Extended Fisher Kolmogorov Equation Research Articles

Superconvergence analysis of a low order expanded conforming mixed finite element method for the extended Fisher–Kolmogorov equation

This paper is devoted to the superconvergence analysis of a low order expanded conforming mixed finite element method(MFEM) for the extended Fisher–Kolmogorov equation. Both semi-discrete and linearized backward Euler fully-discrete schemes are analyzed with bilinear and Nédéleć elements. The present work has two main contributions: One is that some a priori bounds of the approximations are derived, which lead to the unique solvability of the semi-discrete scheme and make the nonlinear term can be dealt with efficiently. By such a way, the superclose properties and superconvergence results can be derived through the high accuracy results of the above two elements and the interpolated postprocessing technique, respectively. The other one is that a novel splitting technique is utilized to deal with the nonlinear term, which can avoid the use of popular error splitting technique and the restrictions between the time step size and mesh size. Besides, the combination technique of interpolation and projection operators also plays an important role in the superclose and superconvergence analysis. Finally, some numerical results are provided to show the validity of the theoretical analysis.

Numerical solution and analysis of extended Fisher–Kolmogorov equation using an improved collocation algorithm

The paper investigated the approximate solution of the extended Fisher–Kolmogorov. For this, an advanced approach: the improved quintic B-spline collocation technique has been employed, which is an enhancement over the conventional B-spline collocation method. The B-spline interpolant of degree five has been refined through the posteriori corrections, leading to the development of the improved B-spline solution. This proposed technique demonstrates superior convergence compared to the standard B-spline method, as assessed both theoretically and numerically. In tackling the extended Fisher–Kolmogorov equation, the space discretization is conducted using the improved quintic B-spline collocation methodology (IQSCM), and for the temporal domain discretization, the Crank-Nicolson scheme is applied. The error bounds and convergence analysis is established using the Green's functions. Von-Neumann stability analysis is carried out to discuss the stability of the technique. A few examples are solved and are represented graphically which helps in determining the nature of the solution. Also, L 2 , L ∞ error norms, and order of convergence are calculated to demonstrate the contribution of the new improved technique over the standard spline collocation technique.

The Reduced-Dimension Method for Crank–Nicolson Mixed Finite Element Solution Coefficient Vectors of the Extended Fisher–Kolmogorov Equation

A reduced-dimension (RD) method based on the proper orthogonal decomposition (POD) technology and the linearized Crank–Nicolson mixed finite element (CNMFE) scheme for solving the 2D nonlinear extended Fisher–Kolmogorov (EFK) equation is proposed. The method reduces CPU runtime and error accumulation by reducing the dimension of the unknown CNMFE solution coefficient vectors. For this purpose, the CNMFE scheme of the above EFK equation is established, and the uniqueness, stability and convergence of the CNMFE solutions are discussed. Subsequently, the matrix-based RDCNMFE scheme is derived by applying the POD method. Furthermore, the uniqueness, stability and error estimates of the linearized RDCNMFE solution are proved. Finally, numerical experiments are carried out to validate the theoretical findings. In addition, we contrast the RDCNMFE method with the CNMFE method, highlighting the advantages of the dimensionality reduction method.

A new spline method on graded mesh for fourth-order time-dependent PDEs: application to Kuramoto-Sivashinsky and extended Fisher-Kolmogorov equations

In this research, we introduce a two-tier non-polynomial spline approach with graded mesh discretization for addressing fourth-order time-dependent partial differential equations, which find applications in various physical scenarios like the nonlinear Kuramoto-Sivashinsky equation and extended Fisher-Kolmogorov equation. Our method involves considering three spatial points at each time step for scheme development, achieving spatial accuracy of three and temporal accuracy of two. Notably, our approach offers the advantage of directly tackling singular biharmonic problems without the need for discretizing boundary conditions or incorporating fictitious points. It demonstrates unconditional stability when tested on fourth-order problems and outperforms previous methods, as evidenced by comparative analyses. Additionally, we present 2D and 3D plots illustrating numerical solutions for different benchmark scenarios, including the depiction of the chaotic nature of the Kuramoto-Sivashinsky equation under Gaussian initial conditions at various time levels.

