Extended Fisher Kolmogorov Research Articles

A new three-level fourth-order compact finite difference scheme for the extended Fisher-Kolmogorov equation

In this paper, a new fourth-order compact finite difference scheme is proposed to solve the extended Fisher-Kolmogorov (EFK) equation. The scheme is three-level and implicit nonlinear. A new time discretization technique is adopted in the new scheme. The priori estimate and convergence of the numerical solution are obtained by using the discrete energy method, and the optimal error estimation in the maximum norm shows that the proposed numerical scheme has second-order accuracy in time and fourth-order accuracy in space. Finally, the theoretical analysis is supported by the numerical results of several examples.

On the exact solutions of nonlinear extended Fisher-Kolmogorov equation by using the He's variational approach

<abstract><p>In this article, we investigate existence and the exact solutions of the extended Fisher-Kolmogorov (EFK) equation. This equation is used in the population growth dynamics and wave propagation. The fourth-order term in this model describes the phase transitions near critical points which are also known as Lipschitz points. He's variational method is adopted to construct the soliton solutions as well as the periodic wave solutions successfully for the extended (higher-order) EFK equation. This approach is simple and has the greatest advantages because it can reduce the order of our equation and make the equation more simple. So, the results that are obtained by this approach are very simple and straightforward. The graphics behavior of these solutions are also sketched in 3D, 2D, and corresponding contour representations by the different choices of parameters.</p></abstract>

Radial basis functions based meshfree schemes for the simulation of non-linear extended Fisher–Kolmogorov model

This work offers two radial basis functions (RBFs) based meshfree schemes for the numerical simulation of non-linear extended Fisher–Kolmogorov model. In the development of the first scheme, first of all, time derivative is discretized by forward finite difference and then stability and convergence of the semi-discrete model is analyzed in L2 and H02 spaces. After that, RBF-differential quadrature method (RBF–DQM) is applied for fully discretization. Then, the obtained system of algebraic linear equations is solved by Gauss-elimination method. In the second numerical scheme, first RBF–DQM is applied for spatial derivatives approximation and then we obtained a system of nonlinear ODEs. After that, the system of ODEs is solved by RK4 Method. Also, the stability of the scheme is discussed via matrix method. In order to prove the meshfree property of the proposed methods, we considered non-uniform angular and rectangular domains with different radius. In the supporting domain, we used 5,10,15,20 and 25 supporting nodes for each nodes. Six instances of the model are considered to examine the reliability and chastity of the proposed schemes and found accurate results.

Orthonormal shifted discrete Legendre polynomials for the variable-order fractional extended Fisher–Kolmogorov equation

This paper presents a numerical technique for solving the variable-order fractional extended Fisher–Kolmogorov equation. The method suggested to solve this problem is based on the orthonormal shifted discrete Legendre polynomials and the collocation method. First, we expand the unknown solution of the problem using the these polynomialss. Also, we approximate the second- and fourth-order classical derivatives, as well as the variable-order fractional derivatives by these basis functions. Then, we substitute these approximations in the equation. Next, we utilize the classical and fractional derivative matrices together with the collocation method to convert the main equation into a system containing nonlinear algebraic equations. We show the correctness of the proposed scheme by providing several numerical examples.

Analysis of a quasi-reversibility method for nonlinear parabolic equations with uncertainty data

In this paper, we study the backward problem of determining initial condition for some class of nonlinear parabolic equations in multidimensional domain where data are given under random noise. This problem is ill-posed—i.e., the solution does not depend continuously on the data. To regularize the instable solution, we develop some new methods to construct some new regularized solution. We also investigate the convergence rate between the regularized solution and the solution of our equations. In particular, we establish results for several equations with constant coefficients and time dependent coefficients. The equations with constant coefficients include the heat equation, extended Fisher–Kolmogorov equation, Swift–Hohenberg equation, and many others. The equations with time dependent coefficients include Fisher-type logistic equations, the Huxley equation, and the Fitzhugh–Nagumo equation. The methods developed in this paper can also be applied to get approximate solutions to several other equations including the 1-D Kuramoto–Sivashinsky equation, 1-D modified Swift–Hohenberg equation, strongly damped wave equation, and 1-D Burgers equation with randomly perturbed operator. Some numerical examples are given which illustrate the effectiveness of our method.

Numerical solution for stochastic extended Fisher-Kolmogorov equation

In this paper, we derived a new compact finite difference scheme in the spatial direction and used the semi-implicit Euler-Maruyama approach in the temporal direction to study a stochastic extended Fisher-Kolmogorov equation with multiplicative noise numerically. Moreover, the analysis of consistency for the stochastic difference scheme was discussed and the stability analysis was proven in the mean square sense and by Fourier analysis. This approach is numerically analyzed to show the effect of random fluctuations occurring in nature and missing from the deterministic version of the equation and this illustrated in a numerical experiment.

