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The system of ordinary differential equations has many uses in contemporary mathematics and engineering. Finding the numerical solution to a system of ordinary differential equations for any arbitrary interval is very appealing to researchers. The numerical solution of a system of fourth-order ordinary differential equations on any finite interval [ a , b ] is found in this work using a symmetric Bernstein approximation. This technique is based on the operational matrices of Bernstein polynomials for solving the system of fourth-order ODEs. First, using Chebyshev collocation nodes, a generalised approximation of the system of ordinary differential equations is discretized into a system of linear algebraic equations that can be solved using any standard rule, such as Gaussian elimination. We obtain the numerical solution in the form of a polynomial after obtaining the unknowns. The Hyers–Ulam and Hyers–Ulam–Rassias stability analyses are provided to demonstrate that the proposed technique is stable under certain conditions. The results of numerical experiments using the proposed technique are plotted in figures to demonstrate the accuracy of the specified approach. The results show that the suggested Bernstein approximation method for any interval is quick and effective. [ABSTRACT FROM AUTHOR]

In this paper, a new numerical technique is introduced to find the solution of the system of Volterra integral equations based on symmetric Bernstein polynomials. The use of Bernstein polynomials to find the numerical solutions of differential and integral equations increased due to its fast convergence. Here, the numerical solution of the system of Volterra integral equations on any finite interval [ m , n ] is obtained by replacing the unknown functions with the generalized Bernstein basis functions. The proposed technique converts the given system of equations into the system of algebraic equations which can be solved by using any standard rule. Further, Hyers–Ulam stability criteria are used to check the stability of the given technique. The comparison between exact and numerical solution for the distinct nodes is demonstrated to show its fast convergence. [ABSTRACT FROM AUTHOR]

Integral equations are extensively used in many physical models appearing in the field of plasma physics, atmosphere–ocean dynamics, fluid mechanics, mathematical physics and many other disciplines of physics and engineering. In this research work, a new numerical technique for the solution of the system of Fredholm integral equations (FIEs) of both first and second kinds is established, which is based on Bernstein basis functions. Here, the system of FIEs of both kinds has been taken, then reduces the equations to an algebraic linear system and can be solved using any standard rule. Convergence analysis of the proposed technique and some useful numerical results are presented so that the reader could understand this idea easily. Further, Hyers-Ulam stability analysis criteria is used for analyzing the stability of the proposed technique. The comparison of exact and approximate solutions of some problems is demonstrated in tables and their graphs are plotted to show the efficiency of the given technique. [ABSTRACT FROM AUTHOR]

For obtaining numerical solutions of the system of ordinary differential equations (ODEs) of third order, a new numerical technique is proposed by using operational matrices of Bernstein polynomials. These operational matrices can be utilized to solve different problems of integral and differential equations. The System of third-order ODEs occur in various physical and engineering models. In this paper, an iterative algorithm is constructed by using operational matrices of Bernstein polynomials for solving the system of third order ODEs. The proposed technique provides a numerical solution by discretizing the system to a system of algebraic equations which can be solved directly. The method will be verified by using appropriate examples which are arising in Physics and some Engineering problems. The comparison of approximate and exact solution of the given examples is demonstrated with the help of tables and graphs. [ABSTRACT FROM AUTHOR]

This paper is devoted to present a simple but efficient numerical method for solving the 2D Volterra-Fredholm integral equation (VFIE). Both mixed and separate types of VFIEs are considered. The under consideration technique rely on approximating the unknown function with 2D Bernstein polynomials. The proposed technique has advantage in reducing the computational burden. To show the accuracy and convergence of this method, some results are also established. Finally the effectiveness of the technique is demonstrated by applying the technique on some numerical tests. [ABSTRACT FROM AUTHOR]

This paper develops a numerical technique for the solution of 2D Volterra Integral Equations (2DVIEs). The proposed method is based on the discretization of 2DVIEs by using two-dimensional Bernstein's approximation. Some results are developed regarding convergence analysis which shows that the given technique converges to the exact solution rapidly. Several examples are also presented to demonstrate the efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

This paper provides an effective numerical technique for obtaining the approximate solution of mixed Volterra-Fredholm Integral Equations (VFIEs) of second kind. The VFIEs arise from parabolic boundary value problems, mathematical modelling of the spatio-temporal development of an epidemic, and from various physical and Engineering models. The proposed method is based on the discretization of VFIEs by Bernstein’s approximation. Some results on convergence are also established which suggests that the technique converges to a smooth approximate solution. Its remarkable accuracy properties are finally demonstrated by several examples with graphical representation. [ABSTRACT FROM AUTHOR]