Fractional (3+1)-dim Jimbo Miwa system: invariance properties, exact solutions, solitary pattern solutions and conservation laws

2020 ◽  
Vol 21 (7-8) ◽  
pp. 843-854
Author(s):  
Sachin Kumar ◽  
Baljinder Kour
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Explicit Solutions ◽  
Series Method ◽  
Power Series Method ◽  
Invariance Properties ◽  
Residual Power Series ◽  
Feasible Vector ◽  
Residual Power

AbstractThe present article is devoted to scouting invariant analysis and some kind of approximate and explicit solutions of the (3+1)-dimensional Jimbo Miwa system of nonlinear fractional partial differential equations (NLFPDEs). Feasible vector field of the system is obtained by employing the invariance attribute of one-parameter Lie group of transformation. The reduction of the number of independent variables by this method gives the reduction of Jimbo Miwa system of NLFPDES into a system of nonlinear fractional ordinary differential equations (NLFODEs). Explicit solutions in form of power series are scrutinized by using power series method (PSM). In addition, convergence is also examined. The residual power series method (RPSM) is employed for disquisition of solitary pattern (SP) solutions in form of approximate series. A comparative analysis of the obtained results of the considered problem is provided. The conserved vectors are scrutinized in the form of fractional Noether’s operator. Numerical solutions are represented graphically.

scholarly journals Applications of General Residual Power Series Method to Differential Equations with Variable Coefficients

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Bochao Chen ◽  
Li Qin ◽  
Fei Xu ◽  
Jian Zu
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Variable Coefficients ◽  
Series Solutions ◽  
Series Method ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Residual Power Series ◽  
Analytical Series ◽  
Residual Power

This paper is devoted to studying the analytical series solutions for the differential equations with variable coefficients. By a general residual power series method, we construct the approximate analytical series solutions for differential equations with variable coefficients, including nonhomogeneous parabolic equations, fractional heat equations in 2D, and fractional wave equations in 3D. These applications show that residual power series method is a simple, effective, and powerful method for seeking analytical series solutions of differential equations (especially for fractional differential equations) with variable coefficients.

scholarly journals Toward computational algorithm for time-fractional Fokker–Planck models

2019 ◽  
Vol 11 (10) ◽  
pp. 168781401988103 ◽  
Author(s):  
Asad Freihet ◽  
Shatha Hasan ◽  
Mohammad Alaroud ◽  
Mohammed Al-Smadi ◽  
Rokiah Rozita Ahmad ◽  
...  
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fractional Power ◽  
Series Method ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Wide Range ◽  
Residual Power Series ◽  
Residual Power ◽  
Fokker Planck

This article describes an efficient algorithm based on residual power series to approximate the solution of a class of partial differential equations of time-fractional Fokker–Planck model. The fractional derivative is assumed in the Caputo sense. The proposed algorithm gives the solution in a form of rapidly convergent fractional power series with easily computable coefficients. It does not require linearization, discretization, or small perturbation. To test simplicity, potentiality, and practical usefulness of the proposed algorithm, illustrative examples are provided. The approximate solutions of time-fractional Fokker–Planck equations are obtained by the residual power series method are compared with those obtained by other existing methods. The present results and graphics reveal the ability of residual power series method to deal with a wide range of partial fractional differential equations emerging in the modeling of physical phenomena of science and engineering.

scholarly journals Analytical Solution for the Time Fractional BBM-Burger Equation by Using Modified Residual Power Series Method

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Jianke Zhang ◽  
Zhirou Wei ◽  
Longquan Yong ◽  
Yuelei Xiao
Keyword(s):  
Power Series ◽  
Burger Equation ◽  
Numerical Solutions ◽  
Error Function ◽  
Series Method ◽  
Power Series Method ◽  
Transform Method ◽  
Residual Power Series Method ◽  
Residual Power Series ◽  
Residual Power

In this study, a generalized Taylor series formula together with residual error function, which is named the residual power series method (RPSM), is used for finding the series solution of the time fractional Benjamin-Bona-Mahony-Burger (BBM-Burger) equation. The BBM-Burger equation is useful in describing approximately the unidirectional propagation of long waves in certain nonlinear dispersive systems. The numerical solution of the BBM-Burger equation is calculated by Maple. The numerical results show that the RPSM is reliable and powerful in solving the numerical solutions of the BBM-Burger equation compared with the exact solutions as well as the solutions obtained by homotopy analysis transform method through different graphical representations and tables.

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