scholarly journals Reproducing kernel method for solving singularly perturbed differential-difference equations with boundary layer behavior in Hilbert space

2018 ◽  
Vol 328 ◽  
pp. 30-43 ◽  
Author(s):  
Hussein Sahihi ◽  
Saeid Abbasbandy ◽  
Tofigh Allahviranloo
Keyword(s):  
Boundary Layer ◽  
Hilbert Space ◽  
Difference Equations ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Singularly Perturbed ◽  
Reproducing Kernel Method ◽  
Layer Behavior

scholarly journals IMPLEMENTING REPRODUCING KERNEL METHOD TO SOLVE SINGULARLY PERTURBED CONVECTION-DIFFUSION PARABOLIC PROBLEMS

2021 ◽  
Vol 26 (1) ◽  
pp. 116-134
Author(s):  
Saeid Abbasbandy ◽  
Hussein Sahihi ◽  
Tofigh Allahviranloo
Keyword(s):  
Reproducing Kernel ◽  
Kernel Method ◽  
The Other ◽  
Singularly Perturbed ◽  
Parabolic Problems ◽  
Regular Boundary ◽  
Reproducing Kernel Method ◽  
Convection Diffusion ◽  
Linear And Nonlinear ◽  
Layer Behavior

In the present paper, reproducing kernel method (RKM) is introduced, which is employed to solve singularly perturbed convection-diffusion parabolic problems (SPCDPPs). It is noteworthy to mention that regarding very serve singularities, there are regular boundary layers in SPCDPPs. On the other hand, getting a reliable approximate solution could be difficult due to the layer behavior of SPCDPPs. The strategy developed in our method is dividing the problem region into two regions, so that one of them would contain a boundary layer behavior. For more illustrations of the method, certain linear and nonlinear SPCDPP are solved.

scholarly journals Solving Singularly Perturbed Multipantograph Delay Equations Based on the Reproducing Kernel Method

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
F. Z. Geng ◽  
S. P. Qian
Keyword(s):  
Boundary Layer ◽  
Present Method ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Approximate Solutions ◽  
Delay Equations ◽  
Singularly Perturbed ◽  
Regular Domain ◽  
Reproducing Kernel Method ◽  
Boundary Layer Problem

A numerical method is presented for solving the singularly perturbed multipantograph delay equations with a boundary layer at one end point. The original problem is reduced to boundary layer and regular domain problems. The regular domain problem is solved by combining the asymptotic expansion and the reproducing kernel method (RKM). The boundary layer problem is treated by the method of scaling and the RKM. Two numerical examples are provided to illustrate the effectiveness of the present method. The results from the numerical example show that the present method can provide very accurate analytical approximate solutions.

scholarly journals A Novel Method for Solving KdV Equation Based on Reproducing Kernel Hilbert Space Method

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Mustafa Inc ◽  
Ali Akgül ◽  
Adem Kiliçman
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Kdv Equation ◽  
Reproducing Kernel Hilbert Space ◽  
Reproducing Kernel Method ◽  
Numerical Examples ◽  
The Kdv Equation ◽  
Novel Method ◽  
Reproducing Kernel Theory

We propose a reproducing kernel method for solving the KdV equation with initial condition based on the reproducing kernel theory. The exact solution is represented in the form of series in the reproducing kernel Hilbert space. Some numerical examples have also been studied to demonstrate the accuracy of the present method. Results of numerical examples show that the presented method is effective.

scholarly journals Reproducing Kernel Method for Solving Nonlinear Differential-Difference Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Reza Mokhtari ◽  
Fereshteh Toutian Isfahani ◽  
Maryam Mohammadi
Keyword(s):  
Difference Equations ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Reproducing Kernel Method

scholarly journals The Solution of a Class of Singularly Perturbed Two-Point Boundary Value Problems by the Iterative Reproducing Kernel Method

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Zhiyuan Li ◽  
YuLan Wang ◽  
Fugui Tan ◽  
Xiaohui Wan ◽  
Tingfang Nie
Keyword(s):  
Singular Perturbation ◽  
Boundary Layers ◽  
Present Method ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Singularly Perturbed ◽  
Point Boundary ◽  
Reproducing Kernel Method ◽  
Numerical Examples ◽  
Perturbation Problems

In (Wang et al., 2011), we give an iterative reproducing kernel method (IRKM). The main contribution of this paper is to use an IRKM (Wang et al., 2011), in singular perturbation problems with boundary layers. Two numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate that the method is simple and effective.

scholarly journals A REPRODUCING KERNEL METHOD FOR SOLVING A CLASS OF NONLINEAR SYSTEMS OF PDES

2014 ◽  
Vol 19 (2) ◽  
pp. 180-198 ◽  
Author(s):  
Maryam Mohammadi ◽  
Reza Mokhtari
Keyword(s):  
Hilbert Space ◽  
Nonlinear Systems ◽  
System Performance ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Reproducing Kernel Hilbert Space ◽  
Coupled System ◽  
Reproducing Kernel Method ◽  
Nonlinear Hyperbolic System ◽  
Systems Of Pdes

This paper is concerned with a technique for solving a class of nonlinear systems of partial differential equations (PDEs) in the reproducing kernel Hilbert space. The analytical solution is represented in the form of series. An iterative method is given to obtain the approximate solution. The convergence analysis is established theoretically. The proposed method is successfully used for solving a coupled system of viscous Burgers’ equations and a nonlinear hyperbolic system. Performance of the method is tested in terms of various error norms. In the case of non-availability of exact solution, it is compared with the existing methods.

scholarly journals Solving Singularly Perturbed Differential-Difference Equations with Dual Layer using Fourth Order Numerical Method

2019 ◽  
Vol 9 (2) ◽  
pp. 3770-3773
Keyword(s):  
Boundary Layer ◽  
Finite Difference ◽  
Numerical Method ◽  
Fourth Order ◽  
Difference Equations ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Layer Behavior ◽  
Maximum Absolute Errors

In this paper, we presented a fourth-order numerical method to solve SPDDE with the dual-layer. The answer to the problem shows dual-layer behavior. A fourth-order finite difference plan on a uniform mesh is developed. The result of the delay and also advance parameters on the boundary layer(s) has likewise been evaluated as well as represented in charts. The applicability of the planned plan is actually confirmed through executing it on model examples. To show the accuracy of the method, the results are presented in terms of maximum absolute errors.

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