scholarly journals A class of explicit implicit alternating difference schemes for generalized time fractional Fisher equation

2021 ◽  
Vol 6 (10) ◽  
pp. 11449-11466
Author(s):  
Xiao Qin ◽  
◽  
Xiaozhong Yang ◽  
Peng Lyu
Keyword(s):  
Difference Scheme ◽  
Unconditional Stability ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Fisher Equation ◽  
Numerical Techniques ◽  
Implicit Difference Scheme ◽  
Explicit Difference Scheme ◽  
Scientific Significance ◽  
Generalized Time

<abstract><p>The generalized time fractional Fisher equation is one of the significant models to describe the dynamics of the system. The study of effective numerical techniques for the equation has important scientific significance and application value. Based on the alternating technique, this article combines the classical explicit difference scheme and the implicit difference scheme to construct a class of explicit implicit alternating difference schemes for the generalized time fractional Fisher equation. The unconditional stability and convergence with order $ O\left({\tau }^{2-\alpha }+{h}^{2}\right) $ of the proposed schemes are analyzed. Numerical examples are performed to verify the theoretical analysis. Compared with the classical implicit difference scheme, the calculation cost of the explicit implicit alternating difference schemes is reduced by almost $ 60 $%. Numerical experiments show that the explicit implicit alternating difference schemes are also suitable for solving the time fractional Fisher equation with initial weak singularity and have an accuracy of order $ O\left({\tau }^{\alpha }+{h}^{2}\right) $, which verify that the methods proposed in this paper are efficient for solving the generalized time fractional Fisher equation.</p></abstract>

Propagator Method for Numerical Solution of Convective Fisher Equation

2011 ◽  
Vol 222 ◽  
pp. 387-390
Author(s):  
Daiga Zaime ◽  
Janis S. Rimshans ◽  
Sharif E. Guseynov
Keyword(s):  
Difference Scheme ◽  
Iteration Process ◽  
Fisher Equation ◽  
Implicit Difference Scheme ◽  
Nonlinear Convection ◽  
First Order ◽  
Truncation Errors ◽  
Propagator Method ◽  
Left And Right ◽  
The Difference

Propagator numerical method was developed as an effective tool for modeling of linear advective dispersive reactive (ADR) processes [1]. In this work implicit propagator difference scheme for Fisher equation with nonlinear convection (convective Fisher equation) is elaborated. Our difference scheme has truncation errors of the second order in space and of the first order in time. Iteration process for implicit difference scheme is proposed by introducing forcing terms in the left and right sides of the difference equation. Convergence and stability criterions for the elaborated implicit propagator difference scheme are obtained.

scholarly journals A new parallel difference algorithm based on improved alternating segment Crank–Nicolson scheme for time fractional reaction–diffusion equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xiaozhong Yang ◽  
Xu Dang
Keyword(s):  
Parallel Computing ◽  
Diffusion Equation ◽  
Difference Scheme ◽  
Reaction Diffusion ◽  
Difference Schemes ◽  
Reaction Diffusion Equation ◽  
Stability And Convergence ◽  
Parallel Difference ◽  
Fractional Reaction ◽  
Crank Nicolson

Abstract The fractional reaction–diffusion equation has profound physical and engineering background, and its rapid solution research is of important scientific significance and engineering application value. In this paper, we propose a parallel computing method of mixed difference scheme for time fractional reaction–diffusion equation and construct a class of improved alternating segment Crank–Nicolson (IASC–N) difference schemes. The class of parallel difference schemes constructed in this paper, based on the classical Crank–Nicolson (C–N) scheme and classical explicit and implicit schemes, combines with alternating segment techniques. We illustrate the unique existence, unconditional stability, and convergence of the parallel difference scheme solution theoretically. Numerical experiments verify the theoretical analysis, which shows that the IASC–N scheme has second order spatial accuracy and $2-\alpha $ 2 − α order temporal accuracy, and the computational efficiency is greatly improved compared with the implicit scheme and C–N scheme. The IASC–N scheme has ideal computation accuracy and obvious parallel computing properties, showing that the IASC–N parallel difference method is effective for solving time fractional reaction–diffusion equation.

An Numeric Method on the Third Order Term of KDV Equation

2011 ◽  
Vol 135-136 ◽  
pp. 253-255
Author(s):  
Yi Min Tian
Keyword(s):  
Difference Scheme ◽  
Kdv Equation ◽  
Order Term ◽  
Boundary Value ◽  
Difference Schemes ◽  
Boundary Values ◽  
Implicit Difference Scheme ◽  
Third Order ◽  
The Third ◽  
Numeric Experiment

Numeric scheme and numeric result was in this paper. First, We proposes a kind of explicit - implicit difference scheme to solve the initial and boundary value questions of the third order term of KDV equation here,and so we can solve the problem that the additional boundary values must be given first for present difference schemes when we try to realize the calculation by then., second, numeric experiment results was given ay the end of this article.

