A comparative analysis of Soham and Laplace transforms in solving Volterra integral equations

This paper explores the application of Soham transform and Laplace transform to solve Volterra integral equations of the first kind, second kind, and non-homogeneous types. The study focuses on integral equations with convolution-type kernels, widely encountered in mathematical modelling and engineering applications. Analytical solutions are derived to comprehensively understand these transforms in addressing such equations. To validate the accuracy and reliability of the analytical solutions, numerical verification is performed using the trapezoidal rule and an iterative refinement approach. A detailed error analysis is conducted to examine the effect of step size on numerical accuracy, ensuring the robustness of numerical approximations. A real-world application is presented for the one-dimensional non-homogeneous heat conduction problem, which is rigorously formulated with initial and boundary conditions. This case study illustrates the practical utility of integral transforms in solving engineering problems related to heat transfer. The study provides an in-depth discussion of both transforms' numerical accuracy, computational efficiency, and convergence behaviour in addressing integral equations with convolution-type kernels. This comparative analysis offers valuable insights into the suitability of Soham and Laplace transforms for mathematical modelling and engineering applications, emphasising their role in efficiently solving real-world problems.