Heat Equation via Laplace Transform

PDEs · Axiom Academy

LESSON Heat Equation via Laplace Transform A step-by-step approach to solving the heat equation using Laplace transforms in time We start with the one-dimensional heat equation on a finite interval [0, L]: 2. Applying the Laplace Transform We take the Laplace transform in the time variable t, defining: Applying the Laplace transform to the heat equation u_t = ku_xx: Rearranging the transformed equation: This is a second-order linear ODE in x with parameter s. The standard form is: The general solution has the form: 4. Solving with Boundary Conditions Transform the boundary conditions u(0,t) = 0 and u(L,t) = 0: For non-trivial solutions, we need the sine term to vanish at specific eigenvalues: To find u(x,t), we need to invert the Laplace transform. The solution in transform space has the form: Where F(s) depends on the initial condition f(x). Using the convolution theorem and properties of inverse transforms: The final solution is a series expansion: 6. Comparison with Fourier Method Separate into two ODEs with separation constant -lambda Solve spatial ODE to get eigenfunctions sin(n*pi*x/L) Solve temporal ODE to get exp(-k*lambda*t) Superpose solutions and match initial condition Transform PDE in time to get ODE in space Solve spatial ODE with boundary conditions Apply boundary conditions to find eigenvalues Inverse transform to recover time dependence Match initial condition through transform properties

This is the written version of the interactive lesson above. See the full PDEs course.