Chebyshev Wavelets Collocation Method for Extended Fisher Kolmogorov Equations One and Two Space Dimension

Chebyshev Wavelets Collocation Method for Extended Fisher Kolmogorov Equations One and Two Space Dimension

Double layered solutions to the extended Fisher\u2013Kolmogorov P.D.E.

We construct double layered solutions to the extended Fisher–Kolmogorov P.D.E., under the assumption that the set of minimal heteroclinics of the corresponding O.D.E. satisfies a separation condition. The aim of our work is to provide for the extended Fisher–Kolmogorov equation, the first examples of two-dimensional minimal solutions, since these solutions play a crucial role in phase transition models, and are closely related to the De Giorgi conjecture.

Asymptotic behavior of time periodic solutions for extended Fisher-Kolmogorov equations with delays

<p style='text-indent:20px;'>In this paper, we investigate the global existence, uniqueness and asymptotic stability of time periodic classical solution for a class of extended Fisher-Kolmogorov equations with delays and general nonlinear term. We establish a general framework to investigate the asymptotic behavior of time periodic solutions for nonlinear extended Fisher-Kolmogorov equations with delays and general nonlinear function, which will provide an effective way to deal with such kinds of problems. The discussion is based on the theory of compact and analytic operator semigroups and maximal regularization method.</p>

Positive Solutions for Nonlocal Extended Fisher–Kolmogorov and Swift–Hohenberg Equations

In this paper, we study the existence of positive solutions for a class of nonlocal fourth-order semilinear differential equations which can be seen as generalization of extended Fisher–Kolmogorov and Swift–Hohenberg equations. The main result is proved using bifurcation theory.

A three-level linearized high-order accuracy difference scheme for the extended Fisher–Kolmogorov equation

A three-level linearized difference scheme for the extended Fisher–Kolmogorov equation is considered. It is proved that the proposed difference scheme is uniquely solvable and unconditionally convergent. The convergence order in maximum norm is $$O(h^4+k^2)$$ , where k and h are the temporal and spatial grid sizes, respectively. The accurateness and effectiveness of the method are tested by taking various examples. The numerical results of the method are compared with the exact solutions and also compared with earlier published results. It is found that the proposed method produces more accurate results than the others available in the literature.

Blowup and global existence of mild solutions for fractional extended Fisher–Kolmogorov equations

Abstract In this paper, we investigate the blowup, as well as global existence, and uniqueness of mild solutions for the initial-boundary value problem to a class of fractional extended Fisher–Kolmogorov equations with a general nonlinear term. We establish a general framework to find the global mild solutions for fractional extended Fisher–Kolmogorov equations with general nonlinear function, which will provide an effective way to deal with such problems. The results obtained in this paper can be considered as a contribution to this nascent field.

A Finite Pointset Method for Extended Fisher–Kolmogorov Equation Based on Mixed Formulation

In this paper, numerical solutions of the extended Fisher–Kolmogorov equation are obtained using finite pointset method. Finite pointset method is a meshless method which is a local iterative method based on the weighted least square approximation. By employing splitting technique, the extended Fisher–Kolmogorov equation is split into a two coupled system of differential equations by introducing an intermediate function. The method is applied to the resulting coupled system of differential equation. The numerical results confirm the good efficiency of the finite pointset method.

Error analysis of interpolating element free Galerkin method to solve non-linear extended Fisher–Kolmogorov equation

Nonlinear partial differential equations (PDEs) play an important role in the modeling of the natural phenomena as they have great significance in real-world applications. This investigation proposes a new algorithm to find the numerical solution of the non-linear extended Fisher–Kolmogorov equation. Firstly, the time variable is discretized by a second-order finite difference scheme. The rate of convergence and stability of the semi-discrete formulation are studied by the energy method. The existence and uniqueness of the solution of the weak form based on the proposed technique have been proved in detail. Furthermore, the interpolating element free Galerkin approach based on the interpolation moving least-squares approximation is employed to derive a fully discrete scheme. Finally, the error estimate of the full-discrete plan is proposed and its convergence order is O(τ2+δm+1) in which τ, δ and m denote the time step, the radius of the weight function and smoothness of the exact solution of the main problem, respectively.

