New Solution Method for Solving Volterra Integral Equation Using Bernstein Polynomials

Abstract

Abstract Linear and nonlinear differential equations extensively used mathematical models for many interesting and important phenomena observed in numerous areas of science and technology. They are inspired by problems in diverse fields such as economics, biology, fluid dynamics, physics, differential geometry, engineering, control theory, materials science, and quantum mechanics. This special issue aims to highlight recent developments in methods and applications of linear and nonlinear differential equations. In addition, there are papers analyzing equations that arise in engineering, classical and fluid mechanics, and finance. In this paper a novel technique implementing Bernstein polynomials is introduced for the numerical solution of a Volterra integral equation. The obtained solutions are novel, and previous literature lacks such derivations. For this aim, we derive a simple and efficient matrix formulation using Bernstein polynomials as trial functions. Numerical examples are considered to verify the effectiveness of the proposed derivations and numerical solutions are compared with the existing methods available in the literature. The Volterra integral equation can be used to describe the capacitance of the parallel plate capacitor (PPC) in the electrostatic field. It is shown that the numerical results are excellent. The efficiency of the proposed numerical technique is exhibited through graphical illustrations and results drafted in tabular form for specific values of the parameters to validate the numerical investigation.