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Abstract
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The heat equation is one of the most fundamental partial differential equations (PDEs) in mathematical physics, with wide applications in engineering, thermodynamics, material sciences, and biological processes. Its solutions are often required to model heat conduction, diffusion processes, and various transient physical phenomena. However, obtaining exact analytical solutions is possible only for limited cases under simple geometries and boundary conditions. For most practical problems, analytical methods become intractable, and hence, reliable and efficient numerical methods are indispensable.
The necessity of this research arises from the demand for accurate and computationally efficient techniques to approximate the solution of the heat equation. Conventional numerical approaches such as finite difference, finite element, and spectral methods, while effective, may face challenges such as stability issues, high computational cost, or difficulties in handling boundary conditions with sufficient smoothness. In this regard, spline-based methods offer an attractive alternative due to their inherent smoothness, flexibility, and high accuracy with relatively fewer computational resources.
Cubic B-splines, in particular, are highly suitable for collocation methods because of their local support, continuity properties, and ability to provide accurate approximations with fewer basis functions compared to polynomial approaches. Employing a cubic B-spline collocation approach for the heat equation not only ensures numerical stability and accuracy but also reduces computational overhead, making it valuable for solving large-scale or real-time problems.
The importance of this research lies in its potential to advance numerical techniques for PDEs by providing a method that is both mathematically robust and practically efficient. By applying the cubic B-spline collocation method, researchers and practitioners can achieve reliable approximate solutions that can be extended to c
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