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.

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 Iterative Reproducing Kernel Method for Solving Second-Order Integrodifferential Equations of Fredholm Type

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Iryna Komashynska ◽  
Mohammed AL-Smadi
Keyword(s):  
Reproducing Kernel ◽  
Kernel Method ◽  
Simple Algorithm ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Second Order ◽  
Integrodifferential Equations ◽  
Reproducing Kernel Space ◽  
Fredholm Type ◽  
Reproducing Kernel Method

We present an efficient iterative method for solving a class of nonlinear second-order Fredholm integrodifferential equations associated with different boundary conditions. A simple algorithm is given to obtain the approximate solutions for this type of equations based on the reproducing kernel space method. The solution obtained by the method takes form of a convergent series with easily computable components. Furthermore, the error of the approximate solution is monotone decreasing with the increasing of nodal points. The reliability and efficiency of the proposed algorithm are demonstrated by some numerical experiments.

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 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.

Reproducing kernel method to solve non-local fractional boundary value problem

2021 ◽  
Author(s):  
Raziye Mohammad Hosseiny ◽  
Tofigh Allahviranloo ◽  
Saeid Abbasbandy ◽  
Esmail Babolian
Keyword(s):  
Boundary Value Problem ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Reproducing Kernel Method ◽  
Non Local
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