Separation of Variables for PDEs
Assuming u(x,t) = X(x)T(t) splits a linear PDE such as the heat equation into ODEs linked by a separation constant; boundary conditions restrict the constant to discrete eigenvalues, and the full solution is a superposition of the resulting modes with coefficients fixed by Fourier-expanding the initial condition. This is the bridge from ODE technique to the eigenfunction machinery of PDEs.
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Assuming u(x,t) = X(x)T(t) splits a linear PDE such as the heat equation into ODEs linked by a separation constant; boundary conditions restrict the constant to discrete eigenvalues, and the full solution is a superposition of the resulting modes with coefficients fixed by Fourier-expanding the initial condition. This is the bridge from ODE technique to the eigenfunction machinery of PDEs.
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