A Comparative Analytical Study of the Solutions of One-Dimensional Heat and Wave Equations via Separation of Variables and Fourier Series
DOI:
https://doi.org/10.65405/.11.ملحق%2041.2227الكلمات المفتاحية:
Heat equation; Wave equation; Separation of variables; Fourier sine series; Eigenvalue problem; Orthogonal eigenfunctions; Long-time behavior; High-frequency modes.الملخص
This research provides a comparative analytical examination of the one-dimensional heat and wave equations through the method of separation of variables. Both equations are analyzed on a finite interval with homogeneous Dirichlet boundary conditions. This framework results in an identical spatial eigenvalue problem and a common set of orthogonal sine eigenfunctions. The primary difference between the two models manifests in the temporal component of the separated solutions. In the case of the heat equation, each Fourier mode is accompanied by an exponentially decaying factor, which facilitates diffusion, smoothing, and convergence towards equilibrium. Conversely, the wave equation produces oscillatory sine and cosine temporal factors, leading to sustained modal motion in the absence of damping.
The study derives the analytical Fourier sine series solutions for both equations and contrasts their temporal behavior, long-term behavior, and handling of high-frequency modes. To substantiate the theoretical results, numerical and graphical representations based on truncated Fourier series are provided. The findings indicate that, despite the heat and wave equations possessing the same spatial eigenfunctions under the specified boundary conditions, their qualitative behaviors are inherently different. This distinction is influenced by the order and nature of the time derivative: first order in the heat equation and second order in the wave equation. The comparison elucidates the mathematical differences between parabolic and hyperbolic partial differential equations and underscores the divergence between diffusion and wave propagation.
التنزيلات
المراجع
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