Solving Nonlinear Fractional Integro-Differential Equations of Volterra Type Using Novel Mathematical Matrices

2015 ◽  
Vol 10 (6) ◽  
Author(s):  
Farshid Mirzaee ◽  
Saeed Bimesl ◽  
Emran Tohidi
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Collocation Method ◽  
Operational Matrix ◽  
Volterra Type

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.

A Legendre collocation method for fractional integro-differential equations

2011 ◽  
Vol 17 (13) ◽  
pp. 2050-2058 ◽  
Author(s):  
Abbas Saadatmandi ◽  
Mehdi Dehghan
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Numerical Method ◽  
Collocation Method ◽  
Legendre Polynomials ◽  
Integration Method ◽  
Algebraic Equations ◽  
Volterra Type ◽  
Gaussian Integration ◽  
Legendre Collocation Method

A numerical method for solving the linear and non-linear fractional integro-differential equations of Volterra type is presented. The fractional derivative is described in the Caputo sense. The method is based upon Legendre approximations. The properties of Legendre polynomials together with the Gaussian integration method are utilized to reduce the fractional integro-differential equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique and a comparison is made with existing results.

scholarly journals A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Khalid K. Ali ◽  
Mohamed A. Abd El salam ◽  
Emad M. H. Mohamed
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Order ◽  
Linear Functional ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Fractional Order Differential Equations ◽  
The Given ◽  
Functional Argument ◽  
Special Case

AbstractIn this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.

A Comparative Study of Integer and Noninteger Order Wavelets for Fractional Nonlinear Fredholm Integro-Differential Equations

2018 ◽  
Vol 13 (8) ◽  
Author(s):  
F. Mohammadi ◽  
J. A. Tenreiro Machado
Keyword(s):  
Differential Equations ◽  
Comparative Study ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Efficient Algorithm ◽  
Operational Matrix ◽  
Tau Method ◽  
Legendre Wavelets ◽  
Numerical Examples ◽  
Integer Order

This paper compares the performance of Legendre wavelets (LWs) with integer and noninteger orders for solving fractional nonlinear Fredholm integro-differential equations (FNFIDEs). The generalized fractional-order Legendre wavelets (FLWs) are formulated and the operational matrix of fractional derivative in the Caputo sense is obtained. Based on the FLWs, the operational matrix and the Tau method an efficient algorithm is developed for FNFIDEs. The FLWs basis leads to more efficient and accurate solutions of the FNFIDE than the integer-order Legendre wavelets. Numerical examples confirm the superior accuracy of the proposed method.

scholarly journals Systems of nonlinear Volterra integro-differential equations of arbitrary order

2018 ◽  
Vol 36 (4) ◽  
pp. 33-54 ◽  
Author(s):  
Kourosh Parand ◽  
Mehdi Delkhosh
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Approximate Method ◽  
Fractional Order ◽  
Chebyshev Polynomials ◽  
Arbitrary Order ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Arbitrary Integer ◽  
Orthogonal Functions

In this paper, a new approximate method for solving of systems of nonlinear Volterra integro-differential equations of arbitrary (integer and fractional) order is introduced. For this purpose, the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) based on the classical Chebyshev polynomials of the first kind has been introduced that can be used to obtain the solution of the integro-differential equations (IDEs). Also, we construct the fractional derivative operational matrix of order $\alpha$ in the Caputo's definition for GFCFs. This method reduced a system of IDEs by collocation method into a system of algebraic equations. Some examples to illustrate the simplicity and the effectiveness of the propose method have been presented.

scholarly journals COLLOCATION APPROXIMATIONS FOR WEAKLY SINGULAR VOLTERRA INTEGRO‐DIFFERENTIAL EQUATIONS

2003 ◽  
Vol 8 (4) ◽  
pp. 315-328 ◽  
Author(s):  
I. Parts ◽  
A. Pedas
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Order Of Convergence ◽  
Piecewise Polynomial ◽  
Volterra Type ◽  
Weakly Singular ◽  
Uniform Grids ◽  
Attainable Order ◽  
Piecewise Polynomial Collocation Method ◽  
Polynomial Collocation

A piecewise polynomial collocation method for solving linear weakly singular integro‐differential equations of Volterra type is constructed. The attainable order of convergence of collocation approximations on arbitrary and quasi‐uniform grids is studied theoretically and numerically.

scholarly journals Collocation Method Based on Genocchi Operational Matrix for Solving Generalized Fractional Pantograph Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Abdulnasir Isah ◽  
Chang Phang ◽  
Piau Phang
Keyword(s):  
Fractional Derivative ◽  
Collocation Method ◽  
Upper Bound ◽  
Present Method ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Mathematical Tool ◽  
Algebraic Equations ◽  
Genocchi Polynomials

An effective collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations with initial and boundary conditions is presented. Using the properties of Genocchi polynomials, we derive a new Genocchi delay operational matrix which we used together with the Genocchi operational matrix of fractional derivative to approach the problems. The error upper bound for the Genocchi operational matrix of fractional derivative is also shown. Collocation method based on these operational matrices is applied to reduce the generalized fractional pantograph equations to a system of algebraic equations. The comparison of the numerical results with some existing methods shows that the present method is an excellent mathematical tool for finding the numerical solutions of generalized fractional pantograph equations.

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