Abstract
In this paper, we present a practical matrix method for solving nonlinear Volterra-Fredholm integro-differential equations under initial conditions in terms of Bernstein polynomials on the interval [0,1]. The nonlinear part is approximated in the form of matrices’ equations by operational matrices of Bernstein polynomials, and the differential part is approximated in the form of matrices’ equations by derivative operational matrix of Bernstein polynomials. Finally, the main equation is transformed into a nonlinear equations system, and the unknown of the main equation is then approximated. We also give some numerical examples to show the applicability of the operational matrices for solving nonlinear Volterra-Fredholm integro-differential equations (NVFIDEs).
Highlights
Some of the phenomena in physics, electronics, biology, and other applied sciences lead to nonlinear VolterraFredholm integro-differential equations
We present a practical matrix method for solving nonlinear Volterra-Fredholm integro-differential equations under initial conditions in terms of Bernstein polynomials on the interval [0,1]
We give some numerical examples to show the applicability of the operational matrices for solving nonlinear Volterra-Fredholm integro-differential equations (NVFIDEs)
Summary
Some of the phenomena in physics, electronics, biology, and other applied sciences lead to nonlinear VolterraFredholm integro-differential equations. These equations can appear when transforming a differential equation into an integral equation [1,2,3,4,5,6,7]. Bernstein polynomials (B-plynomials) have many useful properties [11,12]. We use Bernstein polynomials and their resulting operational matrices to solve the following nonlinear VolterraFredholm integro-differential equations with the initial conditions indicated as:
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