A reproducing kernel method for solving singularly perturbed delay parabolic partial differential equations

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 formu...

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Published inMathematical modelling and analysis Vol. 28; no. 3; pp. 469 - 486
Main Authors Xie, Ruifeng, Zhang, Jian, Niu, Jing, Li, Wen, Yao, Guangming
Format Journal Article
LanguageEnglish
Published Vilnius Vilnius Gediminas Technical University 04.09.2023
Subjects
Analysis
Approximation
Boundary value problems
collocation method
Collocation methods
Convergence
delay parabolic equation
Differential equations
Exact solutions
Kernels
Mathematical analysis
Mathematical functions
Methods
numerical solution
Parabolic differential equations
Partial differential equations
reproducing kernel method
Singular perturbation
collocation method
numerical solution
delay parabolic equation
reproducing kernel method
Online AccessGet full text
ISSN1392-6292
1648-3510
DOI10.3846/mma.2023.16852

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Abstract 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.
AbstractList 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 [??](s, t) 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 fnal 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. Keywords: delay parabolic equation, reproducing kernel method, collocation method, numerical solution. AMS Subject Classification: 35K20; 46E23; 65L60.
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.
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.
Audience Academic
Author Xie, Ruifeng
Zhang, Jian
Li, Wen
Yao, Guangming
Niu, Jing
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SubjectTerms Analysis
Approximation
Boundary value problems
collocation method
Collocation methods
Convergence
delay parabolic equation
Differential equations
Exact solutions
Kernels
Mathematical analysis
Mathematical functions
Methods
numerical solution
Parabolic differential equations
Partial differential equations
reproducing kernel method
Singular perturbation
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Title A reproducing kernel method for solving singularly perturbed delay parabolic partial differential equations
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