Operator compact exponential approximation for the solution of the system of 2D second order quasilinear elliptic partial differential equations

R. K. Mohanty,Geetan Manchanda,Arshad Khan

doi:10.1186/s13662-019-1968-9

Abstract

In this paper, we suggest a new exponential implicit method based on full step discretization of order four for the solution of quasilinear elliptic partial differential equation of the form A ( x,y,z ) z_{xx} +C ( x,y,z ) z_{yy} =k ( x,y,z, z_{x}, z_{y} ), 0< x,y<1. In this method a single compact cell consisting of nine nodal points is used. Convergence analysis of the said method is discussed in detail. The developed method is successfully applied to solving problems in polar coordinates. The method for scalar equation is eventually applied to solving the system of quasilinear elliptic equations. To measure the rationality and precision, the method is applied to solving several noteworthy problems and numerical results are provided to exhibit the effectiveness of the method.

Highlights

1 Introduction We examine a quasilinear elliptic equation in two space dimensions of the form
We presuppose the following about the boundary value problem (1.1)–(1.2): Mohanty et al Advances in Difference Equations
5 Method for system of equations we extend our method to the system of quasilinear partial differential equations (PDEs) of the form

Summary

Introduction

The following year Ananthakrishnaiah, Saldanha [17] discussed a fourth order finite difference scheme for two-dimensional nonlinear elliptic partial differential equations. 2, we formulate the full step fourth order compact discretization scheme for the solution of a nonlinear elliptic equation with non-constant coefficients. Using the technique discussed in [19], we can derive the fourth order schemes for the system of quasilinear elliptic PDEs. Let us consider the elliptic equation of the form zxx + C(x)zyy = D(x)zx + G(x, y), 0 < x, y < 1,. 5. With the help of boundary values, writing all methods at every interior grid point, one obtains sparse systems of linear algebraic equations for the solution of the multiharmonic equations (7.1), (7.5).

Conclusions

Methods