Two Finite Difference Schemes for Multi-Dimensional Fractional Wave Equations with Weakly Singular Solutions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jinye Shen ◽  
Martin Stynes ◽  
Zhi-Zhong Sun
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Mesh Size ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Initial Time ◽  
Finite Difference Schemes

Abstract A time-fractional initial-boundary value problem of wave type is considered, where the spatial domain is ( 0 , 1 ) d (0,1)^{d} for some d ∈ { 1 , 2 , 3 } d\in\{1,2,3\} . Regularity of the solution 𝑢 is discussed in detail. Typical solutions have a weak singularity at the initial time t = 0 t=0 : while 𝑢 and u t u_{t} are continuous at t = 0 t=0 , the second-order derivative u t ⁢ t u_{tt} blows up at t = 0 t=0 . To solve the problem numerically, a finite difference scheme is used on a mesh that is graded in time and uniform in space with the same mesh size ℎ in each coordinate direction. This scheme is generated through order reduction: one rewrites the differential equation as a system of two equations using the new variable v := u t v:=u_{t} ; then one uses a modified L1 scheme of Crank–Nicolson type for the driving equation. A fast variant of this finite difference scheme is also considered, using a sum-of-exponentials (SOE) approximation for the kernel function in the Caputo derivative. The stability and convergence of both difference schemes are analysed in detail. At each time level, the system of linear equations generated by the difference schemes is solved by a fast Poisson solver, thereby taking advantage of the fast difference scheme. Finally, numerical examples are presented to demonstrate the accuracy and efficiency of both numerical methods.

Numerical solutions of stochastic Fisher equation to study migration and population behavior in biological invasion

2017 ◽  
Vol 10 (07) ◽  
pp. 1750103 ◽  
Author(s):  
S. Singh ◽  
S. Saha Ray
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Wiener Process ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Stability And Convergence ◽  
Fisher Equation ◽  
Sample Paths ◽  
Implicit Finite Difference Scheme ◽  
Cylindrical Wiener Process

In this paper, numerical solutions of the stochastic Fisher equation have been obtained by using a semi-implicit finite difference scheme. The samples for the Wiener process have been obtained from cylindrical Wiener process and Q-Wiener process. Stability and convergence of the proposed finite difference scheme have been discussed scrupulously. The sample paths obtained from cylindrical Wiener process and Q-Wiener process have also been shown graphically.

scholarly journals Image Denoising via Nonlinear Hybrid Diffusion

2013 ◽  
Vol 2013 ◽  
pp. 1-22 ◽  
Author(s):  
Xiaoping Ji ◽  
Dazhi Zhang ◽  
Zhichang Guo ◽  
Boying Wu
Keyword(s):  
Difference Scheme ◽  
Image Denoising ◽  
Heat Diffusion ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Implicit Difference Scheme ◽  
Explicit Difference Scheme ◽  
Long Time ◽  
Diffusivity Function ◽  
Nonlinear Hybrid

A nonlinear anisotropic hybrid diffusion equation is discussed for image denoising, which is a combination of mean curvature smoothing and Gaussian heat diffusion. First, we propose a new edge detection indicator, that is, the diffusivity function. Based on this diffusivity function, the new diffusion is nonlinear anisotropic and forward-backward. Unlike the Perona-Malik (PM) diffusion, the new forward-backward diffusion is adjustable and under control. Then, the existence, uniqueness, and long-time behavior of the new regularization equation of the model are established. Finally, using the explicit difference scheme (PM scheme) and implicit difference scheme (AOS scheme), we do numerical experiments for different images, respectively. Experimental results illustrate the effectiveness of the new model with respect to other known models.

scholarly journals Exact Finite Difference Scheme and Nonstandard Finite Difference Scheme for Burgers and Burgers-Fisher Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Lei Zhang ◽  
Lisha Wang ◽  
Xiaohua Ding
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Burgers Equation ◽  
Solitary Wave Solution ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Fisher Equation ◽  
Nonstandard Finite Difference Scheme ◽  
Nonstandard Finite Difference

We present finite difference schemes for Burgers equation and Burgers-Fisher equation. A new version of exact finite difference scheme for Burgers equation and Burgers-Fisher equation is proposed using the solitary wave solution. Then nonstandard finite difference schemes are constructed to solve two equations. Numerical experiments are presented to verify the accuracy and efficiency of such NSFD schemes.

scholarly journals A Fast Preconditioned Semi-Implicit Difference Scheme for Strongly Nonlinear Space-Fractional Diffusion Equations

2021 ◽  
Vol 5 (4) ◽  
pp. 230
Author(s):  
Yu-Yun Huang ◽  
Xian-Ming Gu ◽  
Yi Gong ◽  
Hu Li ◽  
Yong-Liang Zhao ◽  
...  
Keyword(s):  
Difference Scheme ◽  
Fractional Diffusion ◽  
Euler Method ◽  
Diffusion Equations ◽  
Stability And Convergence ◽  
Implicit Difference Scheme ◽  
Central Difference ◽  
Fractional Diffusion Equations ◽  
Backward Euler ◽  
Difference Formula

In this paper, we propose a semi-implicit difference scheme for solving one-dimensional nonlinear space-fractional diffusion equations. The method is first-order accurate in time and second-order accurate in space. It uses a fractional central difference formula and the backward Euler method to approximate its space and time derivatives, respectively. Stability and convergence properties of the proposed scheme are proved with the help of a discrete Grönwall inequality. Moreover, we extend the method to the solution of two-dimensional nonlinear models. A fast matrix-free implementation based on preconditioned Krylov subspace methods is presented for solving the discretized linear systems. The resulting fast preconditioned semi-implicit difference scheme reduces the memory requirement of conventional semi-implicit difference schemes from O(Ns2) to O(Ns) and the computational complexity from O(Ns3) to O(NslogNs) in each iterative step, where Ns is the number of space grid points. Experiments with two numerical examples are shown to support the theoretical findings and to illustrate the efficiency of our proposed method.

Sign in / Sign up

Export Citation Format

Share Document