scholarly journals Pade Method for Construction of Numerical Algorithms for Fractional Initial Value Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Feng Gao ◽  
Chunmei Chi
Keyword(s):  
Fractional Derivative ◽  
Taylor Series ◽  
Efficient Method ◽  
Initial Value Problem ◽  
Padé Approximation ◽  
Numerical Algorithms ◽  
Pade Approximation ◽  
Initial Value ◽  
Padé Method ◽  
Fractional Derivative Operators

In this paper, we propose an efficient method for constructing numerical algorithms for solving the fractional initial value problem by using the Pade approximation of fractional derivative operators. We regard the Grunwald–Letnikov fractional derivative as a kind of Taylor series and get the approximation equation of the Taylor series by Pade approximation. Based on the approximation equation, we construct the corresponding numerical algorithms for the fractional initial value problem. Finally, we use some examples to illustrate the applicability and efficiency of the proposed technique.

scholarly journals On the sub–diffusion fractional initial value problem with time variable order

2021 ◽  
Vol 10 (1) ◽  
pp. 1301-1315
Author(s):  
Eduardo Cuesta ◽  
Mokhtar Kirane ◽  
Ahmed Alsaedi ◽  
Bashir Ahmad
Keyword(s):  
Asymptotic Behavior ◽  
Fractional Derivative ◽  
Initial Value Problem ◽  
Regularity Result ◽  
Time Variable ◽  
Variable Order ◽  
Initial Value ◽  
Integration By Parts ◽  
Integration By Parts Formula ◽  
Variable Order Fractional Derivative

Abstract We consider a fractional derivative with order varying in time. Then, we derive for it a Leibniz' inequality and an integration by parts formula. We also study an initial value problem with our time variable order fractional derivative and present a regularity result for it, and a study on the asymptotic behavior.

scholarly journals The Limited Validity of the Conformable Euler Finite Difference Method and an Alternate Definition of the Conformable Fractional Derivative to Justify Modification of the Method

2021 ◽  
Vol 26 (4) ◽  
pp. 66
Author(s):  
Dominic Clemence-Mkhope ◽  
Belinda Clemence-Mkhope
Keyword(s):  
Finite Difference ◽  
Fractional Derivative ◽  
Initial Value Problem ◽  
Euler Method ◽  
Market Model ◽  
Alternative Methods ◽  
Conformable Fractional Derivative ◽  
Initial Value ◽  
Relaxation Equation ◽  
Definition Of

A method recently advanced as the conformable Euler method (CEM) for the finite difference discretization of fractional initial value problem Dtαyt = ft;yt, yt0 = y0, a≤t≤b, and used to describe hyperchaos in a financial market model, is shown to be valid only for α=1. The property of the conformable fractional derivative (CFD) used to show this limitation of the method is used, together with the integer definition of the derivative, to derive a modified conformable Euler method for the initial value problem considered. A method of constructing generalized derivatives from the solution of the non-integer relaxation equation is used to motivate an alternate definition of the CFD and justify alternative generalizations of the Euler method to the CFD. The conformable relaxation equation is used in numerical experiments to assess the performance of the CEM in comparison to that of the alternative methods.

scholarly journals Nonlocal Initial Value Problem for Hybrid Generalized Hilfer-type Fractional Implicit Differential Equations

2021 ◽  
Vol 8 (1) ◽  
pp. 87-100
Author(s):  
Abdelkrim Salim ◽  
Mouffak Benchohra ◽  
Jamal Eddine Lazreg ◽  
Juan J. Nieto ◽  
Yong Zhou
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Initial Value Problem ◽  
Banach Algebras ◽  
Existence Results ◽  
Initial Value ◽  
Hilfer Fractional Derivative ◽  
Implicit Differential Equations

Abstract In this paper, we prove some existence results of solutions for a class of nonlocal initial value problem for nonlinear fractional hybrid implicit differential equations under generalized Hilfer fractional derivative. The result is based on a fixed point theorem on Banach algebras. Further, examples are provided to illustrate our results.

