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.

scholarly journals The Time-Fractional Coupled-Korteweg-de-Vries Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Abdon Atangana ◽  
Aydin Secer
Keyword(s):  
Decomposition Method ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Adomian Decomposition Method ◽  
Variational Iteration Method ◽  
Analytical Technique ◽  
Adomian Decomposition ◽  
Homotopy Decomposition ◽  
De Vries ◽  
Homotopy Analysis

We put into practice a relatively new analytical technique, the homotopy decomposition method, for solving the nonlinear fractional coupled-Korteweg-de-Vries equations. Numerical solutions are given, and some properties exhibit reasonable dependence on the fractional-order derivatives’ values. The fractional derivatives are described in the Caputo sense. The reliability of HDM and the reduction in computations give HDM a wider applicability. In addition, the calculations involved in HDM are very simple and straightforward. It is demonstrated that HDM is a powerful and efficient tool for FPDEs. It was also demonstrated that HDM is more efficient than the adomian decomposition method (ADM), variational iteration method (VIM), homotopy analysis method (HAM), and homotopy perturbation method (HPM).

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 An efficient approach for the numerical solution of fifth-order KdV equations

2020 ◽  
Vol 18 (1) ◽  
pp. 738-748 ◽  
Author(s):  
Hijaz Ahmad ◽  
Tufail A. Khan ◽  
Shao-Wen Yao
Keyword(s):  
Decomposition Method ◽  
Numerical Solutions ◽  
Adomian Decomposition Method ◽  
Iteration Algorithm ◽  
Test Problems ◽  
Transform Method ◽  
Kdv Equations ◽  
Adomian Decomposition ◽  
Different Types ◽  
Fifth Order

Abstract The main aim of this article is to use a new and simple algorithm namely the modified variational iteration algorithm-II (MVIA-II) to obtain numerical solutions of different types of fifth-order Korteweg-de Vries (KdV) equations. In order to assess the precision, stability and accuracy of the solutions, five test problems are offered for different types of fifth-order KdV equations. Numerical results are compared with the Adomian decomposition method, Laplace decomposition method, modified Adomian decomposition method and the homotopy perturbation transform method, which reveals that the MVIA-II exceptionally productive, computationally attractive and has more accuracy than the others.

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.

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