Fisher Equation Research Articles

A New Stable, Explicit, Third‐Order Method for Diffusion‐Type Problems

AbstractThis paper reports on a novel explicit numerical method for the spatially discretized diffusion or heat equation. After discretizing the space variables as in conventional finite difference methods, this method does not use a finite difference approximation for the time derivatives, it instead combines constant‐neighbor and linear‐neighbor approximations, which decouple the ordinary differential equations, thus they can be solved analytically. In the obtained three‐stage method, the time step size appears in exponential form with negative coefficients in the final expression. This property guarantees unconditional stability, as it is shown using von Neumann stability analysis. It is also proved that the convergence of the method is third order in the time step size. After verification, by solving Fisher's and Huxley's equations, it is demonstrated that it works for nonlinear equations as well. The new algorithm is tested against widely used numerical solvers for cases where the media is strongly inhomogeneous. According to the results, the new method is significantly more effective than the traditional explicit or implicit methods, especially for extremely large stiff systems. It is believed that this new method is unique in the sense that it is the first unconditionally stable explicit method with third‐order convergence.

Optimal convergence rate of the explicit Euler method for convection–diffusion equations

Revisiting the explicit Euler method of the classical diffusion equation, a new difference scheme with the optimal convergence rate four is achieved under the condition of the specific step-ratio r=1/6. Applying the corrected idea to the convection–diffusion equation, a new corrected numerical scheme is obtained which owns a similar fourth-order optimal convergence rate. Rigorous numerical analysis is carried out by the maximum principle. Compared with the standard difference schemes, the new proposed difference schemes have obvious advantage in accuracy. Extensive numerical examples with and without exact solutions confirm our theoretical results. Moreover, extending our technique to nonlinear problems such as the Fisher equation and viscous Burgers’ equation is available.

Numerical Method for Approximate Solution of Fisher's Equation

In this paper, Fisher's reaction diffusion equation has been solved numerically by Strang splitting technique depending on collocation method with cubic B-spline. For our purpose, the initial and boundary value problem consisting of Fisher's equation is split into two sub-problems to be one linear and the other nonlinear such that each one contains the derivative in terms of time. Then, the whole problem is reduced to the algebraic equation system using finite element collocation method combined with the cubic B-spline for spatial discretization and the convenient classical finite difference approaches for time discretization. The effective and efficiency of the newly given method have been shown on the four examples. In addition, the newly obtained numerical results are shown in formats graphical profiles and tables to compare with studies available in the literature.

NUMERICAL SOLUTION OF TRAVELING WAVES IN CHEMICAL KINETICS: TIME-FRACTIONAL FISHERS EQUATIONS

This paper addresses the numerical solution of nonlinear time-fractional Fisher equations via local meshless method combined with explicit difference scheme. This procedure uses radial basis functions to compute space derivatives while Caputo derivative scheme utilizes for time-fractional integration to semi-discretize the model equations. To assess the accuracy, maximum error norm is used. In order to validate the proposed method, some non-rectangular domains are also considered.

Systematic fuzzy Navier–Stokes equations for aerospace vehicles

PurposeTo prove the effectiveness of the proposed design method, this study aims to propose the Fisher equation and temperature cooling fins that control high-speed aerospace vehicles.Design/methodology/approachA new approach whereby the control of aerospace vehicles can be achieved by fuzzy controller and appropriate Navier–Stokes equations in this article. The design of the controller based on models of Navier–Stokes equations simplified complex mathematical simulations and approximations.FindingsIf the fuzzy controller cannot stabilize the system, the Navier–Stokes fuzzy function is injected into the system as a controller tool, and the system is asymptotically stabilized by adjusting the fuzzy parameters.Originality/valueThe simulation results show that if the tuning frequency is high enough, the fuzzy controller and fuzzy observer can create chaotic movements by adjusting the dither amplitude appropriately. The demonstration of the Fisher equation and the temperature-cooled fin control problem for high-speed aerospace vehicles has displayed the benefits of combining fuzzy control with the Navier–Stokes equation.

