Time-fractional Fisher Equation Research Articles

A second order difference method combined with time two-grid algorithm for two-dimensional time-fractional Fisher equation

In this paper, a finite difference (FD) scheme based on time two-grid algorithm is proposed for solving the two-dimensional time-fractional Fisher equation (2D-TFFE). Firstly, the Caputo fractional derivative and the spatial derivative are discretized by the L 2 − 1 σ formula and the central difference formula, respectively. In order to improve the efficiency of computation, the time two-grid algorithm is then constructed. Secondly, based on the cut-off function technic, stability and convergence of the time two-grid FD scheme are obtained by the energy method, and the global convergence order is O ( τ F 2 + τ C 4 + h x 2 + h y 2 ) , where τ F and τ C represent time-step sizes on the fine and coarse grid, respectively, while h x and h y represent the space-step sizes. Finally, numerical experiments are presented to show the feasibility and efficiency of the algorithm.

Analysis of nonlinear fractional-order Fisher equation using two reliable techniques

Abstract In this article, the solution to the time-fractional Fisher equation is determined using two well-known analytical techniques. The suggested approaches are the new iterative method and the optimal auxiliary function method, with the fractional derivative handled in the Caputo sense. The obtained results demonstrate that the suggested approaches are efficient and simple to use for solving fractional-order differential equations. The approximate and exact solutions of the partial fractional differential equations for integer order were compared. Additionally, the fractional-order and integer-order results are contrasted using simple tables. It has been confirmed that the solution produced using the provided methods converges to the exact solution at the appropriate rate. The primary advantage of the suggested method is the small number of computations needed. Moreover, it may be used to address fractional-order physical problems in a number of fields.

Symmetry Analysis and Wave Solutions of the Fisher Equation Using Conformal Fractional Derivatives

In the present article, the time fractional Fisher equation is considered in conformal form to study the application of the Lie classical method and quantitative analysis. The Lie symmetry method has been applied to find the infinitesimal generators and symmetry reductions of the fractional Fisher equation. The obtained reduced form of the equation is solved by the method of G ′ / G , which gives different forms of solutions. The theory of bifurcation has been utilized in the reduced form to check the stability and nature of critical points by transforming the equations into an autonomous system. Some phase portraits have been drawn at different critical points by the use of maple.

A New Parallelized Computation Method of HASC-N Difference Scheme for Inhomogeneous Time Fractional Fisher Equation

The fractional Fisher equation has a wide range of applications in many engineering fields. The rapid numerical methods for fractional Fisher equation have momentous scientific meaning and engineering applied value. A parallelized computation method for inhomogeneous time-fractional Fisher equation (TFFE) is proposed. The main idea is to construct the hybrid alternating segment Crank-Nicolson (HASC-N) difference scheme based on alternating segment difference technology, using the classical explicit scheme and classical implicit scheme combined with Crank-Nicolson (C-N) scheme. The unique existence, unconditional stability and convergence are proved theoretically. Numerical tests show that the HASC-N difference scheme is unconditionally stable. The HASC-N difference scheme converges to O(τ2−α+h2) under strong regularity and O(τα+h2) under weak regularity of fractional derivative discontinuity. The HASC-N difference scheme has high precision and distinct parallel computing characteristics, which is efficient for solving inhomogeneous TFFE.

Evaluation of time-fractional Fisher's equations with the help of analytical methods

<abstract><p>This article shows how to solve the time-fractional Fisher's equation through the use of two well-known analytical methods. The techniques we propose are a modified form of the Adomian decomposition method and homotopy perturbation method with a Yang transform. To show the accuracy of the suggested techniques, illustrative examples are considered. It is confirmed that the solution we get by implementing the suggested techniques has the desired rate of convergence towards the accurate solution. The main benefit of the proposed techniques is the small number of calculations. To show the reliability of the suggested techniques, we present some graphical behaviors of the accurate and analytical results, absolute error graphs and tables that strongly agree with each other. Furthermore, it can be used for solving fractional-order physical problems in various fields of applied sciences.</p></abstract>

Lie symmetry analysis, explicit solutions, and conservation laws of the time-fractional Fisher equation in two-dimensional space

In these analyses, we consider the time-fractional Fisher equation in two-dimensional space. Through the use of the Riemann-Liouville derivative approach, the well-known Lie point symmetries of the utilized equation are derived. Herein, we overturn the fractional fisher model to a fractional differential equation of nonlinear type by considering its Lie point symmetries. The diminutive equation's derivative is in the Erdélyi-Kober sense, whilst we use the technique of the power series to conclude explicit solutions for the diminutive equations for the first time. The conservation laws for the dominant equation are built using a novel conservation theorem. Several graphical countenances were utilized to award a visual performance of the obtained solutions. Finally, some concluding remarks and future recommendations are utilized.

