Computational Method for Fractional Differential Equations Using Nonpolynomial Fractional Spline

2016 ◽  
Vol 5 (2) ◽  
pp. 131-136 ◽  
Author(s):  
F. K. Hamasalh ◽  
P. O. Muhammed
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Computational Method ◽  
Fractional Differential

High Speed Algorithm for Computation of Fractional Differentiation and Integration

2011 ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Standard Method ◽  
High Speed ◽  
Computational Method ◽  
Integration Time ◽  
Fractional Differentiation ◽  
History Of ◽  
Weighted Integrals ◽  
Fractional Differential

A high speed algorithm for computing fractional differentiations and fractional integrations in fractional differential equations is proposed. In this algorithm the stored data is not the history of the function to be differentiated or integrated but the history of the weighted integrals of the function. It is shown that, by the computational method based on the new algorithm, the integration time only increases in proportion to n log n, different from n2 by a standard method, for n steps of integrations of a differential integration.

scholarly journals Computational method based on Bernstein operational matrices for multi-order fractional differential equations

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 591-601 ◽  
Author(s):  
Davood Rostamy ◽  
Hossein Jafari ◽  
Mohsen Alipour ◽  
Chaudry Khalique
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Operational Matrix ◽  
Adomian Decomposition Method ◽  
Computational Method ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Transform Method ◽  
Adomian Decomposition ◽  
Fractional Differential

In this paper, the Bernstein operational matrices are used to obtain solutions of multi-order fractional differential equations. In this regard we present a theorem which can reduce the nonlinear fractional differential equations to a system of algebraic equations. The fractional derivative considered here is in the Caputo sense. Finally, we give several examples by using the proposed method. These results are then compared with the results obtained by using Adomian decomposition method, differential transform method and the generalized block pulse operational matrix method. We conclude that our results compare well with the results of other methods and the efficiency and accuracy of the proposed method is very good.

scholarly journals APPLICATIONS OF BI-FRAMELET SYSTEMS FOR SOLVING FRACTIONAL ORDER DIFFERENTIAL EQUATIONS

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040051 ◽  
Author(s):  
MUTAZ MOHAMMAD ◽  
CARLO CATTANI
Keyword(s):  
Numerical Analysis ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Sparse Matrices ◽  
Computational Method ◽  
Extension Principle ◽  
Vanishing Moments ◽  
Fractional Order Differential Equations ◽  
Fractional Differential

Framelets and their attractive features in many disciplines have attracted a great interest in the recent years. This paper intends to show the advantages of using bi-framelet systems in the context of numerical fractional differential equations (FDEs). We present a computational method based on the quasi-affine bi-framelets with high vanishing moments constructed using the generalized (mixed) oblique extension principle. We use this system for solving some types of FDEs by solving a series of important examples of FDEs related to many mathematical applications. The quasi-affine bi-framelet-based methods for numerical FDEs show the advantages of using sparse matrices and its accuracy in numerical analysis.

Symmetry properties of fractional order transport equations

2012 ◽  
Vol 9 (1) ◽  
pp. 59-64
Author(s):  
R.K. Gazizov ◽  
A.A. Kasatkin ◽  
S.Yu. Lukashchuk
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lie Group ◽  
Initial Conditions ◽  
Group Analysis ◽  
Equivalence Transformation ◽  
Point Change ◽  
Infinitesimal Operator ◽  
Symmetry Properties ◽  
Fractional Differential

In the paper some features of applying Lie group analysis methods to fractional differential equations are considered. The problem related to point change of variables in the fractional differentiation operator is discussed and some general form of transformation that conserves the form of Riemann-Liouville fractional operator is obtained. The prolongation formula for extending an infinitesimal operator of a group to fractional derivative with respect to arbitrary function is presented. Provided simple example illustrates the necessity of considering both local and non-local symmetries for fractional differential equations in particular cases including the initial conditions. The equivalence transformation forms for some fractional differential equations are discussed and results of group classification of the wave-diffusion equation are presented. Some examples of constructing particular exact solutions of fractional transport equation are given, based on the Lie group methods and the method of invariant subspaces.

Caputo fractional differential equations with non-instantaneous impulses and strict stability by Lyapunov functions

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5217-5239 ◽  
Author(s):  
Ravi Agarwal ◽  
Snehana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Time Intervals ◽  
Dini Derivative ◽  
Comparison Results ◽  
Strict Stability ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

In this paper the statement of initial value problems for fractional differential equations with noninstantaneous impulses is given. These equations are adequate models for phenomena that are characterized by impulsive actions starting at arbitrary fixed points and remaining active on finite time intervals. Strict stability properties of fractional differential equations with non-instantaneous impulses by the Lyapunov approach is studied. An appropriate definition (based on the Caputo fractional Dini derivative of a function) for the derivative of Lyapunov functions among the Caputo fractional differential equations with non-instantaneous impulses is presented. Comparison results using this definition and scalar fractional differential equations with non-instantaneous impulses are presented and sufficient conditions for strict stability and uniform strict stability are given. Examples are given to illustrate the theory.

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