scholarly journals An Operational Matrix Technique for Solving Variable Order Fractional Differential-Integral Equation Based on the Second Kind of Chebyshev Polynomials

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Jianping Liu ◽  
Xia Li ◽  
Limeng Wu
Keyword(s):  
Integral Equation ◽  
Chebyshev Polynomials ◽  
Algebraic System ◽  
Operational Matrix ◽  
Original Equation ◽  
Variable Order ◽  
Computationally Efficient ◽  
Collocation Points ◽  
Differential Integral Equation ◽  
Fractional Differential

An operational matrix technique is proposed to solve variable order fractional differential-integral equation based on the second kind of Chebyshev polynomials in this paper. The differential operational matrix and integral operational matrix are derived based on the second kind of Chebyshev polynomials. Using two types of operational matrixes, the original equation is transformed into the arithmetic product of several dependent matrixes, which can be viewed as an algebraic system after adopting the collocation points. Further, numerical solution of original equation is obtained by solving the algebraic system. Finally, several examples show that the numerical algorithm is computationally efficient.

New Operational Matrix for Solving Multiterm Variable Order Fractional Differential Equations

2017 ◽  
Vol 13 (1) ◽  
Author(s):  
A. M. Nagy ◽  
N. H. Sweilam ◽  
Adel A. El-Sayed
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Algebraic System ◽  
Order Differential Equation ◽  
Fundamental Problem ◽  
Operational Matrix ◽  
Variable Order ◽  
Engineering Problems ◽  
Fractional Differential ◽  
Fourth Kind

The multiterm fractional variable-order differential equation has a massive application in physics and engineering problems. Therefore, a numerical method is presented to solve a class of variable order fractional differential equations (FDEs) based on an operational matrix of shifted Chebyshev polynomials of the fourth kind. Utilizing the constructed operational matrix, the fundamental problem is reduced to an algebraic system of equations which can be solved numerically. The error estimate of the proposed method is studied. Finally, the accuracy, applicability, and validity of the suggested method are illustrated through several examples.

scholarly journals An Operational Matrix of Fractional Differentiation of the Second Kind of Chebyshev Polynomial for Solving Multiterm Variable Order Fractional Differential Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Jianping Liu ◽  
Xia Li ◽  
Limeng Wu
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Chebyshev Polynomial ◽  
Algebraic System ◽  
Operational Matrix ◽  
Main Idea ◽  
Operational Matrices ◽  
Variable Order ◽  
Fractional Differential ◽  
Variable Order Fractional Derivative

The multiterm fractional differential equation has a wide application in engineering problems. Therefore, we propose a method to solve multiterm variable order fractional differential equation based on the second kind of Chebyshev Polynomial. The main idea of this method is that we derive a kind of operational matrix of variable order fractional derivative for the second kind of Chebyshev Polynomial. With the operational matrices, 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 solution of original equation is acquired. Numerical examples show that only a small number of the second kinds of Chebyshev Polynomials are needed to obtain a satisfactory result, which demonstrates the validity of this method.

scholarly journals Operational matrices based on the shifted fifth-kind Chebyshev polynomials for solving nonlinear variable order integro-differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
H. Jafari ◽  
S. Nemati ◽  
R. M. Ganji
Keyword(s):  
Differential Equations ◽  
Chebyshev Polynomials ◽  
Original Problem ◽  
High Accuracy ◽  
Operational Matrices ◽  
Variable Order ◽  
Algebraic Equations ◽  
Numerical Tests ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations

AbstractIn this research, we study a general class of variable order integro-differential equations (VO-IDEs). We propose a numerical scheme based on the shifted fifth-kind Chebyshev polynomials (SFKCPs). First, in this scheme, we expand the unknown function and its derivatives in terms of the SFKCPs. To carry out the proposed scheme, we calculate the operational matrices depending on the SFKCPs to find an approximate solution of the original problem. These matrices, together with the collocation points, are used to transform the original problem to form a system of linear or nonlinear algebraic equations. We discuss the convergence of the method and then give an estimation of the error. We end by solving numerical tests, which show the high accuracy of our results.

scholarly journals An accurate approach based on the orthonormal shifted discrete Legendre polynomials for variable-order fractional Sobolev equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
M. H. Heydari ◽  
A. Atangana
Keyword(s):  
Legendre Polynomials ◽  
Algebraic System ◽  
Operational Matrix ◽  
Order Equation ◽  
Basis Functions ◽  
Variable Order ◽  
System Of Equations ◽  
Sobolev Equation ◽  
Numerical Examples ◽  
Operational Matrix Method

AbstractThis paper applies the Heydari–Hosseininia nonsingular fractional derivative for defining a variable-order fractional version of the Sobolev equation. The orthonormal shifted discrete Legendre polynomials, as an appropriate family of basis functions, are employed to generate an operational matrix method for this equation. A new fractional operational matrix related to these polynomials is extracted and employed to construct the presented method. Using this approach, an algebraic system of equations is obtained instead of the original variable-order equation. The numerical solution of this system can be found easily. Some numerical examples are provided for verifying the accuracy of the generated approach.

scholarly journals Solution of Integral Equation in Two-Dimensional using Spectral Relationships

2020 ◽  
pp. 1-10
Author(s):  
F. M. Alharbi
Keyword(s):  
Integral Equation ◽  
Chebyshev Polynomials ◽  
Algebraic System ◽  
Integral Operators ◽  
Numerical Results ◽  
Two Dimensional ◽  
Linear Algebraic System

This paper concerned using spectral relationships in the solution of the integral equation (IE) in two-dimensional. To discuss that, the (IE) in two-dimensional under certain conditions was considered. The existence of at least one solution of the (IE) was discussed by proving the continuity and compactness of an integral operators. Chebyshev polynomials of the first kind were used to transform the (IE) to a linear algebraic system. Many numerical results and estimating errors were calculated and plotted by the Maple program in different cases.

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