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.

Wavelet Galerkin method for fourth-order multi-dimensional elliptic partial differential equations

2018 ◽  
Vol 16 (05) ◽  
pp. 1850045 ◽  
Author(s):  
B. V. Rathish Kumar ◽  
Gopal Priyadarshi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Computational Cost ◽  
Stability Estimates ◽  
Partial Differential ◽  
Wavelet Galerkin Method

We describe a wavelet Galerkin method for numerical solutions of fourth-order linear and nonlinear partial differential equations (PDEs) in 2D and 3D based on the use of Daubechies compactly supported wavelets. Two-term connection coefficients have been used to compute higher-order derivatives accurately and economically. Localization and orthogonality properties of wavelets make the global matrix sparse. In particular, these properties reduce the computational cost significantly. Linear system of equations obtained from discretized equations have been solved using GMRES iterative solver. Quasi-linearization technique has been effectively used to handle nonlinear terms arising in nonlinear biharmonic equation. To reduce the computational cost of our method, we have proposed an efficient compression algorithm. Error and stability estimates have been derived. Accuracy of the proposed method is demonstrated through various examples.

Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm

2018 ◽  
Vol 28 (4) ◽  
pp. 828-856 ◽  
Author(s):  
Omar Abu Arqub
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Fluid Flows ◽  
Kernel Functions ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Content Type

Purpose The purpose of this study is to introduce the reproducing kernel algorithm for treating classes of time-fractional partial differential equations subject to Robin boundary conditions with parameters derivative arising in fluid flows, fluid dynamics, groundwater hydrology, conservation of energy, heat conduction and electric circuit. Design/methodology/approach The method provides appropriate representation of the solutions in convergent series formula with accurately computable components. This representation is given in the W(Ω) and H(Ω) inner product spaces, while the computation of the required grid points relies on the R(y,s) (x, t) and r(y,s) (x, t) reproducing kernel functions. Findings Numerical simulation with different order derivatives degree is done including linear and nonlinear terms that are acquired by interrupting the n-term of the exact solutions. Computational results showed that the proposed algorithm is competitive in terms of the quality of the solutions found and is very valid for solving such time-fractional models. Research limitations/implications Future work includes the application of the reproducing kernel algorithm to highly nonlinear time-fractional partial differential equations such as those arising in single and multiphase flows. The results will be published in forthcoming papers. Practical implications The study included a description of fundamental reproducing kernel algorithm and the concepts of convergence, and error behavior for the reproducing kernel algorithm solvers. Results obtained by the proposed algorithm are found to outperform in terms of accuracy, generality and applicability. Social implications Developing analytical and numerical methods for the solutions of time-fractional partial differential equations is a very important task owing to their practical interest. Originality/value This study, for the first time, presents reproducing kernel algorithm for obtaining the numerical solutions of some certain classes of Robin time-fractional partial differential equations. An efficient construction is provided to obtain the numerical solutions for the equations, along with an existence proof of the exact solutions based upon the reproducing kernel theory.

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 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 A New Mixed Element Method for a Class of Time-Fractional Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Yang Liu ◽  
Hong Li ◽  
Wei Gao ◽  
Siriguleng He ◽  
Zhichao Fang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Error Estimates ◽  
A Priori ◽  
A Priori Error Estimates ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Priori Error Estimates ◽  
Mixed Element ◽  
Element Method

A kind of new mixed element method for time-fractional partial differential equations is studied. The Caputo-fractional derivative of time direction is approximated by two-step difference method and the spatial direction is discretized by a new mixed element method, whose gradient belongs to the simpleL2Ω2space replacing the complexH(div;Ω)space. Some a priori error estimates inL2-norm for the scalar unknownuand inL22-norm for its gradientσ. Moreover, we also discuss a priori error estimates inH1-norm for the scalar unknownu.

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