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    A new application of the reproducing kernel method for solving linear systems of fractional order Volterra integro-differential equations
    (IOP publishing Ltd, 2024) Amoozad, Taher; Abbasbandy, Saeid; Sahihi, Hussein; Allahviranloo, Tofigh
    In this article, a new implementation of the reproducing kernel method is presented for solving systems of fractional-order Volterra integro-differential equations. Unlike previous implementations, this method does not rely on the Gram-Schmidt process. The reproducing kernel method utilizes various components, including space, inner product, bases, and points. Furthermore, the system of fractional-order Volterra integro-differential equations involves Caputo's fractional derivative and Volterra integral. However, when using the reproducing kernel method to solve these systems, challenges such as longer execution time and lower accuracy may arise compared to other methods. The present method has overcome these challenges with features such as easy implementation, high accuracy, and lower execution time.
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    A new application of the reproducing kernel method for solving linear systems of fractional order Volterra integro-differential equations
    (Institute of Physics, 2024) Amoozad, Taher; Abbasbandy, Saeid; Sahihi, Hussein; Allahviranloo, Tofigh
    In this article, a new implementation of the reproducing kernel method is presented for solving systems of fractional-order Volterra integro-differential equations. Unlike previous implementations, this method does not rely on the Gram-Schmidt process. The reproducing kernel method utilizes various components, including space, inner product, bases, and points. Furthermore, the system of fractional-order Volterra integro-differential equations involves Caputo’s fractional derivative and Volterra integral. However, when using the reproducing kernel method to solve these systems, challenges such as longer execution time and lower accuracy may arise compared to other methods. The present method has overcome these challenges with features such as easy implementation, high accuracy, and lower execution time. © 2024 IOP Publishing Ltd.
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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.