Numerical Method for Simulation of Physical Processes Described by Fractional-Order Integro-Differential Equations

2018 ◽  
pp. 109-134
Author(s):  
Seshu Kumar Damarla
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Fractional Order ◽  
Physical Processes

scholarly journals A Numerical Method for Delayed Fractional-Order Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Zhen Wang
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Fractional Order ◽  
Exact Analytical Solution ◽  
Interpolation Method ◽  
Linear Interpolation ◽  
Positive Real ◽  
Time Varying Delay ◽  
Fractional Order Differential Equations ◽  
Varying Delay

A numerical method for nonlinear fractional-order differential equations with constant or time-varying delay is devised. The order here is an arbitrary positive real number, and the differential operator is with the Caputo definition. The general Adams-Bashforth-Moulton method combined with the linear interpolation method is employed to approximate the delayed fractional-order differential equations. Meanwhile, the detailed error analysis for this algorithm is given. In order to compare with the exact analytical solution, a numerical example is provided to illustrate the effectiveness of the proposed method.

Legendre Collocation Solution to Fractional Ordinary Differential Equations

2014 ◽  
Vol 687-691 ◽  
pp. 601-605
Author(s):  
Ting Gang Zhao ◽  
Zi Lang Zhan ◽  
Jin Xia Huo ◽  
Zi Guang Yang
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Fractional Order ◽  
Numerical Results ◽  
Spectral Accuracy ◽  
Collocation Solution

In this paper, we propose an efficient numerical method for ordinary differential equation with fractional order, based on Legendre-Gauss-Radau interpolation, which is easy to be implemented and possesses the spectral accuracy. We apply the proposed method to multi-order fractional ordinary differential equation. Numerical results demonstrate the effectiveness of the approach.

scholarly journals Iterative numerical method for fractional order two-point boundary value problems

2021 ◽  
Vol 38 (1) ◽  
pp. 47-55
Author(s):  
ALEXANDRU MIHAI BICA ◽  
Keyword(s):  
Differential Equations ◽  
Error Estimate ◽  
Boundary Value Problems ◽  
Numerical Method ◽  
Fractional Order ◽  
Boundary Value ◽  
Point Boundary ◽  
Boundary Problems ◽  
Numerical Example

In this paper we develop an iterative numerical method based on Bernstein splines for solving two-point boundary problems associated to differential equations of fractional order $\alpha\in\left( 0,1\right) $. The convergence of the method is proved by providing the error estimate and it is tested on a numerical example.

AN EFFECTIVE NUMERICAL METHOD FOR SOLVING THE NONLINEAR SINGULAR LANE-EMDEN TYPE EQUATIONS OF VARIOUS ORDERS

2016 ◽  
Vol 79 (1) ◽  
Author(s):  
Kourosh Parand ◽  
Mehdi Delkhosh
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Collocation Method ◽  
Fractional Order ◽  
Spectral Methods ◽  
Mathematical Physics ◽  
Infinite Domain ◽  
Orthogonal Functions ◽  
Non Linear

The Lane-Emden type equations are employed in the modeling of several phenomena in the areas of mathematical physics and astrophysics. These equations are categorized as non-linear singular ordinary differential equations on the semi-infinite domain. In this paper, the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) of the first kind have been introduced as a new basis for Spectral methods, and also presented an effective numerical method based on the GFCFs and the collocation method for solving the nonlinear singular Lane-Emden type equations of various orders. Obtained results have compared with other results to verify the accuracy and efficiency of the presented method.

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