Fourier Transform Solution for a One-Dimensional Non-Homogeneous Wave Equation with Boundary and Initial Value Conditions

The one-dimensional non-homogeneous wave equation subject to non-homogeneous boundary and initial conditions presents significant analytical challenges, particularly when external forcing and irregular boundary data are simultaneously present. Classical methods such as separation of variables are restricted to homogeneous settings and fail to accommodate non-homogeneous terms in a unified framework. This study employs the Fourier transform method to derive an exact analytical solution by decomposing the total wave displacement into two components: u(x,t) = V(x,t) + ψ(x), where V(x,t) is the oscillatory component satisfying the homogeneous wave equation and ψ(x) is the spatially adjusted component encoding the influence of non-homogeneous boundary conditions and external forcing. The analytical solution is verified by direct substitution and benchmarked against a second-order explicit finite-difference scheme on a grid of 500 spatial points, yielding a maximum point-wise absolute error below 8×10⁻³, consistent with the second-order truncation error of the numerical scheme. For the representative test case with unit wave speed, unit domain length, constant spatial forcing F(x) = 2, and initial displacement u₀(x) = sin(πx), the steady-state component is recovered exactly as ψ(x) = x − x², and the dominant Fourier coefficient is A₁ ≈ 0.742. A direct point-by-point comparison with published benchmark values further quantifies the sensitivity of wave solutions to boundary condition specification. The proposed framework accommodates both finite and infinite spatial domains and offers a systematic, closed-form alternative to purely numerical approaches for this class of wave propagation problems, with relevance to acoustics, structural dynamics, and materials science.