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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.

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