International Journal of Science, Engineering and TechnologyAuthors: Assistant Professor Rahul Kaushik, Rajkumar Soni
Abstract: Integral equations form the mathematical backbone of many physical models, enabling precise formulation of heat transfer and wave propagation phenomena, particularly in unbounded or complex domains. This paper reviews key numerical approaches—including collocation, Galerkin, Nyström, spectral, and fast boundary-element methods—focusing on their theoretical foundations, convergence properties, and computational efficiency. Benchmark problems in transient conduction and acoustic scattering illustrate each method’s accuracy and cost trade-offs. We find that while spectral and fast-multipole–accelerated solvers offer superior precision for smooth kernels, collocation and Galerkin schemes remain robust for singular and non-smooth geometries. Future work should explore adaptive discretization, machine-learning–enhanced kernels, and multiphysics extensions.
