Computational method for finding various solutions for a Quasilinear Elliptic Equation of Kirchhoff type

This paper is concerned with the various solutions to the class of nonlocal boundary value problem with the help of finite element method for the class of Kirchhoff-type problem-M∫Ω|∇u|2dxΔu=f(x,u)inΩ,u=0on∂Ωwhere Ω⊂RNΩ⊂RN is a bounded smooth domain, M:R+→RM:R+→R is continuous and f:Ω×R→Rf:Ω×R→R has subcritical growth. Kirchhoff equation does not appear to have been previously studied in detail computationally and it is hope that this paper will help to provide a new idea in this direction. Here we solved the nonlinear function f(x,u)f(x,u) with the help of Newton Raphson method and quasi-linear term by using Scaling Iterative Algorithm, forward difference and with the help of these schemes; we used finite element method to solve the Quasilinear Elliptic Equation of Kirchhoff Type and discussed the behaviour. In order to numerically confirm our theoretical results and to demonstrate the performance of the algorithm, we develop a MATLAB program for solving Kirchhoff-type problem.