A new non-polynomial spline method for solution of linear and non-linear third order dispersive equations

Talat Sultana,Arshad Khan,Pooja Khandelwal

doi:10.1186/s13662-018-1763-z

Talat Sultana, Arshad Khan + Show 1 more

Open Access

https://doi.org/10.1186/s13662-018-1763-z

Copy DOI

Abstract

In this paper, a new three-level implicit method is developed to solve linear and non-linear third order dispersive partial differential equations. The presented method is obtained by using exponential quartic spline to approximate the spatial derivative of third order and finite difference discretization to approximate the first order spatial and temporal derivative. The developed method is tested on four examples and the results are compared with other methods from the literature, which shows the applicability and feasibility of the presented method. Furthermore, the truncation error and stability analysis of the presented method are investigated, and graphical comparison between analytical and approximate solution is also shown for each example.

Highlights

[4] Boussinesq and Korteweg-de Vries (KdV) equations described the problem of water waves and the long waves in which dispersive effects are present
We use an exponential quartic spline function to develop a numerical method to approximate the solution of third order homogeneous and non-homogeneous linear dispersive equation in one space dimension with f (x, t) as a source term:
The purpose of this paper is to present a new method to solve third order linear and nonlinear dispersive partial differential equations based on spline function approximation

Summary

Introduction

In [4] Boussinesq and Korteweg-de Vries (KdV) equations described the problem of water waves and the long waves in which dispersive effects are present. Where Tjm = E(3)(xj, tm) is the third order spline derivative at (xj, tm) w.r.t. the space variable, fjm = f (xj, tm), ymj is the approximate solution of (1.1) at (xj, tm), δt is the central difference operator w.r.t. t and σ is a parameter such that finite difference approximation to the time derivative is of O(k) for arbitrary σ Operating x on both sides of (3.1) and after some simplifications, we obtain the following method: δt pymj+1 + qymj + qymj–1 + pymj–2. 3.2 Spline solution for non-linear dispersive equation In the similar manner, Eq(1.4) is discretized as follows: Operating x on both sides of (3.5) and after some simplifications, we obtain the following method:.

Methods

Conclusion