Cubic Spline Approximation for Two-Dimensional Non-Linear Elliptic Boundary Value Problems

ABSTRACT

This paper illustrates the use of cubic spline approximations to solve two-dimensional nonlinear elliptic boundary value problems on a non-uniform mesh using a nine point reduced discretization of order two in y- and three in x-bearings. We discuss the method’s complete deduction process in depth and also discuss how our discretization can solve Poisson’s equation in polar coordinates. The method’s convergence has been established. A few physical examples and their numerical consequences are provided to demonstrate the suggested method’s ease. The second order elliptic equations are obtained as the solutions of the illustrated and wave equations in their consistent state form (ast⟶∞). The solutions to these equations are extremely significant in a variety of branches of research, for example, electromagnetics, astronomy, heat transfer, and fluid mechanics, because they can represent a temperature, an electric or attractive potential, or the displacement of an elastic membrane.

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