Advancing Computation in Solving Non-Homogeneous Parabolic Problem With Non-Linear Integral Boundary Conditions Via the Cubic B-Spline Method

Abstract

The aim of this research is to improve the cubic B-spline method for finding the approximate solution of one-dimensional non-homogeneous reaction-diffusion equation with non-linear integral boundary conditions. In this work, an approximate solution is built by combining a Crank-Nicolson scheme for temporal discretization and a modified cubic B-spline basis with a new i coefficient. This n coefficient is chosen to ensure the suggested technique's convergence to enhance the efficiency of the existing cubic B-spline techniques. Thus, the numerical method employed for dealing with the integral non-linear boundary conditions produces a system that possesses a tridiagonal coefficient matrix, except for the first and last lines. Furthermore, a predictor-corrector approach for solving the resultant non-linear system due to the integral and non-linear boundary conditions is presented as well as some convergence results are then updated numerically. The novelty and originality of this article are that the considered non-linear integral boundary conditions are new conditions of mathematical models as well as the modified basis given in this paper and the outcomes are also new. Finally, the robustness and efficiency of our approach are demonstrated by testing our technique on three examples with integral boundary conditions of order c = 2. Our findings have been shown to be more accurate and to offer challenging upgrades over those found in the literature.