scholarly journals Abundant Wave Accurate Analytical Solutions of the Fractional Nonlinear Hirota–Satsuma–Shallow Water Wave Equation

Fluids ◽  
2021 ◽  
Vol 6 (7) ◽  
pp. 235
Author(s):  
Chen Yue ◽  
Dianchen Lu ◽  
Mostafa M. A. Khater
Keyword(s):  
Wave Equation ◽  
Fractional Derivative ◽  
Shallow Water ◽  
Analytical Solutions ◽  
Water Wave ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Conformable Fractional Derivative ◽  
Adomian Decomposition ◽  
Fractional System

This research paper targets the fractional Hirota’s analytical solutions–Satsuma (HS) equations. The conformable fractional derivative is employed to convert the fractional system into a system with an integer–order. The extended simplest equation (ESE) and modified Kudryashov (MKud) methods are used to construct novel solutions of the considered model. The solutions’ accuracy is investigated by handling the computational solutions with the Adomian decomposition method. The solutions are explained in some different sketches to demonstrate more novel properties of the considered model.

scholarly journals Dynamical control on the Adomian decomposition method for solving shallow water wave equation

2021 ◽  
Vol 25 (5) ◽  
pp. 623-632
Author(s):  
L. Noeiaghdam ◽  
S. Noeiaghdam ◽  
D. N. Sidorov
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Numerical Results ◽  
Stochastic Arithmetic ◽  
Shallow Water Wave ◽  
Adomian Decomposition ◽  
Dynamical Control

The aim of this study is to apply a novel technique to control the accuracy and error of the Adomian decomposition method (ADM) for solving nonlinear shallow water wave equation. The ADM is among semi-analytical and powerful methods for solving many mathematical and engineering problems. We apply the Controle et Estimation Stochastique des Arrondis de Calculs (CESTAC) method which is based on stochastic arithmetic (SA). Also instead of applying mathematical packages we use the Control of Accuracy and Debugging for Numerical Applications (CADNA) library. In this library we will write all codes using C++ programming codes. Applying the method we can find the optimal numerical results, error and step of the ADM and they are the main novelties of this research. The numerical results show the accuracy and efficiency of the novel scheme.

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 Approximate analytical solutions of non-linear local fractional heat equations

2019 ◽  
Vol 23 (Suppl. 3) ◽  
pp. 837-841 ◽  
Author(s):  
Shuxian Deng
Keyword(s):  
Analytical Solutions ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Heat Equations ◽  
Transform Method ◽  
Fractional Complex Transform ◽  
Fractional Heat Equation ◽  
Approximate Analytical Solutions ◽  
Adomian Decomposition ◽  
Non Linear

Consider the non-linear local fractional heat equation. The fractional complex transform method and the Adomian decomposition method are used to solve the equation. The approximate analytical solutions are obtained.

Analytical Solution of a Dynamic System Containing Fractional Derivative of Order One-Half by Adomian Decomposition Method

2005 ◽  
Vol 72 (2) ◽  
pp. 290-295 ◽  
Author(s):  
S. Saha Ray ◽  
B. P. Poddar ◽  
R. K. Bera
Keyword(s):  
Fractional Derivative ◽  
Dynamic System ◽  
Decomposition Method ◽  
Materials Science ◽  
Adomian Decomposition Method ◽  
Expansion Method ◽  
Physical Problems ◽  
Relaxation Functions ◽  
Damping Behavior ◽  
Adomian Decomposition

The fractional derivative has been occurring in many physical problems, such as frequency-dependent damping behavior of materials, motion of a large thin plate in a Newtonian fluid, creep and relaxation functions for viscoelastic materials, the PIλDμ controller for the control of dynamical systems, etc. Phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry, and materials science are also described by differential equations of fractional order. The solution of the differential equation containing a fractional derivative is much involved. Instead of an application of the existing methods, an attempt has been made in the present analysis to obtain the solution of an equation in a dynamic system whose damping behavior is described by a fractional derivative of order 1/2 by the relatively new Adomian decomposition method. The results obtained by this method are then graphically represented and compared with those available in the work of Suarez and Shokooh [Suarez, L. E., and Shokooh, A., 1997, “An Eigenvector Expansion Method for the Solution of Motion Containing Fraction Derivatives,” ASME J. Appl. Mech., 64, pp. 629–635]. A good agreement of the results is observed.

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