A new operational matrix based on Bernoulli polynomials

In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized to transform the differential equation to a matrix equation which corresponds to a system of algebraic equations with unknown Bernoulli coefficients. This method can be used for many problems such as differential equations, integral equations and so on. Numerical examples show the method is computationally simple and also illustrate the efficiency and accuracy of the method

Sumudu-Bernstein solution of differential, integral and integro-differential equations

A numerical method based on the inverse Sumudu transform and the Bernstein polynomials operational matrix of integration is developed. The derived method is implemented in solving linear differential, integral and integro-differential equations. Also, a procedure for overcoming nonlinearity is developed and implemented to solve nonlinear Volterra integral equations. The approximate results are compared with the exact solutions and an existing method. Error estimation shows that the proposed method has elevated level of accuracy for just a few terms of the polynomial

COMPARISON OF VARIOUS FRACTIONAL BASIS FUNCTIONS FOR SOLVING FRACTIONAL-ORDER LOGISTIC POPULATION MODEL

Three types of orthogonal polynomials (Chebyshev, Chelyshkov, and Legendre) are employed as basis functions in a collocation scheme to solve a nonlinear cubic initial value problem arising in population growth models. The method reduces the given problem to a set of algebraic equations consist of polynomial coefficients. Our main goal is to present a comparative study of these polynomials and to asses their performances and accuracies applied to the logistic population equation. Numerical applications are given to demonstrate the validity and applicability of the method. Comparisons are also made between the present method based on different basis functions and other existing approximation algorithms

A novel third kind Chebyshev wavelet collocation method for the numerical solution of stochastic fractional Volterra integro-differential equations

In the formulation of natural processes like emissions, population development, financial markets, and the mechanical systems, in which the past affect both the present and the future, Volterra integro-differential equations appear. Moreover, as many phenomena in the real world suffer from disturbances or random noise, it is normal and healthy for them to go from probabilistic models to stochastic models. This article introduces a new approach to solve stochastic fractional Volterra integro-differential equations based on the operational matrix method of Chebyshev wavelets of third kind and stochastic operational matrix of Chebyshev wavelets of third kind. Also, we have given the convergence and error analysis of the proposed method. A variety of numerical experiments are carried out to demonstrate our theoretical findings.Publisher's Versio

A Shifted Jacobi-Gauss Collocation Scheme for Solving Fractional Neutral Functional-Differential Equations

The shifted Jacobi-Gauss collocation (SJGC) scheme is proposed and implemented to solve the fractional neutral functional-differential equations with proportional delays. The technique we have proposed is based upon shifted Jacobi polynomials with the Gauss quadrature integration technique. The main advantage of the shifted Jacobi-Gauss scheme is to reduce solving the generalized fractional neutral functional-differential equations to a system of algebraic equations in the unknown expansion. Reasonable numerical results are achieved by choosing few shifted Jacobi-Gauss collocation nodes. Numerical results demonstrate the accuracy, and versatility of the proposed algorithm

Numerical Solution for Solving Linear Fractional Differential Equations using Chebyshev Wavelets

In this paper, a numerical method for solving linear fractional differential equations using Chebyshev wavelets matrices has been presented. Fractional differential equations have received great attention in the recent period due to the expansion of their uses in many applications, It is difficult to find a solution to them by the analytical method due to the presence of derivatives with fractional orders. Therefore, we resort to numerical solutions. The use of wavelets in solving these equations is a relatively new method, as it was found to give more accurate results than other methods. We created Chebyshev matrices by utilizing Chebyshev sequences, where these matrices can be created in different sizes, and the larger the matrix size, The results are more accurate. Chebyshev wavelet matrices are characterized by their speed when compared to other wavelet matrices. The algorithm converts fractional differential equations into algebraic equations by using the derivative of an operational matrix of the pulsing mass of the fractional integral with Chebyshev matrices. Then, the solution is found by applying the algorithm and comparing it with the exact solution. The results are convergent with very small errors. To prove the effectiveness and applicability of the algorithm, for validation, and show how the results are close to the exact solution, several examples have been solved