scholarly journals On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Erdal Karapinar ◽  
Ho Duy Binh ◽  
Nguyen Hoang Luc ◽  
Nguyen Huu Can
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Initial Value Problem ◽  
Caputo Fractional Derivative ◽  
Parabolic Systems ◽  
Initial Value ◽  
Fractional Order Derivative ◽  
Nonlinear Parabolic ◽  
Need To Evaluate ◽  
Derivative Order

AbstractIn this work, we study an initial value problem for a system of nonlinear parabolic pseudo equations with Caputo fractional derivative. Here, we discuss the continuity which is related to a fractional order derivative. To overcome some of the difficulties of this problem, we need to evaluate the relevant quantities of the Mittag-Leffler function by constants independent of the derivative order. Moreover, we present an example to illustrate the theory.

Effect of Large Structural Perturbations on Response of Structural-Acoustic Systems

1999 ◽  
Author(s):  
Steven R. Hahn ◽  
Aldo A. Ferri
Keyword(s):  
Taylor Series ◽  
Padé Approximation ◽  
Taylor Series Expansion ◽  
Convergence Criteria ◽  
Pade Approximation ◽  
Scattering Problems ◽  
Structural Perturbation ◽  
The Poor ◽  
Rational Polynomial ◽  
Poor Convergence

Abstract Due to high sensitivity, first-order perturbation analysis of structural-acoustic systems can be inaccurate for large perturbations. Using high-order derivatives to create a Taylor series approximation for the perturbed solution can result in slow convergence or even divergence. A rational polynomial or Padé approximation may overcome the poor convergence of the Taylor series by canceling out the poles causing the poor convergence. In this study, a finite element framework is used to describe the structural-acoustic system. External radiation and scattering problems are accommodated by truncating the infinite fluid region using exponential decay infinite elements. Changes to a nominal model are introduced through a structural perturbation to the nominal structural stiffness and/or mass matrices. An efficient method for calculating the solution derivatives with respect to the structural perturbation is presented. A Taylor series expansion is constructed using the derivative information and the convergence criteria for the series is examined. The local solution and derivatives are then used to construct a Padé approximation. The method is illustrated by considering what effect the addition of a rib stiffener to a plate in a rigid baffle has on the scattering of a plane wave. The approximation is shown to be quite accurate for large perturbations even when there are one or more nearby poles and the Taylor series fails to converge.

scholarly journals On fuzzy Volterra-Fredholm integrodifferential equation associated with Hilfer-generalized proportional fractional derivative

2021 ◽  
Vol 6 (10) ◽  
pp. 10920-10946
Author(s):  
Saima Rashid ◽  
◽  
Fahd Jarad ◽  
Khadijah M. Abualnaja ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Fractional Derivative ◽  
Integrodifferential Equation ◽  
Initial Value Problem ◽  
Initial Conditions ◽  
Equivalent Form ◽  
Initial Value ◽  
Uniqueness Of The Solution ◽  
Generalized Lipschitz Condition ◽  
Fuzzy Framework

<abstract><p>This investigation communicates with an initial value problem (IVP) of Hilfer-generalized proportional fractional ($ \mathcal{GPF} $) differential equations in the fuzzy framework is deliberated. By means of the Hilfer-$ \mathcal{GPF} $ operator, we employ the methodology of successive approximation under the generalized Lipschitz condition. Based on the proposed derivative, the fractional Volterra-Fredholm integrodifferential equations $ (\mathcal{FVFIE}s) $ via generalized fuzzy Hilfer-$ \mathcal{GPF} $ Hukuhara differentiability ($ \mathcal{HD} $) having fuzzy initial conditions are investigated. Moreover, the existence of the solution is proposed by employing the fixed-point formulation. The uniqueness of the solution is verified. Furthermore, we derived the equivalent form of fuzzy $ \mathcal{FVFIE}s $ which is supposed to demonstrate the convergence of this group of equations. Two appropriate examples are presented for illustrative purposes.</p></abstract>

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