In this paper Operational matrix of Bernstein Polynomials (BPs) is used to solve Bratu equation. This nonlinear equation appears in the particular elecotrospun nanofibers fabrication process framework. Elecotrospun organic nanofibers have been used for a large variety of filtration applications such as in non-wovens and filtration industries. By using Operational matrix of integration and multiplication the investigated equations are turned into set of algebraic equations. Numerical solutions show both accuracy and simplicity of the suggested approach.

This paper presents a computational method for solving two types of integro-differential equations, system of nonlinear high order Volterra-Fredholm integro-differential equation (VFIDEs) and nonlinear fractional order integro-differential equations. Our tools for this aims is Operational matrices of integration and fractional integration. By this method the given problems reduce to solve a system of algebraic equations. Illustrative examples are included to demonstrate the efficiency and high accuracy of the method.

In this paper, a numerical method for solving the constrained optimal control of time-varying singular systems with quadratic performance index is presented. Presented method is based on Bernstein polynomials. Operational matrices of integration, differentiation and product are introduced and utilized to reduce the solution of optimal control problems with time-varying singular systems to the solution of algebraic equations set. The strength of the method is shown by exhibiting a numerical implementation using Operational matrices that solves the determined control problem by solving an equation set. The method converges rapidly to the exact solution and gives very accurate results even by low value of . Illustrative examples are included to demonstrate the validity and efficiency of the technique and convergence of method to the exact solution especially for unstable singular systems.

In this paper, an effective direct method to determine the numerical solution of linear and nonlinear Fredholm and Volterra integral and integro-differential equations is proposed. The method is based on expanding the required approximate solution as the elements of Chebyshev cardinal functions. The Operational matrices for the integration and product of the Chebyshev cardinal functions are described in detail. These matrices play the important role of reducing an integral equation to a system of algebraic equations. Illustrative examples are shown, which confirms the validity and applicability of the presented technique.

We offer a method for solving the fractional optimal control problems of multi-dimensional. We obtain a fractional derivative and multiplication Operational matrix for Mott polynomials (M-polynomials). In the proposed method, the Caputo sense of the fractional derivative is applied on dynamical system. The main feature of this method is to reduce the problem into a system of algebraic equations in order to simplify it. We also show that by increasing the approximation points, the responses converge to the real answer. When the degree of fractional derivative approaches to 1, then the obtained solution approaches to the classical solution as well.

This article proposes an efficient numerical method for solving nonlinear stochastic differential equations. Using the Operational matrices of block pulse functions, stochastic differential equations can be reduced to a system of algebraic equations. Computation of presented method is very simple and attractive. In addition, convergence analysis and numerical examples that illustrate accuracy and efficiency of the method are presented.

This article proposes an efficient method based on the Fibonacci functions for solving nonlinear stochastic Ito-Volterra integral equations. For this purpose, we obtain stochastic Operational matrix of Fibonacci functions. We use the proposed basis function in combination with stochastic Operational matrix. This problem is then reduced into a system of nonlinear equations which can be solved by Newton's method. Also, the existence, uniqueness, and convergence of the proposed method are discussed. Furthermore, in order to show the accuracy and reliability of the proposed method, the new approach is applied to some practical problems.

In this article, a new numerical method based on triangular functions for solving nonlinear stochastic di erential equations is presented. For this, the stochastic Operational matrix of triangular functions for It^o integral are determined. Computation of presented method is very simple and attractive. In addition, convergence analysis and numerical examples that illustrate accuracy and e ciency of the method are presented.