Fisher Kolmogorov Research Articles

A Boundary Value Problem for Algebraically Accelerated Frequency of Mutations Governed by Fisher–Kolmogorov–Petrovskii–Piskunov Equation

A new generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation [Formula: see text] is introduced and subjected to Lie’s Classical Method to establish the existence of the traveling waves. Indeed, exact expressions for both the solution [Formula: see text], with [Formula: see text] and the wave speed [Formula: see text] are subsequently derived by the regular perturbation method and are shown to contain both the reports of Ablowitz and Zeppetella [Explicit solutions of fisher’s equation for a special wave speed, Bulletin of Mathematical Biology 41(6) (1979) 835–840] when [Formula: see text] and Kaliappan [An exact solution for the traveling waves of [Formula: see text], Physica D: Nonlinear Phenomena 11(3) (1984) 368–374] when [Formula: see text], respectively. Remarkably, the generalized Huxley equation is a special case of our model equation. The form of the traveling waves for the Fisher–Kolmogorov–Petrovskii–Piskunov equations is identified with the regular perturbation method and the Binomial series. Our study is in agreement with the conclusion of Fisher [The wave of advance of advantageous genes, Annals of Eugenics 7(4) (1937) 355–369] that the increase of frequency of mutant genes results in the diffusion of the mutant genes into the long range.

Existence of Solutions to Periodic Boundary Value Problem of the Extended Fisher–Kolmogorov Equation Near Resonance

Existence of Solutions to Periodic Boundary Value Problem of the Extended Fisher–Kolmogorov Equation Near Resonance

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.

Reaction-diffusion for fish populations with realistic mobility

Nonlinear reaction-diffusion equations, with Fisher logistic growth and constant diffusion coefficient, have been used in fisheries research to estimate sustainable harvesting rates and critical domain sizes of no-take areas. However, constant diffusivity in a population density corresponds to standard Brownian motion of individuals, with a normal distribution for displacement over a fixed time interval. For available good data sets on mobile fish populations, the distribution is certainly not normal. The data can be fitted with a long-tailed stable Lévy distribution that results from diffusion by fractional Laplacian. Exact multidimensional solutions are developed here for realistic Fisher–Kolmogorov–Petrovski-Piscounov models with diffusion by fractional Laplacian. These can also account for a delay in the reaction term. It is then shown how to modify critical domain sizes of protected areas with Dirichlet and Robin boundary conditions for populations.

Dynamics behaviours of N-kink solitons in conformable Fisher–Kolmogorov–Petrovskii–Piskunov equation

PurposeThis manuscript is related to compute $N$-kink soliton solutions for conformable Fisher–Kolmogorov equation (CFKE) by using the generalized extended direct algebraic method (EDAM). The considered problem has important applications in mathematical biology and reaction diffusion processes. Also, the mentioned problem has significant applications in population dynamics. The fractional order conformable derivative has many features as compared to the other fractional order differential operators. For instance, the chain, product and quotient procedures do not satisfy by other fractional differential operators, but conformable operators obey the mentioned rules. Hence, we compute the soliton solutions for the mentioned problem and present its various dynamical behaviours graphically.Design/methodology/approachThe generalized EDAM is used in this article to examine the calculation of N-kink soliton solutions for the CFKE. In mathematical biology and reaction-diffusion processes, the topic under consideration holds great significance, especially when considering population dynamics.FindingsThe results highlight the benefits of utilising conformable derivatives in mathematical modelling and further our understanding of fractional differential equations and their applications.Research limitations/implicationsThe work focuses primarily on N-kink soliton solutions, which may limit the examination of alternative types of solutions (e.g., multi-soliton or periodic solutions) that might give new insights into the dynamics of the CFKE.Practical implicationsThe generated N N-kink soliton solutions can enhance mathematical models in biological contexts, notably in modelling population dynamics, disease propagation and ecological interactions, leading to better forecasts and interventions.Social implicationsPublic health initiatives can benefit from the understanding of disease transmission and intervention efficacy that comes from modelling population dynamics and reaction-diffusion processes.Originality/valueThe use of the generalized EDAM to obtain solutions for N-kink soliton problems is an innovative method for solving the conformable Fisher–Kolmogorov equation, demonstrating the power of this mathematical tool.