Nonlinear anomalous information diffusion model in social networks

Modeling information diffusion in social networks is very important for predicting and controlling diffusions. Various types of diffusion models exist considering different properties of information diffusion. Recently it was shown that information diffusion in social networks like Digg and Twitter is anomalous, but it is not addressed in any information diffusion model. In this paper, a new mathematical information diffusion model is introduced based on anomalous diffusion characteristic that is not addressed before to the best of our knowledge. The proposed model is a nonlinear time-fractional Fisher's equation with Neumann boundary condition. The proposed model can model super-diffusion and sub-diffusion due to anomalous diffusion consideration. It is numerically solved using Haar wavelet and time discretization and validated with real datasets of Digg and Twitter social networks. The results show high precision in modeling the density of neighboring influenced users in time and space for different types and scales of diffusions. To predict non-neighbor influenced users, continuous-time random walk with jump is used with high precision. Combining the two models, the proposed nonlinear fractional diffusion model is a more realistic mathematical model and can model information diffusion in different types of social networks.

Computational analysis of fuzzy fractional order non-dimensional Fisher equation

In recent decades, fuzzy differential equations of integer and arbitrary order are extensively used for analyzing the dynamics of a mathematical model of the physical process because crisp operators of integer and arbitrary order are not able to study the model being studied when there is uncertainty in values used in modeling. In this article, we have considered the time-fractional Fisher equation in a fuzzy environment. The basic aim of this article is to deduce a semi-analytical solution to the fuzzy fractional-order non-dimensional model of the Fisher equation. Since the Laplace-Adomian method has a good convergence rate. We use the Laplace- Adomian decomposition method (LADM) to determine a solution under a fuzzy concept in parametric form. We discuss the convergence and error analysis of the proposed method. For the validity of the proposed scheme, we provide few examples with detailed solutions. We provide comparisons between exact and approximate solutions through graphs. In the end, the conclusion of the paper is provided.

On the numerical solution of conformable fractional diffusion problem with small delay

AbstractThis study investigates approximate solution of the time‐fractional Fisher equation with small delay by means of residual the residual power series method (RPSM). Since the delay term is small, the time fractional Fisher equation is expanded in powers series of delay term ϵ which allows to construct the approximate solution of time‐fractional of Fisher equation by means of RPSM. The obtained results illustrative examples show that the proposed method in this study is very powerful and efficient for the solution of the time‐fractional Fisher equation with small delay.

Haar wavelet iteration method for solving time fractional Fisher's equation

In this work, we investigate fractional version of the Fisher equation and solve it by using an efficient iteration technique based on the Haar wavelet operational matrices. In fact, we convert the nonlinear equation into a Sylvester equation by the Haar wavelet collocation iteration method (HWCIM) to obtain the solution. We provide four numerical examples to illustrate the simplicity and efficiency of the technique.

A numerical study on time fractional Fisher equation using an extended cubic B-spline approximation

A numerical study on time fractional Fisher equation using an extended cubic B-spline approximation

An Approximate Solution of the Time-Fractional Fisher Equation with Small Delay by Residual Power Series Method

An analytical solution of the time-fractional Fisher equation with small delay is established by means of residual the residual power series method (RPSM) where the fractional derivative is taken in the Caputo sense. Taking advantage of small delay, the time-fractional Fisher equation is expanded in powers series of delay term ϵ. By using RPSM analytical solution of time-fractional of Fisher equation is constructed. The final results and graphical consequences illustrate that the proposed method in this study is very efficient, effective, and reliable for the solution of the time-fractional Fisher equation with small delay.

A new iterative algorithm on the time-fractional Fisher equation: Residual power series method

In this article, the residual power series method is used to solve time-fractional Fisher equation. The residual power series method gets Maclaurin expansion of the solution. The solutions of prese...

On the Time Fractional Generalized Fisher Equation: Group Similarities and Analytical Solutions

In this letter, the Lie point symmetries of the time fractional Fisher (TFF) equation have been derived using a systematic investigation. Using the obtained Lie point symmetries, TFF equation has been transformed into a different nonlinear fractional ordinary differential equations with the Erdélyi–Kober fractional derivative which depends on the parameter α. After that some invariant solutions of underlying equation are reported.

A fully discrete local discontinuous Galerkin method for one-dimensional time-fractional Fisher's equation

In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one-dimensional time-fractional Fisher's equation, which is obtained from the standard one-dimensional Fisher's equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0<α<1). The proposed LDG is based on the LDG finite element method for space and finite difference method for time. We prove that the method is stable, and the numerical solution converges to the exact one with order O(hk+1+τ2−α), where h, τ and k are the space step size, time step size, polynomial degree, respectively. The numerical experiments reveal that the LDG is very effective.

Application of Fractional Variational Iteration Method to Time-Fractional Fisher Equation

Application of Fractional Variational Iteration Method to Time-Fractional Fisher Equation