Efficient Computational Approaches for Solving Integro-Differential Equations in Engineering Applications

Abstract

The use of integro-differential equations (IDEs) is essential in the modeling of engineering systems that have memory and are nonlocal, e.g., heat transfer, viscoelasticity, and control processes. Nevertheless, the current numerical approaches tend to be computationally intensive, lack accuracy and may be unstable particularly in the case of nonlinear and large scale problems. Such predicaments underscore the importance of effective and scalable computational solutions. In this paper, a computational framework to solve IDEs, using an optimized discretization scheme and a structured numerical scheme, is proposed. The method uses adaptive kernel approximation to make computations much less complex, yet accurate. There is a theoretical discussion of stability and convergence to prove the stability of the method. The framework suggested is tested on benchmark problems and opposed to traditional methods. Findings reveal a better accuracy, convergence and a low execution time. Also, its applicability is tested in engineering case studies, such as heat transfer, viscoelastic systems. Generally, the approach offers a scalable and robust solution to IDEs, which is appropriate in large scale and real-time engineering, and has potential to be expanded to more advanced computational models.