Abstract:

Numerous scientific and engineering applications depend heavily on the invention and optimisation of numerical methods for solving integral equations. Integral equations are used to model complicated processes in a variety of disciplines, including electromagnetics, acoustics, fluid dynamics, and image processing. However, due to their intrinsic mathematical complexity and the requirement for effective computer procedures, the solution of integral equations frequently presents substantial obstacles. In this study, we concentrate on the creation and improvement of numerical techniques for resolving integral equations. We investigate a number of methods, including boundary element methods, Galerkin methods, and collocation methods, among others, to effectively and precisely estimate the solutions. Striking a compromise between computational effectiveness and solution precision is the idea, making sure that the numerical methods deliver trustworthy results while utilising suitable processing resources. To do this, we research the use of cutting-edge optimisation strategies to boost the effectiveness of the numerical methods, including adaptive mesh refinement, sparse matrix computations, and quick algorithms. We aim to reduce the computational complexity and memory needs while retaining high solution accuracy by carefully examining the issue structure and taking advantage of its unique features. Through thorough numerical tests, the effectiveness of the suggested numerical approaches and optimisation methodologies is assessed. When analytical solutions are available or there are already existing numerical methodologies, we compare the findings to verify the correctness and effectiveness of our methods. We also examine the effects of several parameters on the overall performance and convergence behaviour, including the mesh density, basis functions, and numerical quadrature.

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