scholarly journals New exact solutions of fractional Hirota-Satsuma coupled Korteweg-de Vries equations

2015 ◽  
Vol 19 (4) ◽  
pp. 1173-1176 ◽  
Author(s):  
Lian-Xiang Cui ◽  
Li-Mei Yan ◽  
Yan-Qin Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Function Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Fractional Complex Transform ◽  
New Exact Solutions ◽  
Non Linear ◽  
De Vries

An improved extended tg-function method, which combines the fractional complex transform and the extended tanh-function method, is applied to find exact solutions of non-linear fractional partial differential equations. Generalized Hirota-Satsuma coupled Korteweg-de Vries equations are used as an example to elucidate the effectiveness and simplicity of the method.

scholarly journals Exp-Function Method for Solving Fractional Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Bin Zheng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Function Method ◽  
Space Time ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
New Exact Solutions ◽  
Complex Transformation ◽  
Liouville Derivative

We extend the Exp-function method to fractional partial differential equations in the sense of modified Riemann-Liouville derivative based on nonlinear fractional complex transformation. For illustrating the validity of this method, we apply it to the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation. As a result, some new exact solutions for them are successfully established.

scholarly journals Application of the extended exp(-φ(ξ))-expansion method to the nonlinear conformable time-fractional partial differential equations

2019 ◽  
Vol 7 (2) ◽  
pp. 81
Author(s):  
Dipankar Kumar ◽  
Samir Chandra Ray
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Expansion Method ◽  
Point Of View ◽  
Hyperbolic Function ◽  
Fractional Partial Differential Equations ◽  
Kawahara Equation ◽  
Partial Differential ◽  
New Exact Solutions

This paper investigates the new exact solutions of the three nonlinear time fractional partial differential equations namely the nonlinear time fractional Clannish Random Walker’s Parabolic (CRWP) equation, the nonlinear time fractional modified Kawahara equation, and the nonlinear time fractional BBM-Burger equation by utilizing an extended form of exp(-φ(ξ))-expansion method in the sense of conformable fractional derivative. As outcomes, some new exact solutions are obtained and signified by hyperbolic function solutions, trigonometric function solutions, and rational function solutions. Some solutions have been plotted by MATLAB software to show the physical significance of our studied equations. In the point of view of our executed method and generated results, we may conclude that extended exp (-φ(ξ))-expansion method is more efficient than exp(-φ(ξ))-expansion method to extract the new exact solutions for solving any types of integer and fractional differential equations arising in mathematical physics.   

scholarly journals Exact solutions of non-linear fractional partial differential equations by fractional sub-equation method

2015 ◽  
Vol 19 (4) ◽  
pp. 1239-1244 ◽  
Author(s):  
Hong-Cai Ma ◽  
Dan-Dan Yao ◽  
Xiao-Fang Peng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Space Time ◽  
Equation Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Non Linear

This paper studies the space-time fractional Whitham-Broer-Kaup equations by the existed fractional sub-equation method, and exact solutions are obtained.

Analytical new soliton wave solutions of the nonlinear conformable time-fractional coupled Whitham–Broer–Kaup equations

2021 ◽  
pp. 2150492
Author(s):  
Delmar Sherriffe ◽  
Diptiranjan Behera ◽  
P. Nagarani
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Function Method ◽  
Visual Representations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Varying Parameters ◽  
Wave Solutions ◽  
Soliton Wave Solutions

The study of nonlinear physical and abstract systems is greatly important in order to determine the behavior of the solutions for Fractional Partial Differential Equations (FPDEs). In this paper, we study the analytical wave solutions of the time-fractional coupled Whitham–Broer–Kaup (WBK) equations under the meaning of conformal fractional derivative. These solutions are derived using the modified extended tanh-function method. Accordingly, different new forms of the solutions are obtained. In order to understand its behavior under varying parameters, we give the visual representations of all the solutions. Finally, the graphs are discussed and a conclusion is given.

The Invariant Subspace Method for Solving Fractional Partial Differential Equations

2020 ◽  
pp. 267-282
Author(s):  
Mohamed Soror Abdel Latif ◽  
Abass Hassan Abdel Kader
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Invariant Subspace ◽  
Three Dimensional ◽  
Variable Coefficients ◽  
Subspace Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Balance Principle ◽  
New Exact Solutions

In this chapter, the authors discuss the effectiveness of the invariant subspace method (ISM) for solving fractional partial differential equations. For this purpose, they have chosen a nonlinear time fractional partial differential equation (PDE) with variable coefficients to be investigated through this method. One-, two-, and three-dimensional invariant subspace classifications have been performed for this equation. Some new exact solutions have been obtained using the ISM. Also, the authors give a comparison between this method and the homogeneous balance principle (HBP).

scholarly journals Invariant subspace method: A tool for solving fractional partial differential equations

2017 ◽  
Vol 20 (2) ◽  
Author(s):  
Sangita Choudhary ◽  
Varsha Daftardar-Gejji
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Invariant Subspace ◽  
Caputo Derivative ◽  
Linear Time ◽  
Subspace Method ◽  
Fractional Partial Differential Equations ◽  
Time And Space ◽  
Partial Differential ◽  
Non Linear

AbstractIn this paper invariant subspace method has been employed for solving linear and non-linear time and space fractional partial differential equations involving Caputo derivative. A variety of illustrative examples are solved to demonstrate the effectiveness and applicability of the method.

Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm

2018 ◽  
Vol 28 (4) ◽  
pp. 828-856 ◽  
Author(s):  
Omar Abu Arqub
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Fluid Flows ◽  
Kernel Functions ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Content Type

Purpose The purpose of this study is to introduce the reproducing kernel algorithm for treating classes of time-fractional partial differential equations subject to Robin boundary conditions with parameters derivative arising in fluid flows, fluid dynamics, groundwater hydrology, conservation of energy, heat conduction and electric circuit. Design/methodology/approach The method provides appropriate representation of the solutions in convergent series formula with accurately computable components. This representation is given in the W(Ω) and H(Ω) inner product spaces, while the computation of the required grid points relies on the R(y,s) (x, t) and r(y,s) (x, t) reproducing kernel functions. Findings Numerical simulation with different order derivatives degree is done including linear and nonlinear terms that are acquired by interrupting the n-term of the exact solutions. Computational results showed that the proposed algorithm is competitive in terms of the quality of the solutions found and is very valid for solving such time-fractional models. Research limitations/implications Future work includes the application of the reproducing kernel algorithm to highly nonlinear time-fractional partial differential equations such as those arising in single and multiphase flows. The results will be published in forthcoming papers. Practical implications The study included a description of fundamental reproducing kernel algorithm and the concepts of convergence, and error behavior for the reproducing kernel algorithm solvers. Results obtained by the proposed algorithm are found to outperform in terms of accuracy, generality and applicability. Social implications Developing analytical and numerical methods for the solutions of time-fractional partial differential equations is a very important task owing to their practical interest. Originality/value This study, for the first time, presents reproducing kernel algorithm for obtaining the numerical solutions of some certain classes of Robin time-fractional partial differential equations. An efficient construction is provided to obtain the numerical solutions for the equations, along with an existence proof of the exact solutions based upon the reproducing kernel theory.

Sign in / Sign up

Export Citation Format

Share Document