Numerical solution of nonlinear Boundary value problems in kernel space

In this paper, we develop an approach based on kernel least mean square algorithm for the numerical solution of nonlinear Boundary value problems(BVP's). The approximate solution is defined as the output of the kernel least mean square algorithm. The algorithm is deduced as the linear combination of the kernel function and the error values. A fitness function consisting of the mean squared error of the nonlinear BVP along with the boundary condition is formulated. The optimization of the obtained fitness function is performed through gradient based algorithm. The reliability and efficacy of the proposed method is confirmed by solving Troesch's boundary value problem and one dimensional Bratu boundary value problems. The obtained numerical solution is found in great agreement with some renowned conventional methods as well as the exact solution.