scholarly journals A CHEBYSHEV PSEUDO-SPECTRAL METHOD FOR SOLVING FRACTIONAL-ORDER INTEGRO-DIFFERENTIAL EQUATIONS

2010 ◽  
Vol 51 (4) ◽  
pp. 464-475 ◽  
Author(s):  
N. H. SWEILAM ◽  
M. M. KHADER
Keyword(s):  
Differential Equations ◽  
Spectral Method ◽  
Fractional Order ◽  
Algebraic Equations ◽  
Computationally Efficient ◽  
Volterra Type ◽  
Nonlinear Algebraic Equations ◽  
Linear And Nonlinear ◽  
Pseudo Spectral Method ◽  
Pseudo Spectral

AbstractA Chebyshev pseudo-spectral method for solving numerically linear and nonlinear fractional-order integro-differential equations of Volterra type is considered. The fractional derivative is described in the Caputo sense. The suggested method reduces these types of equations to the solution of linear or nonlinear algebraic equations. Special attention is given to study the convergence of the proposed method. Finally, some numerical examples are provided to show that this method is computationally efficient, and a comparison is made with existing results.

scholarly journals On the continuous analogue of the Seidel method

2018 ◽  
Vol 20 (4) ◽  
pp. 364-377
Author(s):  
I. V. Boikov ◽  
A. I. Boikova
Keyword(s):  
Differential Equations ◽  
Systems Of Nonlinear Equations ◽  
Algebraic Equations ◽  
Seidel Method ◽  
Linear Algebraic Equations ◽  
Nonlinear Algebraic Equations ◽  
Continuous Analogue ◽  
Solution Methods ◽  
Linear And Nonlinear ◽  
Equations With Delay

Continuous Seidel method for solving systems of linear and nonlinear algebraic equations is constructed in the article, and the convergence of this method is investigated. According to the method discussed, solving a system of algebraic equations is reduced to solving systems of ordinary differential equations with delay. This allows to use rich arsenal of numerical ODE solution methods while solving systems of algebraic equations. The main advantage of the continuous analogue of the Seidel method compared to the classical one is that it does not require all the elements of the diagonal matrix to be non-zero while solving linear algebraic equations’ systems. The continuous analogue has the similar advantage when solving systems of nonlinear equations.

New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Ali Bhrawy ◽  
Yahia Alhamed ◽  
Dumitru Baleanu ◽  
Abdulrahim Al-Zahrani
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Laguerre Polynomials ◽  
Orthogonal Functions ◽  
Spectral Techniques ◽  
Fractional Differential ◽  
Definition Of ◽  
Pseudo Spectral Method ◽  
Pseudo Spectral

AbstractFractional-order generalized Laguerre functions (FGLFs) are proposed depends on the definition of generalized Laguerre polynomials. In addition, we derive a new formula expressing explicitly any Caputo fractional-order derivatives of FGLFs in terms of FGLFs themselves. We also propose a fractional-order generalized Laguerre tau technique in conjunction with the derived fractional-order derivative formula of FGLFs for solving Caputo type fractional differential equations (FDEs) of order ν (0 < ν < 1). The fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on FGLFs and compare them with other methods. Several numerical example are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.

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