Analytical study reveals accurate heat diffusion behaviors in 1D systems with oscillatory boundaries, confirming findings through numerical simulations.
This paper presents an analytical study of the one-dimensional generalized heat equation subject to time-dependent boundary conditions. A transformation technique is employed to convert the original problem with non-homogeneous Dirichlet boundary conditions into a problem with homogeneous boundary conditions, facilitating the application of the separation of variables method. The resulting solution is expressed as a Fourier sine series, where coefficients are determined using orthogonality relations. Comparisons between the analytical solution and finite difference numerical simulations (MATLAB) demonstrate excellent agreement, confirming the accuracy of the analytical approach. Graphical results including 3D surface plots and 2D cross-sections reveal the dynamic behavior of heat diffusion influenced by oscillatory boundaries.
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Pandey et al. (2024) studied this question.
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