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.
