Homotopy analysis Shehu transform method for solving fuzzy differential equations of fractional and integer order derivatives

2021 ◽  
Vol 40 (3) ◽  
Author(s):  
Shehu Maitama ◽  
Weidong Zhao
Keyword(s):  
Differential Equations ◽  
Transform Method ◽  
Integer Order ◽  
Homotopy Analysis

Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions

2010 ◽  
Vol 65 (11) ◽  
pp. 935-949 ◽  
Author(s):  
Mehdi Dehghan ◽  
Jalil Manafian ◽  
Abbas Saadatmandi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Analysis Method ◽  
Partial Differential ◽  
Integer Order ◽  
Homotopy Analysis

In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

scholarly journals Exact solution of time-fractional partial differential equations using Laplace transform

2016 ◽  
Vol 5 (1) ◽  
pp. 86
Author(s):  
Naser Al-Qutaifi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Partial Differential Equations ◽  
Laplace Transform Method ◽  
Partial Differential ◽  
Transform Method ◽  
Integer Order ◽  
First Derivative ◽  
The Laplace Transform

<p>The idea of replacing the first derivative in time by a fractional derivative of order , where , leads to a fractional generalization of any partial differential equations of integer order. In this paper, we obtain a relationship between the solution of the integer order equation and the solution of its fractional extension by using the Laplace transform method.</p>

Toward a New Algorithm for Nonlinear Fractional Differential Equations

2013 ◽  
Vol 5 (2) ◽  
pp. 222-234
Author(s):  
Fadi Awawdeh ◽  
S. Abbasbandy
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Homotopy Analysis Method ◽  
Analytical Techniques ◽  
Analytic Solutions ◽  
Analysis Method ◽  
Transform Method ◽  
Nonlinear Fractional Differential Equations ◽  
Homotopy Analysis ◽  
Fractional Differential

AbstractThis paper is concerned with the development of an efficient algorithm for the analytic solutions of nonlinear fractional differential equations. The proposed algorithm Laplace homotopy analysis method (LHAM) is a combined form of the Laplace transform method with the homotopy analysis method. The biggest advantage the LHAM has over the existing standard analytical techniques is that it overcomes the difficulty arising in calculating complicated terms. Moreover, the solution procedure is easier, more effective and straightforward. Numerical examples are examined to demonstrate the accuracy and efficiency of the proposed algorithm.

scholarly journals New dynamical behaviour of the coronavirus (COVID-19) infection system with nonlocal operator from reservoirs to people

2020 ◽  
Author(s):  
P. Veeresha ◽  
D. G. Prakasha ◽  
Naveen Sanju Malagi ◽  
Haci Mehmet Baskonus ◽  
Wei Gao
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Arbitrary Order ◽  
Dynamical Behaviour ◽  
Nonlocal Operator ◽  
Nonlinear Ordinary Differential Equations ◽  
Transform Method ◽  
Homotopy Analysis ◽  
The Fixed Point Theorem ◽  
The Mathematical Model

Abstract The fundamental aim of the present study is to analyse and find the solution for the system of nonlinear ordinary differential equations describing the deadly and most dangerous virus from the lost three months called coronavirus. The mathematical model consisting of six nonlinear ordinary differential equations are exemplified and the corresponding solution is studied within the frame of 𝑞-homotopy analysis transform method (𝑞-HATM). Moreover, a newly defined fractional operator is employed in order to understand more effectively, known as Atangana-Baleanu (AB) operator. For the obtained results, the fixed point theorem is hired to present the exactness as well as uniqueness. For diverse arbitrary order, the behaviour of the outcomes is presented in terms of plots. Finally, the present study may help to examine the wild class of real-world models and also aid to predict their behaviour with respect to parameters considered in the models.

A new analysis of fractional fish farm model associated with Mittag-Leffler-type kernel

2020 ◽  
Vol 13 (02) ◽  
pp. 2050010 ◽  
Author(s):  
Jagdev Singh ◽  
Devendra Kumar ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equations ◽  
Arbitrary Order ◽  
Nonlinear Differential Equations ◽  
Dynamical Behavior ◽  
Fish Farm ◽  
Uniqueness Of Solution ◽  
Integer Order ◽  
Farm Model ◽  
Complete Study ◽  
Homotopy Analysis

In this paper, we analyze the dynamical behavior of fish farm model related to Atangana–Baleanu derivative of arbitrary order. The model is constituted with the group of nonlinear differential equations having nutrients, fish and mussel. We have included discrete kind gestational delay of fish. The solution of fish farm model is determined by employing homotopy analysis transforms method (HATM). Existence of and uniqueness of solution are studied through Picard–Lindelof approach. The influence of order of new non-integer order derivative on nutrients, fish and mussel is discussed. The complete study reveals that the outer food supplies manage the behavior of the model. Moreover, to show the outcomes of the study, some numerical results are demonstrated through graphs.

scholarly journals An efficient hybrid technique for the solution of fractional-order partial differential equations

2021 ◽  
Vol 13 (3) ◽  
pp. 790-804
Author(s):  
H.K. Jassim ◽  
H. Ahmad ◽  
A. Shamaoon ◽  
C. Cesarano
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Hybrid Technique ◽  
Science And Engineering ◽  
Partial Differential ◽  
Transform Method ◽  
Suggested Technique ◽  
Homotopy Analysis ◽  
Sumudu Transform

In this paper, a hybrid technique called the homotopy analysis Sumudu transform method has been implemented solve fractional-order partial differential equations. This technique is the amalgamation of Sumudu transform method and the homotopy analysis method. Three examples are considered to validate and demonstrate the efficacy and accuracy of the present technique. It is also demonstrated that the results obtained from the suggested technique are in excellent agreement with the exact solution which shows that the proposed method is efficient, reliable and easy to implement for various related problems of science and engineering.

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