Computational method based on Bernstein operational matrices for nonlinear Volterra–Fredholm–Hammerstein integral equations (original) (raw)
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Filomat, 2014
In this paper, the Bernstein operational matrices are used to obtain solutions of multi-order fractional differential equations. In this regard we present a theorem which can reduce the nonlinear fractional differential equations to a system of algebraic equations. The fractional derivative considered here is in the Caputo sense. Finally, we give several examples by using the proposed method. These results are then compared with the results obtained by using Adomian decomposition method, differential transform method and the generalized block pulse operational matrix method. We conclude that our results compare well with the results of other methods and the efficiency and accuracy of the proposed method is very good.
Journal of King Saud University - Science, 2017
An algorithm for approximating solutions to fractional differential equations (FDEs) in a modified new Bernstein polynomial basis is introduced. Writing x ! x a ð0 < a < 1Þ in the operational matrices of Bernstein polynomials, the fractional Bernstein polynomials are obtained and then transformed into matrix form. Furthermore, using Caputo fractional derivative, the matrix form of the fractional derivative is constructed for the fractional Bernstein matrices. We convert each term of the problem to the matrix form by means of fractional Bernstein matrices. A basic matrix equation which corresponds to a system of fractional equations is utilized, and a new system of nonlinear algebraic equations is obtained. The method is given with some priori error estimate. By using the residual correction procedure, the absolute error can be estimated. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.
A New Method Based on Operational Matrices of Bernstein Polynomials for Nonlinear Integral Equations
2011
Abstract—An approximation method based on operational matrices of Bernstein polynomials used for the solution of Hammerstein integral equations. The operational matrices of these functions are utilized to reduce a nonlinear Hammerstein and Volterra Hammerstein integral equation to a system of nonlinear algebraic equations. The method is computationally very simple and attractive, and applications are demonstrated through illustrative examples. The results obtained are compared by the known results.
In this paper, we use Bernstein polynomials to solve multiterm variable order fractional differential equations. The main idea of this paper is that we use Bernstein polynomials and operational matrices to solve such types of equations. The equation is transformed into the products of several dependent matrices, which can also be viewed as an algebraic system by making use of the collocation points. By solving the algebraic system, the numerical solutions will then be obtained. Finally, numerical examples are presented to demonstrate the accuracy of the proposed method. Keywords Variable order caputo fractional derivatives, Bernstein polynomials, operational matrices of variable fractional order derivative of Bernstein polynomials.
Solution of fractional integro-differential equations by Bernstein polynomials
Malaya Journal of Matematik
This paper focus on the study of Bernstein polynomial to approximate the solution of Fractional-integro differential equations (FIDE) with caputo derivative. This method reduces Fractional-integro differential equations into system of linear equations. Illustrations are given to show the accuracy of the method and results are simulated.
Advances in Mathematical Physics, 2013
We present two methods for solving a nonlinear system of fractional differential equations within Caputo derivative. Firstly, we derive operational matrices for Caputo fractional derivative and for Riemann-Liouville fractional integral by using the Bernstein polynomials (BPs). In the first method, we use the operational matrix of Caputo fractional derivative (OMCFD), and in the second one, we apply the operational matrix of Riemann-Liouville fractional integral (OMRLFI). The obtained results are in good agreement with each other as well as with the analytical solutions. We show that the solutions approach to classical solutions as the order of the fractional derivatives approaches 1.
Journal of Computational and Nonlinear Dynamics, 2015
In this paper, the operational matrix of Euler functions for fractional derivative of order β in the Caputo sense is derived. Via this matrix, we develop an efficient collocation method for solving nonlinear fractional Volterra integro-differential equations. Illustrative examples are given to demonstrate the validity and applicability of the proposed method, and the comparisons are made with the existing results.
Entropy
We extend the operational matrices technique to design a spectral solution of nonlinear fractional differential equations (FDEs). The derivative is considered in the Caputo sense. The coupled system of two FDEs is considered, subjected to more generalized integral type conditions. The basis of our approach is the most simple orthogonal polynomials. Several new matrices are derived that have strong applications in the development of computational scheme. The scheme presented in this article is able to convert nonlinear coupled system of FDEs to an equivalent S-lvester type algebraic equation. The solution of the algebraic structure is constructed by converting the system into a complex Schur form. After conversion, the solution of the resultant triangular system is obtained and transformed back to construct the solution of algebraic structure. The solution of the matrix equation is used to construct the solution of the related nonlinear system of FDEs. The convergence of the proposed...