Laplace transform method for linear sequential Riemann Liouville and Caputo fractional differential equations

2017 ◽  
Author(s):  
Aghalaya S. Vatsala ◽  
M. Sowmya
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

scholarly journals On matrix fractional differential equations

2017 ◽  
Vol 9 (1) ◽  
pp. 168781401668335
Author(s):  
Adem Kılıçman ◽  
Wasan Ajeel Ahmood
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Convolution Product ◽  
Operational Matrices ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Fractional Differential

The aim of this article is to study the matrix fractional differential equations and to find the exact solution for system of matrix fractional differential equations in terms of Riemann–Liouville using Laplace transform method and convolution product to the Riemann–Liouville fractional of matrices. Also, we show the theorem of non-homogeneous matrix fractional partial differential equation with some illustrative examples to demonstrate the effectiveness of the new methodology. The main objective of this article is to discuss the Laplace transform method based on operational matrices of fractional derivatives for solving several kinds of linear fractional differential equations. Moreover, we present the operational matrices of fractional derivatives with Laplace transform in many applications of various engineering systems as control system. We present the analytical technique for solving fractional-order, multi-term fractional differential equation. In other words, we propose an efficient algorithm for solving fractional matrix equation.

scholarly journals Hyers–Ulam Stability and Existence of Solutions to the Generalized Liouville–Caputo Fractional Differential Equations

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 955
Author(s):  
Kui Liu ◽  
Michal Fečkan ◽  
Jinrong Wang
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fixed Point Theorems ◽  
Uniqueness Of Solutions ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Caputo Fractional Differential Equations ◽  
The Cauchy Problem ◽  
Fractional Differential ◽  
The Stability

The aim of this paper is to study the stability of generalized Liouville–Caputo fractional differential equations in Hyers–Ulam sense. We show that three types of the generalized linear Liouville–Caputo fractional differential equations are Hyers–Ulam stable by a ρ -Laplace transform method. We establish existence and uniqueness of solutions to the Cauchy problem for the corresponding nonlinear equations with the help of fixed point theorems.

scholarly journals Conformable Laplace Transform of Fractional Differential Equations

Axioms ◽  
2018 ◽  
Vol 7 (3) ◽  
pp. 55 ◽  
Author(s):  
Fernando Silva ◽  
Davidson Moreira ◽  
Marcelo Moret
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Classical Model ◽  
Transform Method ◽  
Fractional Models ◽  
Constant Coefficients ◽  
Fractional Differential ◽  
Linear Differential ◽  
Fractional Orders

In this paper, we use the conformable fractional derivative to discuss some fractional linear differential equations with constant coefficients. By applying some similar arguments to the theory of ordinary differential equations, we establish a sufficient condition to guarantee the reliability of solving constant coefficient fractional differential equations by the conformable Laplace transform method. Finally, the analytical solution for a class of fractional models associated with the logistic model, the von Foerster model and the Bertalanffy model is presented graphically for various fractional orders. The solution of the corresponding classical model is recovered as a particular case.

scholarly journals Numerical Solutions of Fractional Differential Equations by Using Laplace Transformation Method and Quadrature Rule

2021 ◽  
Vol 5 (3) ◽  
pp. 111
Author(s):  
Samaneh Soradi-Zeid ◽  
Mehdi Mesrizadeh ◽  
Carlo Cattani
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Numerical Method ◽  
Fractional Differential Equations ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
Quadrature Rule ◽  
Differential Problem ◽  
Transform Method ◽  
Fractional Differential

This paper introduces an efficient numerical scheme for solving a significant class of fractional differential equations. The major contributions made in this paper apply a direct approach based on a combination of time discretization and the Laplace transform method to transcribe the fractional differential problem under study into a dynamic linear equations system. The resulting problem is then solved by employing the numerical method of the quadrature rule, which is also a well-developed numerical method. The present numerical scheme, which is based on the numerical inversion of Laplace transform and equal-width quadrature rule is robust and efficient. Some numerical experiments are carried out to evaluate the performance and effectiveness of the suggested framework.

scholarly journals Homotopy Perturbation Transform method for solving the partial and the time-fractional differential equations with variable coefficients

2020 ◽  
Vol 26 (1) ◽  
pp. 35-55
Author(s):  
Abdelkader Kehaili ◽  
Ali Hakem ◽  
Abdelkader Benali
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Variable Coefficients ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

In this paper, we present the exact solutions of the Parabolic-like equations and Hyperbolic-like equations with variable coefficients, by using Homotopy perturbation transform method (HPTM). Finally, we extend the results to the time-fractional differential equations. Keywords: Caputo’s fractional derivative, fractional differential equations, homotopy perturbation transform method, hyperbolic-like equation, Laplace transform, parabolic-like equation.

scholarly journals Conformable Laplace Transform of Fractional Differential Equations

2018 ◽  
Author(s):  
Fernando S. Silva ◽  
Davidson M. Moreira ◽  
Marcelo A. Moret
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Classical Model ◽  
Transform Method ◽  
Fractional Models ◽  
Constant Coefficients ◽  
Fractional Differential ◽  
Linear Differential ◽  
Fractional Orders

In this paper we use the conformable fractional derivative to discuss some fractional linear differential equations with constant coefficients. By applying some similar arguments to the theory of ordinary differential equations, we establish a sufficient condition to guarantee the reliability of solving constant coefficient fractional differential equations by the conformable Laplace transform method. Finally, we analyze the analytical solution for a class of fractional models associated with Logistic model, Von Foerster model and Bertalanffy model is presented graphically for various fractional orders and solution of corresponding classical model is recovered as a particular case.

A New Method for Laplace Transforms of Multi-Term Fractional Differential Equations of the Caputo Type

2021 ◽  
Author(s):  
Masataka Fukunaga
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Initial Conditions ◽  
Expansion Method ◽  
Laplace Transforms ◽  
Transform Method ◽  
Change Of Variables ◽  
The Laplace Transform ◽  
Fractional Differential

Abstract The Laplace transform method is one of the powerful tools in studying the frac- tional differential equations (FDEs). In this paper, it is shown that the Heaviside expansion method for integer order differential equations is also applicable to the Laplace transforms of multi-term Caputo fractional differential equations (FDEs) of zero initial conditions if the orders of Caputo derivatives are integer multiples of a common real number. The particular solution of a linear multi-term Caputo FDE is obtained by its Laplace transform and the Heaviside expansion method. A Caputo FDE of non zero initial conditions is transformed to an Caputo FDE of zero initial conditions by an appropriate change of variables. In the latter, the terms originated from the initial conditions appear as nonhomogeneous terms. Thus, the Caputo FDE of nonzero initial conditions is obtained as the particular solutions to the equivalent Caputo FDE of zero initial conditions. The solutions of a linear multi-term Caputo FDEs of nonzero initial conditions are expressed through the two parameter Mittag-Leffler functions.

scholarly journals Series Solutions of High-Dimensional Fractional Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2021
Author(s):  
Jing Chang ◽  
Jin Zhang ◽  
Ming Cai
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Salient Feature ◽  
Approximate Solutions ◽  
Series Solutions ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Time Space ◽  
Sumudu Transform ◽  
Fractional Differential

In the present paper, the series solutions and the approximate solutions of the time–space fractional differential equations are obtained using two different analytical methods. One is the homotopy perturbation Sumudu transform method (HPSTM), and another is the variational iteration Laplace transform method (VILTM). It is observed that the approximate solutions are very close to the exact solutions. The solutions obtained are very useful and significant to analyze many phenomena, and the solutions have not been reported in previous literature. The salient feature of this work is the graphical presentations of the third approximate solutions for different values of order α.

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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