Solution Techniques Summary

PDEs · Axiom Academy

FORMULA SHEET Solution Techniques Summary Quick reference guide for selecting and applying PDE solution methods Simple geometric domains (rectangular, cylindrical, spherical) Homogeneous boundary conditions Time-independent or separable domains Non-homogeneous BCs require shifting Forgetting orthogonality conditions Incomplete eigenfunction expansions Infinite or semi-infinite domains Forgetting to check integrability Mishandling derivative transforms Not verifying decay conditions Time-evolution problems (t ≥ 0) Discontinuous forcing functions Forgetting initial conditions in transform Complex inversion contour errors Not checking convergence region Not checking for crossing characteristics Ignoring boundary compatibility Incorrect symmetry assumptions Wrong boundary conditions on G No closed-form solution exists Stability violations (CFL condition) Boundary condition implementation Many problems require combining multiple techniques. Here are common combinations: Use separation of variables to reduce PDE to ODEs, then expand solution in Fourier series to satisfy initial/boundary conditions. Apply Laplace in time, Fourier in space for problems on infinite spatial domains with time evolution. Green's + Eigenfunction Expansion Construct Green's function using eigenfunction expansions from separation of variables. Characteristics + Weak Solutions Use characteristics until shock forms, then switch to weak formulation or numerical shock-capturing schemes.

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