Solving a New Type of Fractional Differential Equation by Reproducing Kernel Method

2020 ◽  
pp. 34-43
Author(s):  
Ali Akgül ◽  
Esra Karatas Akgül
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Reproducing Kernel Method ◽  
New Type ◽  
Fractional Differential

scholarly journals A Unified Reproducing Kernel Method and Error Estimation for Solving Linear Differential Equation with Functional Constraints

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Xinjian Zhang ◽  
Xiongwei Liu
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Linear Differential Equation ◽  
Error Estimation ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Inner Product ◽  
Reproducing Kernel Method ◽  
Linear Differential ◽  
Polynomial Reproducing

A unified reproducing kernel method for solving linear differential equations with functional constraint is provided. We use a specified inner product to obtain a class of piecewise polynomial reproducing kernels which have a simple unified description. Arbitrary order linear differential operator is proved to be bounded about the special inner product. Based on space decomposition, we present the expressions of exact solution and approximate solution of linear differential equation by the polynomial reproducing kernel. Error estimation of approximate solution is investigated. Since the approximate solution can be described by polynomials, it is very suitable for numerical calculation.

scholarly journals On the solutions of electrohydrodynamic flow with fractional differential equations by reproducing kernel method

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 685-689 ◽  
Author(s):  
Ali Akgül ◽  
Dumitru Baleanu ◽  
Mustafa Inc ◽  
Fairouz Tchier
Keyword(s):  
Approximate Solution ◽  
Fractional Differential Equations ◽  
Numerical Experiments ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
High Accuracy ◽  
Reproducing Kernel Method ◽  
Fractional Differential ◽  
Kernel Model ◽  
Theoretical Results

AbstractIn this manuscript we investigate electrodynamic flow. For several values of the intimate parameters we proved that the approximate solution depends on a reproducing kernel model. Obtained results prove that the reproducing kernel method (RKM) is very effective. We obtain good results without any transformation or discretization. Numerical experiments on test examples show that our proposed schemes are of high accuracy and strongly support the theoretical results.

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.

Sign in / Sign up

Export Citation Format

Share Document