A reproducing kernel method for solving heat conduction equations with delay

Jing, Ning; Sun, Liang; Xu, Minqiang; Hou, Jinjiao

doi:10.1016/j.aml.2019.106036

Cited by 22 publications

(5 citation statements)

References 12 publications

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“…Xu and Lin [38] applied the RKM for solving the delay fractional differential equations. The RKM proposed by Geng and Cui [11] can be used to solve presented the RKM to solve the nonlocal fractional boundary value problems, in addition to the partial integro-differential equation, multi-point boundary value problems and so on, see [8,10,13,16,21,23,24,25,28,31,33,34,35,36,37,39,41] for more details. The aim of this article is to seek the approximate solutions of Equation (1.1) by the RKM and collocation method.…”

Section: Introductionmentioning

confidence: 99%

A Reproducing Kernel Method for Solving Singularly Perturbed Delay Parabolic Partial Differential Equations

Xie,

Zhang,

Niu

et al. 2023

Mathematical Modelling and Analysis

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In this article, we put forward an efficient method on the foundation of a few reproducing kernel spaces(RK-spaces) and the collocation method to seek the solution of delay parabolic partial differential equations(PDEs) with singular perturbation. The approximated solution to the equations is formulated and proved the exact solution is uniformly convergent by the solution. Furthermore, the partial differentiation of the approximated solution is also proved the partial derivatives of the exact solution is uniformly convergent by the solution. Meanwhile, we show that the accuracy of our method is in the order of T/n where T is the final time and n is the number of spatial (and time) discretization in the domain of interests. Three numerical examples are put forward to demonstrate the effectiveness of our presented scheme.

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Section: Introductionmentioning

confidence: 99%

A Reproducing Kernel Method for Solving Singularly Perturbed Delay Parabolic Partial Differential Equations

Xie,

Zhang,

Niu

et al. 2023

Mathematical Modelling and Analysis

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“…Many researchers use the RKHS method to nd approximate solutions to various problems [28][29][30], and also some new applications of reproducing kernel methods and neural networks in machine learning are found in [31][32][33]. Very recently, RKHS is applied on fractional di erential equations [34][35][36]. In this paper, the reproducing kernel method is presented for the fractional differential equations with periodic conditions in the Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

Applying the Reproducing Kernel Method to Fractional Differential Equations with Periodic Conditions in Hilbert Space

Saky

Abbasbandy

Shivanian

2022

Journal of Mathematics

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In this article, the reproducing kernel method is presented for the fractional differential equations with periodic conditions in the Hilbert space. This method gives an approximate solution to the problem. The approximate and exact solutions are displayed in the form of series in the reproduction kernel space. In addition, we provide an error analysis for this technique. The presented method is tested by some examples to show its precision.

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“…Here we extend QNM to solve nonlinear systems (1). Many improved versions of RKM have been developed to solve linear second-order BVPs [14][15][16][17][18][19][20][21]. However, the high-order RKM has been seldom discussed.…”

Section: Introductionmentioning

confidence: 99%

A High-Order Numerical Method for a Nonlinear System of Second-Order Boundary Value Problems

Jing

Guo

2020

Mathematical Problems in Engineering

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This paper is concerned with a high-order numerical scheme for nonlinear systems of second-order boundary value problems (BVPs). First, by utilizing quasi-Newton’s method (QNM), the nonlinear system can be transformed into linear ones. Based on the standard Lobatto orthogonal polynomials, we introduce a high-order Lobatto reproducing kernel method (LRKM) to solve these linear equations. Numerical experiments are performed to investigate the reliability and efficiency of the presented method.

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A reproducing kernel method for solving heat conduction equations with delay

Cited by 22 publications

References 12 publications

A Reproducing Kernel Method for Solving Singularly Perturbed Delay Parabolic Partial Differential Equations

A Reproducing Kernel Method for Solving Singularly Perturbed Delay Parabolic Partial Differential Equations

Applying the Reproducing Kernel Method to Fractional Differential Equations with Periodic Conditions in Hilbert Space

A High-Order Numerical Method for a Nonlinear System of Second-Order Boundary Value Problems

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