Element-Free Galerkin (EFG) Method for Time Fractional Partial Differential Equations

2011 ◽  
Vol 101-102 ◽  
pp. 343-347
Author(s):  
Yong Qing Liu ◽  
Rong Jun Cheng ◽  
Hong Xia Ge
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Time Derivative ◽  
Discrete Equation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Element Free Galerkin ◽  
Element Free ◽  
Caputo Fractional Order Derivative ◽  
Good Agreement

In this paper, the first order time derivative of time fractional partial differential equations are replaced by the Caputo fractional order derivative. We derive the numerical solution of this equation using the Element-free Galerkin (EFG) method. In order to obtain the discrete equation, a various method is used and the essential boundary conditions are enforced by the penalty method. Numerical examples are presented and the results are in good agreement with exact solutions.

scholarly journals On a Novel Numerical Scheme for Riesz Fractional Partial Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 2014
Author(s):  
Junjiang Lai ◽  
Hongyu Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
A Priori ◽  
Time Derivative ◽  
Stability Estimates ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Sharp Stability

In this paper, we consider numerical solutions for Riesz space fractional partial differential equations with a second order time derivative. We propose a Galerkin finite element scheme for both the temporal and spatial discretizations. For the proposed numerical scheme, we derive sharp stability estimates as well as optimal a priori error estimates. Extensive numerical experiments are conducted to verify the promising features of the newly proposed method.

scholarly journals Meshless Technique for the Solution of Time-Fractional Partial Differential Equations Having Real-World Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-17 ◽  
Author(s):  
Mehnaz Shakeel ◽  
Iltaf Hussain ◽  
Hijaz Ahmad ◽  
Imtiaz Ahmad ◽  
Phatiphat Thounthong ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Difference Scheme ◽  
Caputo Derivative ◽  
Time Derivative ◽  
Order Derivative ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Central Difference ◽  
Definition Of

In this article, radial basis function collocation scheme is adopted for the numerical solution of fractional partial differential equations. This method is highly demanding because of its meshless nature and ease of implementation in high dimensions and complex geometries. Time derivative is approximated by Caputo derivative for the values of α ∈ 0 , 1 and α ∈ 1 , 2 . Forward difference scheme is applied to approximate the 1st order derivative appearing in the definition of Caputo derivative for α ∈ 0 , 1 , whereas central difference scheme is used for the 2nd order derivative in the definition of Caputo derivative for α ∈ 1 , 2 . Numerical problems are given to judge the behaviour of the proposed method for both the cases of α . Error norms are used to asses the accuracy of the method. Both uniform and nonuniform nodes are considered. Numerical simulation is carried out for irregular domain as well. Results are also compared with the existing methods in the literature.

scholarly journals Solving Fractional Partial Differential Equations with Corrected Fourier Series Method

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Nor Hafizah Zainal ◽  
Adem Kılıçman
Keyword(s):  
Fourier Series ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Time Derivative ◽  
Finite Domain ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Fourier Series Method ◽  
Fractional Time Derivative

The corrected Fourier series (CFS) is proposed for solving partial differential equations (PDEs) with fractional time derivative on a finite domain. In the previous work, we have been solving partial differential equations by using corrected Fourier series. The fractional derivatives are described in Riemann sense. Some numerical examples are presented to show the solutions.

scholarly journals Oscillation criteria of certain fractional partial differential equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Di Xu ◽  
Fanwei Meng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Riccati Transformation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation

Abstract In this article, we regard the generalized Riccati transformation and Riemann–Liouville fractional derivatives as the principal instrument. In the proof, we take advantage of the fractional derivatives technique with the addition of interval segmentation techniques, which enlarge the manners to demonstrate the sufficient conditions for oscillation criteria of certain fractional partial differential equations.

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