scholarly journals Localized modes in time-fractional modified coupled Korteweg-de Vries equation with singular and non-singular kernels

2021 ◽  
Vol 7 (2) ◽  
pp. 1580-1602
Author(s):  
Khalid Khan ◽  
◽  
Amir Ali ◽  
Manuel De la Sen ◽  
Muhammad Irfan ◽  
...  
Keyword(s):  
Decomposition Method ◽  
Vries Equation ◽  
Coupled System ◽  
Approximate Solutions ◽  
Coupled Systems ◽  
Double Laplace Transform ◽  
Proposed Model ◽  
De Vries ◽  
Singular Kernels ◽  
The Waves

<abstract><p>In this article, the modified coupled Korteweg-de Vries equation with Caputo and Caputo-Fabrizio time-fractional derivatives are considered. The system is studied by applying the modified double Laplace transform decomposition method which is a very effective tool for solving nonlinear coupled systems. The proposed method is a composition of the double Laplace and decomposition method. The results of the problems are obtained in the form of a series solution for $ 0 &lt; \alpha\leq 1 $, which is approaching to the exact solutions when $ \alpha = 1 $. The precision and effectiveness of the considered method on the proposed model are confirmed by illustrated with examples. It is observed that the proposed model describes the nonlinear evolution of the waves suffered by the weak dispersion effects. It is also observed that the coupled system forms the wave solution which reveals the evolution of the shock waves because of the steeping effect to temporal evolutions. The error analysis is performed, which is comparatively very small between the exact and approximate solutions, which signifies the importance of the proposed method.</p></abstract>

Nonlinear singular one dimensional thermo-elasticity coupled system and double Laplace decomposition methods

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6269-6280
Author(s):  
Hassan Gadain
Keyword(s):  
Laplace Transform ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Coupled System ◽  
Decomposition Methods ◽  
One Dimensional ◽  
Convergence Proof ◽  
Double Laplace Transform ◽  
Adomian Decomposition

In this work, combined double Laplace transform and Adomian decomposition method is presented to solve nonlinear singular one dimensional thermo-elasticity coupled system. Moreover, the convergence proof of the double Laplace transform decomposition method applied to our problem. By using one example, our proposed method is illustrated and the obtained results are confirmed.

Asymptotic stepwise solutions of the Korteweg--de Vries equation with a singular perturbation and their accuracy

2021 ◽  
Vol 8 (3) ◽  
pp. 410-421
Author(s):  
S. I. Lyashko ◽  
◽  
V. H. Samoilenko ◽  
Yu. I. Samoilenko ◽  
I. V. Gapyak ◽  
...  
Keyword(s):  
Singular Perturbation ◽  
Small Parameter ◽  
Asymptotic Approximation ◽  
Vries Equation ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Main Term ◽  
Asymptotic Approximations ◽  
De Vries ◽  
The Given

The paper deals with the Korteweg-de Vries equation with variable coefficients and a small parameter at the highest derivative. The asymptotic step-like solution to the equation is obtained by the non-linear WKB technique. An algorithm of constructing the higher terms of the asymptotic step-like solutions is presented. The theorem on the accuracy of the higher asymptotic approximations is proven. The proposed technique is demonstrated by example of the equation with given variable coefficients. The main term and the first asymptotic approximation of the given example are found, their analysis is done and statement of the approximate solutions accuracy is presented.

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.

Coupled system of Korteweg-de Vries equation-type in domains with oscillatory moving boundaries

PAMM ◽  
2007 ◽  
Vol 7 (1) ◽  
pp. 2040043-2040044
Author(s):  
Octavio Vera ◽  
Mauricio Sepúlveda ◽  
Vanilde Bisognin
Keyword(s):  
Vries Equation ◽  
Coupled System ◽  
Moving Boundaries ◽  
De Vries

scholarly journals Analytical Solutions for the Mathematical Model Describing the Formation of Liver Zones via Adomian’s Method

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Abdelhalim Ebaid
Keyword(s):  
Differential Equations ◽  
Decomposition Method ◽  
Applied Mathematics ◽  
Mathematical Formulation ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Stationary Solutions ◽  
Proposed Model ◽  
Adomian Decomposition ◽  
The Mathematical Model