An Efficient Technique to Study Time Fractional Extended Fisher-Kolmogorov Equation

Reaction-diffusion partial differential equations are among the most widely used equations in applied mathematical modelling. In this study, we examine the solutions of one such equation, namely, the time-fractional extended Fisher-Kolmogorov equation. This equation is widely used in the study of population growth and wave propagation dynamics. The technique we consider is a combination of the natural transform and the Adomian decomposition method. Fractional derivatives are considered with singular and non-singular kernels. The existence and uniqueness of the solutions are presented. We analyse two different cases of the proposed problem to determine the validity and efficacy of the proposed scheme. Additionally, numerical simulation is shown, and the nature of the achieved solution is captured in terms of plots for various fractional orders. The outcome demonstrates that the method is straightforward, efficient, and dependable. The proposed method does not require any predetermined assumptions, linearization, perturbation, or discretization, and it prevents rounding errors. Therefore, the technique is ready to be implemented for a variety of nonlinear time fractional partial differential equations.

An adaptive low-rank splitting approach for the extended Fisher–Kolmogorov equation

The extended Fisher–Kolmogorov (EFK) equation has been used to describe some phenomena in physical, material and biological systems. In this paper, we propose a full-rank splitting scheme and a rank-adaptive splitting approach for this equation. We first use a finite difference method to approximate the space derivatives. Then, the resulting semi-discrete system is split into two stiff linear parts and a nonstiff nonlinear part. This leads to our full-rank splitting scheme. The convergence of the proposed scheme is proved rigorously. Based on the frame of the full-rank splitting scheme, we design a rank-adaptive splitting approach for obtaining a low-rank solution of the EFK equation. Numerical examples show that our methods are robust and accurate. They can also preserve the energy dissipation.

Numerical discretization for Fisher-Kolmogorov problem with nonlocal diffusion based on mixed Galerkin BDF2 scheme

Nonlocal problems involving fourth-order terms pose several difficulties such as numerical discretization and its related convergences analysis. In this paper, the well-posedness of the extended Fisher-Kolmogorov equation with nonlocal diffusion is first analyzed using the Faedo-Galerkin technique and the classical compactness arguments. Moreover, we adopt a BDF2 scheme for time discretization and a mixed Galerkin scheme for spatial discretization. Then, we derive the optimal order convergence rates of the fully discrete system. Finally, some numerical simulations and convergence results are provided to confirm the theoretical results and the accuracy of the proposed scheme.

A NEW MULTI-STEP BDF ENERGY STABLE TECHNIQUE FOR THE EXTENDED FISHER-KOLMOGOROV EQUATION

The multi-step backward difference formulas of order k (BDF-k) for 3 ≤ k ≤ 5 are proposed for solving the extended Fisher–Kolmogorov equation. Based upon the careful discrete gradient structures of the BDF-k formulas, the suggested numerical schemes are proved to preserve the energy dissipation laws at the discrete levels. The maximum norm priori estimate of the numerical solution is established by means of the energy stable property. With the help of discrete orthogonal convolution kernels techniques, the L2 norm error estimates of the implicit BDF-k schemes are established. Several numerical experiments are included to illustrate our theoretical results.

Finite element analysis of extended Fisher-Kolmogorov equation with Neumann boundary conditions

This paper delves into the numerical analysis of the extended Fisher-Kolmogorov (EFK) equation within open bounded convex domains R⊂Rd, where d≤3. Two distinct finite element schemes are introduced, namely the semi-discrete and fully-discrete finite element approximations. The existence and uniqueness of solutions are established for both the semi-discrete and fully-discrete finite element approximations. Error bounds are investigated across different scenarios, including comparisons between the semi-discrete and exact solutions, as well as between the semi-discrete and fully-discrete solutions, along with the fully-discrete and exact solutions. An effective algorithm has been proposed to solve the nonlinear system resulting from the fully-discrete finite element approximation at each time step. The paper also provides computed numerical error results and showcases a variety of numerical experiments to further illustrate and support the findings.

