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partial-differential-equations

The separation-of-variables ansatz

taught

Substitute a product trial solution u(x,t) = X(x)T(t) into a linear homogeneous PDE and split it into two ordinary differential equations joined by a separation constant.

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Local knowledge map

Local knowledge map

Choosing which series a boundary-condition pair calls for Deriving the Poisson kernel Energy conservation for the wave equation Finite differences for Laplace's equation: the five-point stencil Finite differences for the wave equation Laplace's equation in a box by 3-D separation Laplace's equation on a wedge and an annulus Separating the Laplacian eigenvalue problem in polar… Solving the heat equation on a finite rod with Dirichlet… Superposing four one-sided rectangle problems The Neumann problem's solvability condition The vibrating circular drumhead The vibrating rectangular membrane From boundary conditions to a Sturm-Liouville… Laplace's equation by separation of variables on a disk Laplace's equation by separation of variables on a… Solving the heat equation on a finite rod with Neumann… Solving the heat equation on a finite rod with Robin… Solving the vibrating-string problem by separation… The separation-of-variables ansatz Fourier cosine series Orthogonality of Sturm-Liouville eigenfunctions Singular Sturm-Liouville problems, previewed The full Fourier series The regular Sturm-Liouville problem Boundary-value problems, contrasted with IVPs Eigenvalues and the characteristic equation The linear differential operator L

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