scholarly journals Exact Solutions for Some Fractional Partial Differential Equations by the Method

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Bin Zheng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Wave Equations ◽  
Dimensional Space ◽  
Space Time ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Liouville Derivative ◽  
Dimensional Space Time

We apply the method to seek exact solutions for several fractional partial differential equations including the space-time fractional (2 + 1)-dimensional dispersive long wave equations, the (2 + 1)-dimensional space-time fractional Nizhnik-Novikov-Veselov system, and the time fractional fifth-order Sawada-Kotera equation. The fractional derivative is defined in the sense of modified Riemann-liouville derivative. Based on a certain variable transformation, these fractional partial differential equations are transformed into ordinary differential equations of integer order. With the aid of mathematical software, a variety of exact solutions for them are obtained.

scholarly journals A New Fractional Subequation Method and Its Applications for Space-Time Fractional Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Fanwei Meng ◽  
Qinghua Feng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Periodic Wave ◽  
Space Time ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Kdv Equations ◽  
Wave Solutions ◽  
Liouville Derivative

A new fractional subequation method is proposed for finding exact solutions for fractional partial differential equations (FPDEs). The fractional derivative is defined in the sense of modified Riemann-Liouville derivative. As applications, abundant exact solutions including solitary wave solutions as well as periodic wave solutions for the space-time fractional generalized Hirota-Satsuma coupled KdV equations are obtained by using this 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 The Jacobi Elliptic Equation Method for Solving Fractional Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Bin Zheng ◽  
Qinghua Feng
Keyword(s):  
Partial Differential Equations ◽  
Elliptic Equation ◽  
Differential Equations ◽  
Exact Solutions ◽  
Equation Method ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Liouville Derivative

Based on a nonlinear fractional complex transformation, the Jacobi elliptic equation method is extended to seek exact solutions for fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. For demonstrating the validity of this method, we apply it to solve the space fractional coupled Konopelchenko-Dubrovsky (KD) equations and the space-time fractional Fokas equation. As a result, some exact solutions for them including the hyperbolic function solutions, trigonometric function solutions, rational function solutions, and Jacobi elliptic function solutions are successfully found.

scholarly journals Improved Fractional Subequation Method and Exact Solutions to Fractional Partial Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Jun Jiang ◽  
Yuqiang Feng ◽  
Shougui Li
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Burgers Equation ◽  
Space Time ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Biological Population ◽  
Regularized Long Wave Equation ◽  
Generalized Time

In this paper, the improved fractional subequation method is applied to establish the exact solutions for some nonlinear fractional partial differential equations. Solutions to the generalized time fractional biological population model, the generalized time fractional compound KdV-Burgers equation, the space-time fractional regularized long-wave equation, and the (3+1)-space-time fractional Zakharov-Kuznetsov equation are obtained, respectively.

scholarly journals A New Approach for Solving Fractional Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Fanwei Meng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dimensional Space ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
New Approach ◽  
New Exact Solutions ◽  
Biological Population ◽  
Complex Transformation ◽  
Dimensional Space Time

We propose a new approach for solving fractional partial differential equations based on a nonlinear fractional complex transformation and the general Riccati equation and apply it to solve the nonlinear time fractional biological population model and the (4+1)-dimensional space-time fractional Fokas equation. As a result, some new exact solutions for them are obtained. This approach can be suitable for solving fractional partial differential equations with more general forms than the method proposed by S. Zhang and H.-Q. Zhang (2011).

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.

scholarly journals A Table Lookup Method for Exact Analytical Solutions of Nonlinear Fractional Partial Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Ji Juan-Juan ◽  
Guo Ye-Cai ◽  
Zhang Lan-Fang ◽  
Zhang Chao-Long
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Space Time ◽  
Hyperbolic Function ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Table Lookup ◽  
Exact Analytical Solutions ◽  
Function Solution

A table lookup method for solving nonlinear fractional partial differential equations (fPDEs) is proposed in this paper. Looking up the corresponding tables, we can quickly obtain the exact analytical solutions of fPDEs by using this method. To illustrate the validity of the method, we apply it to construct the exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the (2+1)-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation. As a result, many new types of exact analytical solutions are obtained including triangular periodic solution, hyperbolic function solution, singular solution, multiple solitary wave solution, and Jacobi elliptic function solution.

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.

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