PDE Reference

Differential Equations · Axiom Academy

FORMULA SHEET Partial Differential Equations Classification criteria, the standard PDEs, separation of variables, Fourier series, and the heat/wave/Laplace series solutions. STANDARD PARTIAL DIFFERENTIAL EQUATIONS SEPARATION OF VARIABLES METHOD Assume a product solution u(x,t) = X(x)T(t) Substitute into the PDE and separate variables Each side depends on a different variable, so both must equal the same constant (the separation constant). Apply boundary conditions to find eigenvalues Boundary conditions determine the allowed values of . Solve each ODE for every eigenvalue Get eigenfunctions and corresponding Form the general solution as a series Apply the initial condition to find the coefficients c_n FOURIER SERIES COEFFICIENT FORMULAS Identify the PDE type and verify it is separable. Apply separation of variables: u = X(x)T(t) . Use the boundary conditions to find the eigenvalues. Solve the temporal ODE for each eigenvalue. Form the general solution as an infinite series. Apply the initial conditions using a Fourier series. Compute the coefficients via orthogonality integrals.

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