A Discrete Collocation Technique using the Thin Plate Splines for Solving a Certain Class of Integro-Differential Equations Arising in Biology Models (Advances in Applied Mathematics and Mechanics, 2025)
The primary motivation of the current article is to study an approximate algorithm to solve a type of Volterra integro-differential equations involving nonlinearterms. These integro-differential equations have been utilized to simulate several realistic modeling arising in biological sciences, for example, the growth model of the toxins’ cumulative effects on a population residing in a closed system. Thin plate splines (TPSs), known as a subclass of radial basis functions (RBFs) independent of the shape parameter, create an efficient scheme to interpolate a function. Due to this desirable property, we use the discrete collocation approach with the TPS basis for estimating the solution of the mentioned integro-differential equations. Furthermore, the Gauss-Legendre integration formula is utilized to calculate the integrals emerging in the scheme. Remarkably, the algorithm of the offered approach can be comfortably performed on a PC with typical specifications derived from the simplicity of using TPSs. The researchers also investigate the convergence of the presented scheme. Some test examples including Volterra integro-differential equations are considered to make sure the veracity of the proposed scheme and the validity of the theoretical error analysis.