USE OF REPRODUCING KERNEL HILBERT SPACE FUNCTIONS TO SOLVE BOUNDARY VALUE PROBLEMS

Abstract

The aim of the thesis is to present iterative reproducing kernel methods, based on reproducing kernel space, for the solution of linear and nonlinear second order to eighth order boundary value problems. The main advantage of reproducing kernel methods is that many boundary values problems which are not simple to solve, can be solved easily in the reproducing kernel spaces. The boundary value problems are solved by combining the properties of reproducing kernel spaces with the computational techniques. The reproducing kernel method is applied directly for the solution of linear BVPs. For the solution of nonlinear boundary value problems, homotopy perturbation - reproducing kernel method, searching least valuereproducing kernel method and two more iterative reproducing kernel methods are proposed. The exact solution is represented in the form of series and the approximate solution converges uniformly to the exact solution. The proposed methods have an advantage that it is possible to pick any point in the interval of integration and the approximate solution will be applicable. The methods proposed reduce the computational cost in solving problems and improve the accuracy of computation by preventing accumulating error of calculation. The performance of the methods proposed is shown to be very encouraging by the numerical results when compared to the results obtained from other existing methods.