Numerical solutions to differential equations using the reproducing kernel Hilbert space method

In this PhD thesis, which consists of six chapters, approximate solutions of nonlinear differential equations containing ordinary and partial derivatives were obtained using the Reproducing Kernel Hilbert Space Method. Firstly, some basic definitions and theorems about differential equations and functional analysis are given in order to better understand the main subject of the thesis and provide context for this research. The basics of Reproducing Kernel Hilbert Space Method is explained. The theoretical structures of the $V_2^m[a,b] $ and $W_2^{(m,n)}(\Omega)$ reproducing kernel spaces and reproducing kernels are presented in chapter three. In the fourth chapter, solutions to ordinary differential equations using RKHSM were investigated, and approximate solutions of inhomogeneous initial and boundary value problems in different intervals were obtained. By using RKHSM, the solutions of partial differential equations were examined in the fifth chapter and approximate solutions of Burgers, Fisher, Burgers-Fisher, Burgers-Huxley equations are obtained. For all the problems in the study, the reprorducing kernel Hilbert spaces were constructed for each problem, the kernel functions of these spaces were calculated, and convergence analyzes were performed. The numerical results are presented in the thesis. Tables and graphs of exact solutions, approximate solutions and absolute-relative errors are given. Finally, the validity and reliability of the Reproducing Kernel Hilbert Space Method is interpreted by evaluating the results obtained.