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>

New Solitary Wave Solutions of the Fisher Equation

In this paper, we use Riccati equation to find new solitary wave solutions of Fisher equation, which describes the process of interaction between diffusion and reaction. It is of great importance to comprehend the equation to solve the problems in chemical kinetics and population dynamics. We resolve the Ricatti equation through diverse function transformation and many types of exact solutions are obtained. Then it is used as an auxiliary equation to solve Fisher equation. In the process, we select different coefficients in the Racatti equation, as a result, abundant solitary wave solutions are obtained, some of which haven’t been found in other documents yet. Moreover, these solutions we got in this paper will be favorable for understanding the Fisher equation.

Analytical Approach to Fractional Fisher Equations by Laplace-Adomian Decomposition Method

This article implements the Laplace-Adomian decomposition method to obtain approximate analytical solutions in series form for non-linear time-fractional Fisher's equations with initial conditions. The fractional derivatives are given in the sense of Caputo. In addition, the results of this investigation are represented graphically, and they are simple yet highly accurate and compare favourably with the solutions reported in the earliest literature.

Stability estimates for singularly perturbed Fisher's equation using element-free Galerkin algorithm

<abstract><p>In the present article, a mesh-free technique has been presented to study the behavior of nonlinear singularly perturbed Fisher's problem, which exhibits the traveling wave propagation phenomenon. Some narrow regions adjacent to the left and right lateral boundary may possess rapid variations when the singular perturbation parameter $ \epsilon\rightarrow 0 $, which are not captured nicely by the traditional numerical schemes. In the current work, a robust numerical strategy is proposed, which comprises the implicit Crank-Nicolson scheme to discretize the time derivative term and the element-free Galerkin (EFG) scheme to discretize the spatial derivative terms with nodes densely distributed in the boundary layer regions. The stability of the semi-discrete scheme has been analyzed, and the rate of convergence is shown to be $ \mathcal{O}(\tau^{2}) $. The nonlinear nature of the considered problem has been tackled by employing the quasilinearization process, and its convergence rate has been discussed. Some numerical experiments have been performed to verify the computational uniformity and robustness of the suggested method, rate of convergence as well $ L_{\infty} $ errors have been presented, which depicts the effectiveness of the proposed method.</p></abstract>

Efficient algorithm to study the class of Burger's Fisher equation

Efficient algorithm to study the class of Burger's Fisher equation

Analyzing Similarity Solution of Modified Fisher Equation

In this paper, we first examine the type of structure of the solutions to the modified form of a nonlinear Fisher’s reaction‐diffusion equation. The existence of the traveling wave solution to the equation in the long term is observed by using dynamical system theory and exhibiting a phase space analysis of its stable points. In parallel, we represent radial basis functions (RBFs)‐based differential quadrature methods (DQMs) to close the solution of the equation. The stability analysis of the recommended method is demonstrated. Some initial‐boundary value problems are considered test problems. The numerical results indicate extremely exact and stable initial and boundary conditions in the same domain with dissimilar time ranges.

Numerical solutions of space-fractional diffusion equations via the exponential decay kernel

<abstract><p>The main object of this paper is to investigate the spectral collocation method for three new models of space fractional Fisher equations based on the exponential decay kernel, for which properties of Chebyshev polynomials are utilized to reduce these models to a set of differential equations. We then numerically solve these differential equations using finite differences, with the resulting algebraic equations solved using Newton 's method. The accuracy of the numerical solution is verified by computing the residual error function. Additionally, the numerical results are compared with other results obtained using the power law kernel and the Mittag-Leffler kernel. The advantage of the present work stems from the use of spectral methods, which have high accuracy and exponential convergence for problems with smooth solutions. The numerical solutions based on Chebyshev polynomials are in remarkably good agreement with numerical solutions obtained using the power law and the Mittag-Leffler kernels. Mathematica was used to obtain the numerical solutions.</p></abstract>

Analysis of the Impact of Asteroid Mining on Equality

To study the impact of asteroid mining on equality, we need to measure the global equity, so we developed the NRI (National Resources Index) model, with 18 secondary indicators and 6 primary indicators to measure the NRI for 17 sample countries. We simulated the complete process of asteroid mining and combine linear programming models and Monte Carlo methods to simulate the development and profitability of each country over the next 100 years. we built the AMP (Asteroid Mining Profit) model. Then we took into account the devaluation of minerals. By using the Fisher equation to quantify the gold price changes.

