scholarly journals A Novel Treatment of Fuzzy Fractional Swift–Hohenberg Equation for a Hybrid Transform within the Fractional Derivative Operator

2021 ◽  
Vol 5 (4) ◽  
pp. 209
Author(s):  
Saima Rashid ◽  
Rehana Ashraf ◽  
Fatimah S. Bayones
Keyword(s):  
Fractional Derivative ◽  
Decomposition Method ◽  
Numerical Solutions ◽  
Integral Transform ◽  
Initial Conditions ◽  
Adomian Decomposition Method ◽  
Initial Value ◽  
Adomian Decomposition ◽  
In Series ◽  
Fractional Orders

This article investigates the semi-analytical method coupled with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). In addition, we apply this method to the time-fractional Swift–Hohenberg equation (SHe) with various initial conditions (IC) under gH-differentiability. Some aspects of the fuzzy Caputo fractional derivative (CFD) with the Elzaki transform are presented. Moreover, we established the general formulation and approximate findings by testing examples in series form of the models under investigation with success. With the aid of the projected method, we establish the approximate analytical results of SHe with graphical representations of initial value problems by inserting the uncertainty parameter 0≤℘≤1 with different fractional orders. It is expected that fuzzy EADM will be powerful and accurate in configuring numerical solutions to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures.

scholarly journals Analytic Fuzzy Formulation of a Time-Fractional Fornberg–Whitham Model with Power and Mittag–Leffler Kernels

2021 ◽  
Vol 5 (3) ◽  
pp. 113 ◽  
Author(s):  
Saima Rashid ◽  
Rehana Ashraf ◽  
Ahmet Ocak Akdemir ◽  
Manar A. Alqudah ◽  
Thabet Abdeljawad ◽  
...  
Keyword(s):  
Fractional Derivative ◽  
Decomposition Method ◽  
Numerical Solutions ◽  
Integral Transform ◽  
Initial Conditions ◽  
Adomian Decomposition Method ◽  
Algorithmic Approach ◽  
Closed Form Solutions ◽  
Adomian Decomposition ◽  
Fractional Orders

This manuscript assesses a semi-analytical method in connection with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). Moreover, we use the aforesaid strategy to address the time-fractional Fornberg–Whitham equation (FWE) under gH-differentiability by employing different initial conditions (IC). Several algebraic aspects of the fuzzy Caputo fractional derivative (CFD) and fuzzy Atangana–Baleanu (AB) fractional derivative operator in the Caputo sense, with respect to the Elzaki transform, are presented to validate their utilities. Apart from that, a general algorithm for fuzzy Caputo and AB fractional derivatives in the Caputo sense is proposed. Some illustrative cases are demonstrated to understand the algorithmic approach of FWE. Taking into consideration the uncertainty parameter ζ∈[0,1] and various fractional orders, the convergence and error analysis are reported by graphical representations of FWE that have close harmony with the closed form solutions. It is worth mentioning that the projected approach to fuzziness is to verify the supremacy and reliability of configuring numerical solutions to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures.

scholarly journals Solution to the Linear Fractional Differential Equation Using Adomian Decomposition Method

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Jin-Fa Cheng ◽  
Yu-Ming Chu
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Fractional Derivative ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Initial Value ◽  
Adomian Decomposition ◽  
Constant Coefficients ◽  
Fractional Differential ◽  
Linear Fractional Differential Equations

We obtain the analytical general solution of the linear fractional differential equations with constant coefficients by Adomian decomposition method under nonhomogeneous initial value condition, which is in the sense of the Caputo fractional derivative.

Solution of jamming transition problem using adomian decomposition method

2018 ◽  
Vol 35 (5) ◽  
pp. 1950-1964 ◽  
Author(s):  
Erman Şentürk ◽  
Safa Bozkurt Coşkun ◽  
Mehmet Tarık Atay
Keyword(s):  
Decomposition Method ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Adomian Decomposition Method ◽  
Numerical Function ◽  
Content Type ◽  
Jamming Transition ◽  
Transition Problem ◽  
Adomian Decomposition ◽  
Highly Nonlinear

