scholarly journals Approximate analytical solution for modified Korteweg-de Vries equation with local fractional derivative via new iterative method

2020 ◽  
Vol 24 (6 Part B) ◽  
pp. 4027-4032
Author(s):  
Shu-Xian Deng ◽  
Zhi-Jun Wang
Keyword(s):  
Analytical Solution ◽  
Iterative Method ◽  
Fractional Derivative ◽  
Vries Equation ◽  
Approximate Analytical Solution ◽  
Variable Coefficients ◽  
Local Fractional Derivative ◽  
De Vries ◽  
New Iterative Method

In this paper, we obtain the approximate analytical solution of variable coefficients modified Korteweg-de Vries equation with local fractional derivative by using new iterative method.

scholarly journals Fractal complex transform technology for fractal Kkorteweg-de Vries equation within a local fractional derivative

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 841-845
Author(s):  
Jinze Xu ◽  
Zeng-Shun Chen ◽  
Jian-Hong Wang ◽  
Ping Cui ◽  
Yunru Bai
Keyword(s):  
Fractional Derivative ◽  
Traveling Wave ◽  
Vries Equation ◽  
Special Functions ◽  
Traveling Wave Solutions ◽  
Cantor Sets ◽  
Local Fractional Derivative ◽  
Linear Partial Differential Equations ◽  
Wave Solutions ◽  
De Vries

In this paper, we present the fractal complex transform via a local fractional derivative. The traveling wave solutions for the fractal Korteweg-de Vries equations within local fractional derivative are obtained based on the special functions defined on Cantor sets. The technology is a powerful tool for solving the local fractional non-linear partial differential equations.

scholarly journals Homotopy Analysis Method for Conformable Burgers-Korteweg-de Vries Equation

2016 ◽  
Vol 17 ◽  
pp. 17-23 ◽  
Author(s):  
Ali Kurt ◽  
Orkun Tasbozan ◽  
Yücel Cenesiz
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Homotopy Analysis Method ◽  
Vries Equation ◽  
Approximate Analytical Solution ◽  
Analysis Method ◽  
Conformable Derivative ◽  
De Vries ◽  
Homotopy Analysis

The main goal of this paper is nding the approximate analytical solution of Burgers-Korteweg-de Vries with newly de ned conformable derivative by using homotopy analysis method (HAM). Then the approximate analytical solution is compared with the exact solution and comparative tables are given.

scholarly journals Nonlinear Vibration of a Nonlocal Nanobeam Resting on Fractional-Order Viscoelastic Pasternak Foundations

2018 ◽  
Vol 2 (3) ◽  
pp. 21 ◽  
Author(s):  
Guy Eyebe ◽  
Gambo Betchewe ◽  
Alidou Mohamadou ◽  
Timoleon Kofane
Keyword(s):  
Analytical Solution ◽  
Fractional Derivative ◽  
Nonlinear Vibration ◽  
Fractional Order ◽  
Nonlocal Elasticity ◽  
Multiple Scales ◽  
Approximate Analytical Solution ◽  
Nonlocal Elasticity Theory ◽  
Pasternak Foundation ◽  
D’Alembert Principle

In the present study, the nonlinear vibration of a nanobeam resting on the fractional order viscoelastic Winkler–Pasternak foundation is studied using nonlocal elasticity theory. The D’Alembert principle is used to derive the governing equation and the associated boundary conditions. The approximate analytical solution is obtained by applying the multiple scales method. A detailed parametric study is conducted, and the effects of the variation of different parameters belonging to the application problems on the system are calculated numerically and depicted. We remark that the order and the coefficient of the fractional derivative have a significant effect on the natural frequency and the amplitude of vibrations.

Global Asymptotic Step-Type Solutions to Singularly Perturbed Korteweg-De Vries Equation with Variable Coefficients

2020 ◽  
Vol 52 (9) ◽  
pp. 27-38
Author(s):  
Sergey I. Lyashko ◽  
Valeriy H. Samoilenko ◽  
Yulia I. Samoilenko ◽  
Igor V. Gapyak ◽  
Nataliya I. Lyashko ◽  
...  
Keyword(s):  
Vries Equation ◽  
Variable Coefficients ◽  
Singularly Perturbed ◽  
De Vries ◽  
Step Type

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.

Painlevé Test, Bäcklund Transformation and Consistent Riccati Expansion Solvability for two Generalised Cylindrical Korteweg-de Vries Equations with Variable Coefficients

2018 ◽  
Vol 73 (3) ◽  
pp. 207-213 ◽  
Author(s):  
Rehab M. El-Shiekh
Keyword(s):  
Vries Equation ◽  
Dusty Plasmas ◽  
Variable Coefficients ◽  
Bäcklund Transformations ◽  
Painlevé Test ◽  
New Exact Solutions ◽  
Backlund Transformations ◽  
Integrability Conditions ◽  
De Vries ◽  
Consistent Riccati Expansion

AbstractIn this paper, the integrability of the (2+1)-dimensional cylindrical modified Korteweg-de Vries equation and the (3+1)-dimensional cylindrical Korteweg-de Vries equation with variable coefficients arising in dusty plasmas in its generalised form was studied by two different techniques: the Painlevé test and the consistent Riccati expansion solvability. The integrability conditions and Bäcklund transformations are constructed. By using Bäcklund transformations and the solutions of the Riccati equation many new exact solutions are found for the two equations in this study. Finally, the application of the obtained solutions in dusty plasmas is investigated.

scholarly journals Fifth order semi analytical solution of exact Korteweg-de Vries equation

2018 ◽  
Vol 1116 ◽  
pp. 022002 ◽  
Author(s):  
Afriadi ◽  
Yulia Zahara ◽  
Vera Halfiani ◽  
Harish Abdillah Mardi ◽  
Marwan Ramli
Keyword(s):  
Analytical Solution ◽  
Vries Equation ◽  
De Vries ◽  
Fifth Order
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