Gegenbauer wavelet collocation method for the extended Fisher‐Kolmogorov equation in two dimensions

Gegenbauer wavelets operational matrices play an important role in the numeric solution of differential equations. In this study, operational matrices of rth integration of Gegenbauer wavelets are derived and used to obtain an approximate solution of the nonlinear extended Fisher‐Kolmogorov (EFK) equation in two‐space dimension. Nonlinear EFK equation is converted into the linear partial differential equation by quasilinearization technique. Numerical examples have shown that present method is convergent even in the case of a small number of grid points. The results of the presented method are in a good agreement with the results in literature.

A new high-resolution two-level implicit method based on non-polynomial spline in tension approximations for time-dependent quasi-linear biharmonic equations with engineering applications

In this present work, a new two-level implicit non-polynomial spline in tension method is proposed for the numerical solution of the 1D unsteady quasi-linear biharmonic problem. Using the continuity of the first-order derivative of the spline in tension function, a fourth-order accurate implicit finite-difference method is developed in this manuscript. By considering the linear model biharmonic problem, the given implicit spline method is unconditionally stable. Since the proposed method is based on half-step grid points, so it can be directly applied to 1D singular biharmonic problems without alteration in the scheme. Finally, the numerical experiments of the various biharmonic equation such as the generalized Kuramoto–Sivashinsky equation, extended Fisher–Kolmogorov equation, and 1D linear singular biharmonic problems are carried out to show the efficacy, accuracy, and reliability of the method. From the computational experiments, improved numerical results obtained as compared to the results obtained in earlier research work.

Analysis and application of the interpolating element-free Galerkin method for extended Fisher–Kolmogorov equation which arises in brain tumor dynamics modeling

In this paper, the interpolating element-free Galerkin method is applied for solving the nonlinear biharmonic extended Fisher–Kolmogorov equation which arises in brain tumor dynamics modeling. At first, a finite difference formula is utilized for obtaining a time-discrete scheme. The unconditional stability and convergence of the time-discrete method are proved by the energy method. Then, we use the interpolating element-free Galerkin method to approximate the spatial derivatives. An error analysis of the interpolating element-free Galerkin method is proposed for this nonlinear equation. Moreover, this method is compared with some other meshless local weak-form techniques. The main aim of this paper is to show that the interpolating element-free Galerkin is a suitable technique for solving the nonlinear fourth-order partial differential equations especially extended Fisher–Kolmogorov equation. The numerical experiments confirm the analytical results and show the good efficiency of the interpolating element-free Galerkin method for solving this nonlinear biharmonic equation.

Local radial basis function–finite-difference method to simulate some models in the nonlinear wave phenomena: regularized long-wave and extended Fisher–Kolmogorov equations

In this investigation, we concentrate on solving the regularized long-wave (RLW) and extended Fisher–Kolmogorov (EFK) equations in one-, two-, and three-dimensional cases by a local meshless method called radial basis function (RBF)–finite-difference (FD) method. This method at each stencil approximates differential operators such as finite-difference method. In each stencil, it is necessary to solve a small-sized linear system with conditionally positive definite coefficient matrix. This method is relatively efficient and has low computational cost. How to choose the shape parameter is a fundamental subject in this method, since it has a palpable effect on coefficient matrix. We will employ the optimal shape parameter which results from algorithm of Sarra (Appl Math Comput 218:9853–9865, 2012). Then, we compare the approximate solutions acquired by RBF–FD method with theoretical solution and also with results obtained from other methods. The numerical results show that the RBF–FD method is suitable and robust for solving the RLW and EFK equations.

Orbital dynamics satisfying the 4th-order stationary extended Fisher-Kolmogorov equation

In this study, we discuss the central force problem by using the nonlocal-in-time kinetic energy approach. At low length scales, the system is dominated by the generalized 4th-order extended Fisher-Kolmogorov stationary equation and by the 4th-order stationary Swift-Hohenberg differential equation under explicit conditions. The energy is a conserved quantity along orbits of the extended Fisher-Kolmogorov stationary equation. The system is quantized, the system is stable, and the ground energy problem is solved.

A numerical study of two-dimensional coupled systems and higher order partial differential equations

In this paper, a new approach and methodology is developed by incorporating differential quadrature technique with Bernstein polynomials. In differential quadrature method, approximations are done in a way that the derivatives of the function are replaced by a linear sum of functional values at the grid points of the given domain. In Bernstein differential quadrature method (BDQM), Bernstein polynomials are employed for spatial discretization so that a system of ordinary differential equations (ODE’s) is obtained which is solved by SSPRK-43 method. The stability of the method is also studied. The accuracy of the present method is checked by performing numerical experiments on two-dimensional coupled Burgers’ and Brusselator systems and fourth-order extended Fisher Kolmogorov (EFK) equation. Implementation of the method is very easy, efficient and capable of reducing the size of computational efforts.