat equation heat flow homogeneous boundary conditions homogeneous solution infinite series infinite space Green's initial condition initial value problem integral Laplace transform Laplace's equation linear combination membrane method of eigenfunction method of separation mrx/L nirx nonhomogeneous boundary conditions obtain ordinary differential equations orthogonal oscillation partial differential equation piecewise smooth Poisson's equation product solutions Rayleigh quotient result satisfies separation constant separation of variables series of f(x singular sinh sketched in Fig space Green's function Sturm-Liouville eigenvalue problem temperature distribution thermal energy time-dependent two-dimensional valid vector velocity wave equation yields zero