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 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 Solving a Class of Singularly Perturbed Partial Differential Equation by Using the Perturbation Method and Reproducing Kernel Method

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Yu-Lan Wang ◽  
Hao Yu ◽  
Fu-Gui Tan ◽  
Shanshan Qu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Perturbation Method ◽  
Present Method ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Singularly Perturbed ◽  
Partial Differential ◽  
Reproducing Kernel Method

We give the analytical solution and the series expansion solution of a class of singularly perturbed partial differential equation (SPPDE) by combining traditional perturbation method (PM) and reproducing kernel method (RKM). The numerical example is studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.

scholarly journals An Improved Approach for Solving Partial Differential Equation Based on Reproducing Kernel Method

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Mingjing Du
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Present Method ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Kernel Method ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Reproducing Kernel Method ◽  
First Time

The traditional reproducing kernel method (TRKM) cannot obtain satisfactory numerical results for solving the partial differential equation (PDE). In this study, for the first time, the abovementioned problems are solved by adaptive piecewise interpolation reproducing kernel method (APIRKM) to obtain the exact and approximate solutions of partial differential equations by means of series expansion using reconstructed kernel function. The highlight of this paper is to obtain more accurate approximate solution and save more time through adaptive discovery. Numerical solutions of the three examples show that the present method is more advantageous than TRKM.

scholarly journals Solving Delay Differential Equations by an Accurate Method with Interpolation

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Ali Akgül ◽  
Adem Kiliçman
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Approximation ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Approximate Solutions ◽  
Accurate Method ◽  
Reproducing Kernel Method ◽  
Comparison Of The Results ◽  
Delay Differential

We use the reproducing kernel method (RKM) with interpolation for finding approximate solutions of delay differential equations. Interpolation for delay differential equations has not been used by this method till now. The numerical approximation to the exact solution is computed. The comparison of the results with exact ones is made to confirm the validity and efficiency.

scholarly journals A modified reproducing kernel method for a time-fractional telegraph equation

2017 ◽  
Vol 21 (4) ◽  
pp. 1575-1580 ◽  
Author(s):  
Yulan Wang ◽  
Mingjing Du ◽  
Chaolu Temuer
Keyword(s):  
Numerical Solution ◽  
Present Method ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Telegraph Equation ◽  
Reproducing Kernel Method ◽  
Numerical Examples ◽  
Fractional Telegraph Equation

The aim of this work is to obtain a numerical solution of a time-fractional telegraph equation by a modified reproducing kernel method. Two numerical examples are given to show that the present method overcomes the drawback of the traditional reproducing kernel method and it is an easy and effective method.

scholarly journals Improved reproducing kernel method to solve space-time fractional advection-dispersion equation

2021 ◽  
Author(s):  
Tofigh Allahviranloo ◽  
Hussein Sahihi ◽  
Soheil Salahshour ◽  
D. Baleanu
Keyword(s):  
Dispersion Equation ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Space Time ◽  
Finite Domain ◽  
Reproducing Kernel Method ◽  
Advection Dispersion ◽  
Schmidt Orthogonalization

In this paper, we consider the Space-Time Fractional Advection-Dispersion equation on a finite domain with variable coefficients. Fractional Advection- Dispersion equation as a model for transporting heterogeneous subsurface media as one approach to the modeling of the generally non-Fickian behavior of transport. We use a semi-analytical method as Reproducing kernel Method to solve the Space-Time Fractional Advection-Dispersion equation so that we can get better approximate solutions than the methods with which this problem has been solved. The main obstacle to solve this problem is the existence of a Gram-Schmidt orthogonalization process in the general form of the reproducing kernel method, which is very time-consuming. So, we introduce the Improved Reproducing Kernel Method, which is a different implementation for the general form of the reproducing kernel method. In this method, the Gram-Schmidt orthogonalization process is eliminated to significantly reduce the CPU-time. Also, the present method increases the accuracy of approximate solutions.

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