Solving second order non-linear elliptic partial differential equations using generalized finite difference method

We investigate methods for obtaining exact solutions of the (3 + 1)-dimensional nonlinear space-time fractional Jimbo-Miwa equation in the sense of the modified Riemann-Liouville derivative. The methods employed to analytically solve the equation are the ( / , 1/ )-expansion method and the novel ( / )-expansion method. To the best of our knowledge, there are no researchers who have applied these methods to obtain exact solutions of the equation. The application of the methods is simple, elegant, efficient, and trustworthy. In particular, applying the novel ( / )-expansion method to the equation, we obtain more exact solutions than using other existing methods such as the ( / )-expansion method and the exp(−Φ( ))-expansion method. The exact solutions of the equation, obtained using the two methods, can be categorized in terms of hyperbolic, trigonometric, and rational functions. Some of the results obtained by the two methods are new and reported here for the first time. In addition, the obtained exact explicit solutions of the equation characterize many physical meanings such as soliton solitary wave solutions, periodic wave solutions, and singular multiple-soliton solutions.

Mathematical Problems in Engineering

Section: The ( / 1/ )-Expansionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Two Reliable Methods for Solving the (3 + 1)‐Dimensional Space‐Time Fractional Jimbo‐Miwa Equation

Sirisubtawee

Koonprasert

Khaopant

et al. 2017

“…In a recent work, L. Gavete et al . apply the Newton Raphson method to GFDM to solve nonlinear partial derivative equations.…”

Section: Introductionmentioning

confidence: 99%

Application of generalised finite differences method to reflection and transmission problems in seismic SH waves propagation

Benito

Math Methods in App Sciences

et al. 2017

A matrix formulation of the generalised finite difference method is introduced. A necessary and sufficient condition for the uniqueness of the solution is demonstrated, and important practical consequences are obtained. A generalised finite differences scheme for SH wave is obtained, the stability of the scheme is analysed and the formula for the velocity of the wave due to the scheme is obtained in order to deal with the numerical dispersion. The method is applied to seismic waves propagation problems, specifically to the problem of reflection and transmission of plane waves in heterogeneous media. A heterogeneous approach without nodes at the interface is chosen to solve the problem in heterogeneous media. Copyright © 2017 John Wiley & Sons, Ltd.

“…In the recent years, the GFDM has been applied with different purposes, to solve the wave equation, to solve the advecion‐diffusion equation, to solve nonlinear partial differential equations, to solve the electrical conductivity of a tissue, to solve numerical modeling of casting solidification, to solve 2‐dimensional nonlinear obstacle problems, to solve the propagation of nonlinear water waves in numerical wave flume, to solve the shock‐induced 2‐dimensional coupled non‐Fickian diffusion‐elasticity, to simulate the 2‐dimensional sloshing phenomenon, to solve 2‐dimensional inverse Cauchy problems, to solve shallow water equations in 2 dimensions, to solve inverse Cauchy problems associated with 3‐dimensional Helmholtz‐type equations, or to solve inverse biharmonic boundary‐value problems …”

Section: Introductionmentioning

confidence: 99%

Adaptive strategies to improve the application of the generalized finite differences method in 2D and 3D

Benito

Math Methods in App Sciences

et al. 2017

The generalized finite differences method allows the use of irregular clouds of nodes. The optimal values of the key parameters of the method vary depending on how the nodes in the cloud are distributed, and this can be complicated especially in 3D. Therefore, we establish 2 criteria to allow the automation of the selection process of the key parameters. These criteria depend on 2 discrete functions, one of them penalizes distances and the other one penalizes imbalances. In addition, we show how to generate irregular clouds of nodes more efficient than finer regular clouds of nodes. We propose an improved and more versatile h‐adaptive method that allows adding, moving, and removing nodes. To decide which nodes to act on, we use an indicator of the error a posteriori. This h‐adaptive method gives results more accurate than those ones presented for the generalized finite differences method so far and, in addition, with fewer nodes. In addition, this method can be used in time‐dependant problems to increase the temporal step or to avoid instabilities. As an example, we apply it in problems of seismic waves propagation.