The formation of liver zones is modeled by a system of two integropartial differential equations. In this research, we introduce the mathematical formulation of these integro-partial differential equations obtained by Bass et al. in 1987. For better understanding of this mathematical formulation, we present a medical introduction for the liver in order to make the formulation as clear as possible. In applied mathematics, the Adomian decomposition method is an effective procedure to obtain analytic and approximate solutions for different types of operator equations. This Adomian decomposition method is used in this work to solve the proposed model analytically. The stationary solutions (as time tends to infinity) are also obtained through it, which are in full agreement with those obtained by Bass et al. in 1987.

scholarly journals Haar wavelet method for solving coupled system of fractional order partial differential equations

2021 ◽  
Vol 21 (3) ◽  
pp. 1444
Author(s):  
Abbas AL-Shimmary ◽  
Sajeda Kareem Radhi ◽  
Amina Kassim Hussain
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Coupled System ◽  
Approximate Solutions ◽  
Haar Wavelet ◽  
Coupled Systems ◽  
Operational Matrices ◽  
Orthogonal Matrices ◽  
Partial Differential

<p><span><span><br /></span></span></p><p><span id="docs-internal-guid-231c3b14-7fff-77a8-4252-ec4aa31140d7"><span>This paper deal with the numerical method, based on the operational matrices of the Haar wavelet orthonormal functions approach to approximate solutions to a class of coupled systems of time-fractional order partial differential equations (FPDEs.). By introducing the fractional derivative of the Caputo sense, to avoid the tedious calculations and to promote the study of wavelets to beginners, we use the integration property of this method with the aid of the aforesaid orthogonal matrices which convert the coupled system under some consideration into an easily algebraic system of Lyapunov or Sylvester equation type. The advantage of the present method, including the simple computation, computer-oriented, which requires less space to store, time-efficient, and it can be applied for solving integer (fractional) order partial differential equations. Some specific and illustrating examples have been given; figures are used to show the efficiency, as well as the accuracy of the, achieved approximated results. All numerical calculations in this paper have been carried out with MATLAB.</span></span></p>

scholarly journals NUMERICAL SIMULATION OF IRREGULAR BIMODAL WAVE SYSTEM DYNAMICS WITHIN THE FRAMEWORK OF KORTEWEG-DE VRIES EQUATION

2019 ◽  
Vol 47 (1) ◽  
pp. 38-40
Author(s):  
E.G. Didenkulova ◽  
A.V. Slunyaev ◽  
E.N. Pelinovsky
Keyword(s):  
Numerical Simulation ◽  
Shallow Water ◽  
Vries Equation ◽  
Kdv Equation ◽  
Wave Spectrum ◽  
Wave System ◽  
Quasi Equilibrium ◽  
Long Wave ◽  
De Vries ◽  
The Waves

The dynamics of wave ensembles in shallow water is studied within the framework of the nonlinear dispersive Korteweg – de Vries (KdV) equation by numerical simulation. Bimodal wave systems whose energy is distributed over two spectral domains are considered: the “additional” lobe which corresponds to the system of longer or shorter waves is added to the “main” spectral peak. The concerned problem describes, for example, the interaction between wind waves and swell in shallow water. The case of the unimodal waves (considered in (Pelinovsky, Sergeeva, 2006) is used as the reference. The limitations of the implied assumptions and the relationship of the idealized model to the realistic conditions in the ocean were discussed in the recent paper (Wang et al, 2018). Based on the detailed consideration of the 6 simulated cases, the following general conclusions may be formulated. The transition from the initial state to the quasi-equilibrium one is accompanied by strong variations of the wave characteristics, when the waves exhibit the most extreme features. In particular, the wave kurtosis grows suddenly and the abnormal heavy tails in the wave amplitude probability distributions appear. These processes are observed in all the cases of the bimodal spectra and are quite similar to the single-mode regime. The coexistence of a long-wave system smoothens the rapid oscillations of the wave extremes and kurtosis which take place during the transition stage. The presence of a short-wave system makes the waves on average more symmetric. Skewness attains the minimum value compared to the other cases. The co-existence of shorter waves practically does not change the wave kurtosis or the probability of the wave heights. In contrast, the presence of a long-wave system makes the waves more asymmetric and more extreme. The probability of large waves increases in the bimodal systems with a low-frequency component. The initial wave spectrum expands as a result of the wave interaction and tends to a quasistationary state. One may anticipate that the formulated conclusions are applicable beyond the limits of the Korteweg-de Vries equation to other kindred frameworks and corresponding phenomena. This work was supported by the Russian Science Foundation (project No. 18-77-00063).

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