Two-grid dimension reduction method of Crank-Nicolson mixed finite element solution coefficient vectors for the fourth-order extended Fisher-Kolmogorov equation

In this article, we mainly resort to a proper orthogonal decomposition (POD) method to reduce the dimensionality of unknown mixed finite element (MFE) solution coefficient vectors in the two-grid Crank-Nicolson MFE (TGCNMFE) method for the fourth-order extended Fisher-Kolmogorov (FOEFK) equation and to build a new two-grid reduced-dimension extrapolated Crank-Nicolson MFE (TGRDECNMFE) method. For this purpose, we first propose a new TGCNMFE method for the FOEFK equation and demonstrate the existence, unconditional stability, and error estimates of the TGCNMFE solutions. Thereafter, we use the POD method to reduce the dimensionality of the unknown TGCNMFE solution coefficient vectors of the FOEFK equation to develop the new TGRDECNMFE method, and use matrix analysis to analyze the existence, unconditional stability, and errors of TGRDECNMFE solutions. Finally, we perform numerical experiments to confirm our theoretical results and show the advantages of the TGRDECNMFE method.

A space-time second-order method based on modified two-grid algorithm with second-order backward difference formula for the extended Fisher–Kolmogorov equation

In this paper, a modified two-grid algorithm based on block-centred finite difference method is developed for the fourth-order nonlinear extended Fisher–Kolmogorov equation. To further improve the computational efficiency, an effective second-order accurate backward difference formula is considered. The modified two-grid method based on Newton iteration is constructed to linearize the nonlinear system. The method solves a miniature nonlinear system on a coarse grid accompanying a larger time step to get the numerical solution, then computes a linear system constructed by the previous result with the Taylor expansion on a fine grid accompanying a smaller time step to get the correct numerical solution. Theoretical analysis shows that the modified two-grid algorithm can achieve second-order convergence accuracy both in time and space domain. Several numerical experiments are provided to verify the theoretical result and the high efficiency of this approach. The practical problem illustrates the actual applicable value of the algorithm.

Numerical investigation of nonlinear extended Fisher-Kolmogorov equation via quintic trigonometric B-spline collocation technique

<abstract><p>In this article, a collocation technique based on quintic trigonometric B-spline (QTB-spline) functions was presented for homogeneous as well as the nonhomogeneous extended Fisher-Kolmogorov (F-K) equation. This technique was used for space integration, while the time-derivative was discretized by the usual finite difference method (FDM). To handle the nonlinear term, the process of Rubin-Graves (R-G) type linearization was employed. Three examples of the homogeneous extended F-K equation and one example of the nonhomogeneous extended F-K equation were considered for the analysis. Stability analysis and numerical convergence were also discussed. It was found that the discretized system of the extended F-K equation was unconditionally stable, and the projected technique was second order accurate in space. The consequences were portrayed graphically to verify the accuracy of the outcomes and performance of the projected technique, and a relative investigation was accomplished graphically. The figured results were found to be extremely similar to the existing results.</p></abstract>

Analysis of Extended Fisher-Kolmogorov Equation in 2D Utilizing the Generalized Finite Difference Method with Supplementary Nodes

Analysis of Extended Fisher-Kolmogorov Equation in 2D Utilizing the Generalized Finite Difference Method with Supplementary Nodes

Explicit Integrating Factor Runge–Kutta Method for the Extended Fisher–Kolmogorov Equation

The extended Fisher–Kolmogorov (EFK) equation is an important model for phase transitions and bistable phenomena. This paper presents some fast explicit numerical schemes based on the integrating factor Runge–Kutta method and the Fourier spectral method to solve the EFK equation. The discrete global convergence of these new schemes is analyzed rigorously. Three numerical examples are presented to verify the theoretical analysis and the efficiency of the proposed schemes.

