A Numerical Solution of Parabolic Integro-Differential Equations Using the Left Rectangular Rule

This study presents a numerical technique for addressing parabolic integro-differential equations encountered in various scientific and engineering domains. The backward Euler method for temporal discretization with the left rectangular rule for estimating the memory integral term, is integrated into this proposed method. The semi-discrete and fully discrete formulations of the problem are used to obtain a finite element framework. To evaluate the accuracy of the numerical approximation, the mathematical formulation is established, and error estimates are analyzed. Numerous numerical experiments are conducted and juxtaposed with established exact solutions to assess the efficacy of the proposed method. The results show that numerical solutions are in good agreement with analytical solutions and highlight the expected convergence trend. The graphs are presented to highlight the accuracy of numerical approximations and the effect of network optimization on the calculated solutions. To execute the technique and evaluate the numerical results, MATLAB was used. This study demonstrates that the method used provides an accurate and efficient tool for the numerical solution of parabolic integro-differential equations.