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To solve the heat equation using the finite Fourier transform, we'll follow these steps: ### Given Equation: \[ \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}, \quad 0 < x < 4, \quad t > 0 \] ### Boundary Conditions: \[ u(0, t) = u(4, t) = 0 \] ### Initial Condition: \[ u(x, 0) = 2x, \quad 0 < x < 4 \] ### Step 1: Apply the Finite Fourier Transform The finite Fourier transform (sine transform) is suitable for this problem because the boundary conditions are homogeneous (Dirichlet conditions). The sine transform is defined as: \[ U_n(t) ______ _________ ___ ___ _____ _______ ________.
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