A new third-order energy stable technique and error estimate for the extended Fisher–Kolmogorov equation

A new third-order energy stable technique and error estimate for the extended Fisher–Kolmogorov equation

Sharp L2 Norm Convergence of Variable-Step BDF2 Implicit Scheme for the Extended Fisher–Kolmogorov Equation

A variable-step BDF2 time-stepping method is investigated for simulating the extended Fisher-Kolmogorov equation. The time-stepping scheme is shown to preserve a discrete energy dissipation law if the adjacent time-step ratios r n ≔ Τ n / Τ n − 1 < 3 + 17 / 2 ≈ 3.561 . With the aid of discrete orthogonal convolution kernels, concise L 2 norm error estimates are proved, for the first time, under the mild step ratios constraint 0 < r n < 3.561 . Our error estimates are almost independent of the step ratios r n so that the proposed numerical scheme is robust with respect to the variations of time steps. An adaptive time-stepping strategy based on solution accuracy is then applied to update the computational efficiency. Numerical examples are included to illustrate our theoretical results.

Some efficient numerical schemes for approximating the nonlinear two-space dimensional extended Fisher-Kolmogorov equation

This paper develops some efficient numerical schemes based on the hybridization of finite difference and orthogonal cubic spline collocation (OCSC) techniques to approximate the nonlinear two-dimensional extended Fisher-Kolmogorov model. This model represents a strong nonlinear fourth-order reaction diffusion evolution model. The proposed strategy is based on second-order backward differentiation formula and θ-scheme (including the Crank-Nicolson and backward Euler methods). It is shown that both the semi-discrete and full discrete approaches are unconditionally stable and convergent by means of the energy analysis procedure in an appropriate Sobolev space. In addition, the optimal convergence rates are obtained including fourth-order and second-order accuracies in space and time directions, respectively. Finally, numerical results assess the accuracy of the proposed methods and verify theoretical estimates.

An efficient computational approach for solving two-dimensional extended Fisher–Kolmogorov equation

The extended Fisher–Kolmogorov (EFK) system is a strong nonlinear fourth-order reaction diffusion evolution equation. So far, the numerical methods for solving this problem are only a few. This paper deals with the finite difference solution to the two-dimensional EFK equation. The scheme is proved to be uniquely solvable and second-order convergent in both time and space in maximum norm. Finally, some numerical examples have been illustrated to verify the efficiency and simplicity of the proposed technique.

Three-dimensional application of the meshless generalized finite difference method for solving the extended Fisher–Kolmogorov equation

In this article, a novel meshless local collocation method is proposed for the numerical solution of the three-dimensional (3D) extended Fisher–Kolmogorov (EFK) equation. The second-order Crank–Nicolson scheme and the meshless generalized finite difference method (GFDM) are respectively adopted to discretize the time and spatial derivatives of the EFK equation. A different setting of collocation nodes is introduced to the meshless GFDM for solving the nonlinear fourth order system resulting after the time discretization process. Two numerical experiments are carried out to verify the accuracy and the convergence of the developed numerical meshless algorithm.

A convex splitting BDF2 method with variable time-steps for the extended Fisher–Kolmogorov equation

A variable time-step BDF2 scheme combined with the convex splitting strategy is proposed for the extended Fisher–Kolmogorov equation. Under the zero-stability condition rk:=τk/τk−1≤ruser, (ruser<4.864), the suggested BDF2 scheme is shown to be uniquely solvable unconditionally, and preserve a discrete (modified) energy dissipation law. With the help of the recent discrete orthogonal convolution kernels technique, a concise L2 norm error estimate for the convex splitting BDF2 scheme is established under the time-step ratios restriction 0<rk≤ruser. Numerical examples together with an adaptive time-stepping procedure are provided to demonstrate the robustness and effectiveness of the proposed methods.