Separation of Variables

PDEs · Axiom Academy

A powerful technique for solving PDEs by converting them into ordinary differential equations 1 The Key Assumption: Product Solutions For the heat equation, we assume the solution u(x,t) can be written as a product of two functions: one depending only on x and another depending only on t . Separation of Variables Assumption We seek solutions of the form: where X(x) depends only on the spatial variable and T(t) depends only on time. This assumption is powerful because it reduces the complexity of a PDE (which involves derivatives with respect to multiple variables) into simpler ODEs (which involve derivatives with respect to a single variable). 2 Substitute into the Heat Equation We substitute our assumed form u(x,t) = X(x)T(t) into the heat equation and compute the necessary partial derivatives. Time derivative: Since X(x) is constant with respect to t , we get ∂u/∂t = X(x)·T'(t) Space derivative: Since T(t) is constant with respect to x , we get ∂²u/∂x² = X''(x)·T(t) Now we algebraically rearrange the equation to isolate all terms involving t on one side and all terms involving x on the other side. Since the left side depends only on t and the right side depends only on x , and they must be equal for all values of x and t , both sides must equal the same constant. We call this the separation constant and denote it as -λ .

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