Efficient algorithm to study the class of Burgers Fisher equation

Efficient algorithm to study the class of Burgers Fisher equation

Weak Approximations of the Wright–Fisher Process

In this paper, we construct first- and second-order weak split-step approximations for the solutions of the Wright–Fisher equation. The discretization schemes use the generation of, respectively, two- and three-valued random variables at each discretization step. The accuracy of constructed approximations is illustrated by several simulation examples.

On an Econophysics Model

Krugman has proposed a continuous in time and space model for the emergence of polycentric urban areas in the regional space. By using a discrete version of this model on spatial networks, we predicted the distribution of jobs among the different localities inside a couple of economic regions in Ohio and Texas. The time evolution of the distribution of jobs is governed by a market potential. Employment gradually moves towards locations considered relatively attractive if their market potential is above the spatial average. Similarly, jobs move away from locations with below-average market potential. First, I will show that the market potential satisfies the Fisher equation of natural selection. Second, I will determine the stationary distribution.

A modification to the first integral method and its applications

In this article, we modify Feng’s first integral method (FIM) for the purpose of enlarging its applications. Compared with original FIM, our modified FIM is more straight and can also be employed to find first integral of higher-order ordinary differential equations (ODEs). We employ our modified FIM into five differential equations, namely, the density-dependent conformable fractional diffusion-reaction equation, the Duffing-van der Pol oscillator, the complex cubic-quintic Ginzburg–Landau equation, the well-known nonlinear evolution equation for description of surface waves in a convecting liquid, and the KdV–Burgers–Fisher equation. Consequently, we get the same first integral obtained by Feng’s FIM for the first equation; for the second equation, we reobtain certain important first integral reported previously; for the third equation, we construct a new first integral of complex cubic-quintic Ginzburg–Landau equation; and for the fourth and fifth equations, we show the effectiveness of our approach to third-order ODEs and reobtain the same first integral recently presented by Kudryashov for the fourth equation, and for the fifth equation, two new first integrals are presented. All the above fully reveal the effectiveness of our modification.

Global Existence of Solution for the Fisher Equation via Faedo–Galerkin’s Method

In this study, we consider the Fisher equation in bounded domains. By Faedo–Galerkin’s method and with a homogeneous Dirichlet conditions, the existence of a global solution is proved.

Rigorous verification of Hopf bifurcations in functional differential equations of mixed type

We propose a rigorously validated numerical method to prove the existence of Hopf bifurcations in functional differential equations of mixed type. The eigenvalue transversality and steady state conditions are verified using the Newton–Kantorovich theorem. The non-resonance condition and simplicity of the critical eigenvalues are verified by either computing a pair of generalized Morse indices of the step map, or by applying the argument principle to the characteristic equation and a suitable contour in the complex plane, computing the contour integral using the equivalence with a winding number. As a first application and test problem, we prove the existence of Hopf bifurcations in the Lasota–Wazewska–Czyzewska model and a pair of two such coupled equations. We then use our method to prove the existence of periodic traveling waves in the Fisher equation with nonlocal reaction. These periodic traveling waves are solutions of an ill-posed functional differential equation of mixed type.

The Sine-Cosine Function Method for Exact Solutions of Nonlinear Partial Differential Equations

The Sine-Cosine function algorithm is applied for solving nonlinear partial differential equations. The method is used to obtain the exact solutions for different types of nonlinear partial differential equations such as, The K(n + 1, n + 1) equation, Schrödinger-Hirota equation, Gardner equation, the modified KdV equation, perturbed Burgers equation, general Burger’s-Fisher equation, and Cubic modified Boussinesq equation which are the important Soliton equations.Keywords: Nonlinear PDEs, Exact Solutions, Nonlinear Waves, Gardner equation, Sine-Cosine function method, The Schrödinger-Hirota equation.