Separation of Variables

Fourier Analysis · Axiom Academy

LESSON Separation of Variables A powerful technique for solving partial differential equations by reducing them to ordinary differential equations We begin with the fundamental assumption that the solution u(x,t) can be written as a product of two functions, one depending only on x and the other only on t: This ansatz (educated guess) seems restrictive, but it's remarkably powerful. If we can find solutions of this form, we can later combine them using superposition to satisfy more general conditions. 2. Substituting into the PDE and Separating Consider the heat equation as an example. After substituting u(x,t) = X(x)T(t) into the PDE and computing the necessary derivatives: By dividing both sides appropriately, we can separate the variables so that the left side depends only on x and the right side only on t. Since these must be equal for all x and t, they must both equal a constant. 3. The Separation Constant and Eigenvalue Problem The separation constant, which we typically denote as -lambda, leads to two separate ODEs: The spatial equation, combined with boundary conditions, forms an eigenvalue problem. The values of lambda for which nontrivial solutions exist are called eigenvalues, and the corresponding solutions X(x) are eigenfunctions. For each eigenvalue lambda_n, we solve both ODEs. For example, with boundary conditions X(0) = X(L) = 0, we find:

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