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    Application of the reproducing kernel method for solving linear Volterra integral equations with variable coefficients
    (Iop Publishing Ltd, 2024) Amoozad, Taher; Allahviranloo, Tofigh; Abbasbandy, Saeid; Malkhalifeh, Mohsen Rostamy
    This article proposes a new approach for solving linear Volterra integral equations with variable coefficients using the Reproducing Kernel Method (RKM). This method eliminates the need for the Gram-Schmidt process. However, the accuracy of RKM is influenced by various factors, including the selection of points, bases, space, and implementation method. The main objective of this article is to introduce a generalized method based on the Reproducing Kernel, which is successful in solving a special type of singular weakly nonlinear boundary value problems (BVPs). The easy implementation, elimination of the Gram-Schmidt process, fewer calculations, and high accuracy of the present method are interesting. The conformity of numerical results, including tables and figures, with theorems related to error analysis and convergence order, confirms the practicality of the present method.
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    Using a new implementation of reproducing kernel Hilbert space method to solve a system of second-order BVPs
    (Springernature, 2023) Amoozad, Taher; Allahviranloo, Tofigh; Abbasbandy, Saeid; Rostamy Malkhalifeh, Mohsen
    In this paper, a new implementation based on the reproducing kernel method (RKM) without the Gram-Schmidt orthogonalization for solving linear and nonlinear systems of second-order boundary value problems is presented. In the RKM method, components such as points, space, inner product, bases, and a suitable method have an effect on increasing the accuracy. The easy implementation, elimination of the Gram-Schmidt process, fewer calculations, and high accuracy of the present method are interesting. The compatibility of numerical results and theorems demonstrates that the Present method is effective.