Mehmet Giyas Sakar

In this study, a reproducing kernel method based on shifted Legendre polynomials is proposed for obtaining numerical solutions to the nonlinear Duffing equation involving both integral and non-integral forcing terms subject to three types of boundary conditions. Unlike the classical reproducing kernel method, the proposed method employs Legendre polynomials to construct the reproducing kernel function. Furthermore, instead of homogenising the boundary conditions, the method incorporates additional basis functions to handle the three types of boundary conditions considered: Bitsadze–Samarskii boundary conditions, separated boundary conditions, and integral boundary conditions. The construction of the applied method is established by theorems, and the iteration process of the approximate solution is presented in detail. To show the accuracy and effectiveness of the presented method, the comparison of the numerical results obtained for six different problems is illustrated through tables and graphs.

This article presents an efficient numerical method for obtaining numerical solutions to multi-point boundary problems with multi-fractional orders of Caputo derivatives.The novelty of the suggested method is based on the application of shifted Legendre polynomials in the reproducing kernel theory, as well as the use of extra basis functions for homogeneous or nonhomogeneous boundary conditions. The approximate solution for both linear and nonlinear cases of the considered problem is given in the form of a finite series sum. Convergence and error analysis of the presented method are given. The numerical outcomes obtained with the proposed method are compared with other results found in the literature. The obtained results for linear and nonlinear examples support the method’s efficiency.

This article aims to achieve robust numerical results by applying the Chebyshev reproducing kernel method without homogenizing the initial‐boundary conditions of the Emden–Fowler (E‐F) equation, thereby introducing a new perspective to the literature. A novel numerical approach is presented for solving the initial‐boundary value problem of third‐order E‐F equations using Chebyshev reproducing kernel theory. Unlike previous applications, which were confined to homogeneous initial‐boundary value problems or required homogenization, the proposed method is effective for both homogeneous and nonhomogeneous cases. To handle the initial‐boundary conditions of the E‐F equations, additional basis functions are introduced rather than imposing conditions on the reproducing kernel Hilbert space. The method's effectiveness is demonstrated through five examples, which validate the theoretical analysis. Overall, the results emphasize the method's efficiency.

In this study, an effective approach is presented to obtain a numerical solution of linear and nonlinear singular boundary value problems. The proposed method is constructed by combining reproducing kernel and Legendre polynomials. Legendre basis functions are used to get the kernel function, and then the approximate solution is obtained as a finite series sum. Comparison of numerical results is made with the results obtained by other methods available in the literature. Furthermore, efficiency and accuracy of the method are demonstrated in tabulated results and plotted graphs. The numerical outcomes demonstrate that our method is very effective, applicable, and convenient.

The purpose of this paper is to present a uniform finite difference method for numerical solution of a initial value problem for semilinear second order singularly perturbed delay differential equation. A numerical method is constructed for this problem which involves appropriate piecewise-uniform Shishkin mesh on each time subinterval. The method is shown to uniformly convergent with respect to the perturbation parameter. A numerical experiment illustrate in practice the result of convergence proved theoretically.