Numerical approximation of the system of fractional differential equations with delay and its applications

2020 ◽  
Vol 135 (3) ◽  
Author(s):  
Kazem Nouri ◽  
Marjan Nazari ◽  
Leila Torkzadeh
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Approximation ◽  
Fractional Differential ◽  
Equations With Delay

scholarly journals A Nonlinear Weakly Singular Retarded Henry-Gronwall Type Integral Inequality and Its Application

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Yuanhua Lin ◽  
Shanhe Wu ◽  
Wu-Sheng Wang
Keyword(s):  
Differential Equations ◽  
Singular Integral ◽  
Fractional Differential Equations ◽  
Integral Inequality ◽  
Analysis Techniques ◽  
Weakly Singular ◽  
Type Integral ◽  
Explicit Bounds ◽  
Fractional Differential ◽  
Equations With Delay

We establish a class of new nonlinear retarded weakly singular integral inequality. Under several practical assumptions, the inequality is solved by adopting novel analysis techniques, and explicit bounds for the unknown functions are given clearly. An application of our result to the fractional differential equations with delay is shown at the end of the paper.

ψ-Haar wavelets method for numerically solving fractional differential equations

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Amjid Ali ◽  
Teruya Minamoto ◽  
Umer Saeed ◽  
Mujeeb Ur Rehman
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Approximation ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Nonlinear Problems ◽  
Mathematical Tool ◽  
Content Type ◽  
Fractional Models ◽  
Fractional Differential

Purpose The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear fractional differential equations involving ψ-Caputo derivative. Design/methodology/approach An operational matrix to find numerical approximation of ψ-fractional differential equations (FDEs) is derived. This study extends the method to nonlinear FDEs by using quasi linearization technique to linearize the nonlinear problems. Findings The error analysis of the proposed method is discussed in-depth. Accuracy and efficiency of the method are verified through numerical examples. Research limitations/implications The method is simple and a good mathematical tool for finding solutions of nonlinear ψ-FDEs. The operational matrix approach offers less computational complexity. Originality/value Engineers and applied scientists may use the present method for solving fractional models appearing in applications.

scholarly journals Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus

2021 ◽  
Vol 5 (2) ◽  
pp. 43
Author(s):  
Gerd Baumann
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Numerical Approximation ◽  
Laplace Transforms ◽  
Inverse Laplace Transform ◽  
Exact Methods ◽  
Sinc Methods ◽  
Fractional Differential

We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approximation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace transform converges exponentially and does not use Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions incorporating one, two, and three parameters. The three parameter Mittag-Leffler function represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential equations of constant and variable differentiation order.

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