Purpose The purpose of the study is to obtain an analytical approximate solution for jamming transition problem (JTP) using Adomian decomposition method (ADM). Design/methodology/approach In this study, the jamming transition is presented as a result of spontaneous deviations of headway and velocity that is caused by the acceleration/breaking rate to be higher than the critical value. Dissipative dynamics of traffic flow can be represented within the framework of the Lorenz scheme based on the car-following model in the one-lane highway. Through this paper, an analytical approximation for the solution is calculated via ADM that leads to a solution for headway deviation as a function of time. Findings A highly nonlinear differential equation having no exact solution due to JTP is considered and headway deviation is obtained implementing a number of different initial conditions. The results are discussed and compared with the available data in the literature and numerical solutions obtained from a built-in numerical function of the mathematical software used in the study. The advantage of using ADM for the problem is presented in the study and discussed on the basis of the results produced by the applied method. Originality/value This is the first study to apply ADM to JTP.

scholarly journals Iterative Analysis of Nonlinear BBM Equations under Nonsingular Fractional Order Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Gauhar Ali ◽  
Israr Ahmad ◽  
Kamal Shah ◽  
Thabet Abdeljawad
Keyword(s):  
Fixed Point ◽  
Fractional Order ◽  
Decomposition Method ◽  
Integral Transform ◽  
Research Work ◽  
Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Iterative Analysis ◽  
Fractional Orders ◽  
Variational Type

The present research work is devoted to investigate fractional order Benjamin-Bona-Mahony (FBBM) as well as modified fractional order FBBM (FMBBM) equations under nonlocal and nonsingular derivative of Caputo-Fabrizio (CF). In this regards, some qualitative results including the existence of at least one solution are established via using some fixed point results of Krasnoselskii and Banach. Further on using an iterative method, some semianalytical results are also studied. The concerned tool is formed when the Adomian decomposition method is coupled with some integral transform like Laplace. Graphical presentations are given for various fractional orders. Also, the concerned method is also compared with some variational-type perturbation method to demonstrate the efficiency of the proposed method.

Numerical solutions of the initial value problem for fractional differential equations by modification of the Adomian decomposition method

2014 ◽  
Vol 17 (2) ◽  
Author(s):  
Neda Khodabakhshi ◽  
S. Mansour Vaezpour ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problem ◽  
Decomposition Method ◽  
Numerical Solutions ◽  
Adomian Decomposition Method ◽  
Initial Value ◽  
Adomian Decomposition ◽  
Fractional Differential

AbstractIn this paper, we extend a reliable modification of the Adomian decomposition method presented in [34] for solving initial value problem for fractional differential equations.In order to confirm the applicability and the advantages of our approach, we consider some illustrative examples.

scholarly journals Approximate solution of nonlinear ordinary differential equation using ZZ decomposition method

2020 ◽  
Vol 4 (1) ◽  
pp. 448-455
Author(s):  
Mulugeta Andualem ◽  
◽  
Atinafu Asfaw ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Approximate Solution ◽  
Initial Value Problems ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Nonlinear Ordinary Differential Equation ◽  
Initial Value ◽  
Transform Method ◽  
Adomian Decomposition

Nonlinear initial value problems are somewhat difficult to solve analytically as well as numerically related to linear initial value problems as their variety of natures. Because of this, so many scientists still searching for new methods to solve such nonlinear initial value problems. However there are many methods to solve it. In this article we have discussed about the approximate solution of nonlinear first order ordinary differential equation using ZZ decomposition method. This method is a combination of the natural transform method and Adomian decomposition method.

scholarly journals Computational and Numerical Solutions for 2+1-Dimensional Integrable Schwarz–Korteweg–de Vries Equation with Miura Transform

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Raghda A. M. Attia ◽  
S. H. Alfalqi ◽  
J. F. Alzaidi ◽  
Mostafa M. A. Khater ◽  
Dianchen Lu
Keyword(s):  
Solitary Waves ◽  
Decomposition Method ◽  
Numerical Solutions ◽  
Vries Equation ◽  
Adomian Decomposition Method ◽  
Simplest Equation Method ◽  
Adomian Decomposition ◽  
Tanh Method ◽  
De Vries ◽  
Parameter Values

This paper investigates the analytical, semianalytical, and numerical solutions of the 2+1–dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation. The extended simplest equation method, the sech-tanh method, the Adomian decomposition method, and cubic spline scheme are employed to obtain distinct formulas of solitary waves that are employed to calculate the initial and boundary conditions. Consequently, the numerical solutions of this model can be investigated. Moreover, their stability properties are also analyzed. The solutions obtained by means of these techniques are compared to unravel relations between them and their characteristics illustrated under the suitable choice of the parameter values.

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