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.

Positivity preserving and unconditionally stable numerical scheme for the three-dimensional modified Fisher–Kolmogorov–Petrovsky–Piskunov equation

This paper introduces a numerical approach for the practical solution of the modified Fisher–Kolmogorov–Petrovsky–Piskunov equation that describes population dynamics. The diffusion term and nonlinear term is based on the operator splitting method and interpolation method, respectively. The analytic proof of the discrete maximum principle and positivity preserving for the numerical algorithm is demonstrated. Numerical solution calculated using the proposed method remains stable without blowing up, which implies that the proposed method is unconditionally stable. Numerical studies show that the proposed method is second-order convergence in space and first-order convergence in time. The performance and applicability of the proposed scheme are studied through various computational tests that present the effects of model parameters and evolution dynamics.

Nonlinear evolution equations in dispersive media: Analyzing the semilinear dispersive-fisher model

The nonlinear Semilinear Dispersive-Fisher (SDF) model is an important equation describing wave dynamics in dispersive media influenced by nonlinear and diffusive effects. This study enhances both analytical and numerical methods to solve the SDF model and validate the solutions. We utilize the Modified Khater (MKhat) and Unified (UF) methods for analytical solutions, and He’s Variational Iteration (HVI) scheme for numerical approximations. Our investigation connects the SDF model to other nonlinear equations like the Korteweg–de Vries (KdV) and Fisher–Kolmogorov (FK) equations, demonstrating the consistency between analytical and numerical solutions. This research contributes to the understanding and modeling of wave phenomena in dispersive media with nonlinear and diffusive dynamics, offering refined analytical and numerical techniques specific to the SDF model. Key contributions include introducing the MKhat and UF methods for analytical solutions and employing the HVI scheme for numerical approximations, which are less represented in current literature. This work is positioned in mathematical physics, focusing on nonlinear evolution equations.

Exploring tau protein and amyloid-beta propagation: A sensitivity analysis of mathematical models based on biological data

Alzheimer’s disease is the most common dementia worldwide. Its pathological development is well known to be connected with the accumulation of two toxic proteins: tau protein and amyloid-β. Mathematical models and numerical simulations can predict the spreading patterns of misfolded proteins in this context. However, the calibration of the model parameters plays a crucial role in the final solution. In this work, we perform a sensitivity analysis of heterodimer and Fisher–Kolmogorov models to evaluate the impact of the equilibrium values of protein concentration on the solution patterns. We adopt advanced numerical methods such as the IMEX-DG method to accurately describe the propagating fronts in the propagation phenomena in a polygonal mesh of sagittal patient-specific brain geometry derived from magnetic resonance images. We calibrate the model parameters using biological measurements in the brain cortex for the tau protein and the amyloid-β in Alzheimer’s patients and controls. Finally, using the sensitivity analysis results, we discuss the applicability of both models in the correct simulation of the spreading of the two proteins.Statement of significance: Alzheimer’s disease is related to the accumulation of tau protein and amyloid-β. Mathematical models to predict the spreading patterns require accurate parameter calibration. In this work, we perform a sensitivity analysis of heterodimer and Fisher–Kolmogorov models to evaluate the impact of the equilibrium values of protein concentration on the solution patterns obtained with advanced numerical simulations on patient-specific brain geometry derived from magnetic resonance images. By using biological measurements in the brain cortex for the proteins in Alzheimer’s patients and controls, we use sensitivity analysis to discuss the applicability of models in simulating protein spreading.

Dominant hippocampus segmentation with brain atrophy analysis-based AD subtype classification using KLW-RU-Net and T1FL

A progressive neurodegenerative disorder that leads to cognitive decline and memory loss is known as Alzheimer's Disease (AD). Researchers propose an integrated framework centered on brain atrophy analysis and hippocampus segmentation to predict AD and its subtypes. This approach augments the accurate prediction of AD subtype classification, thus addressing the lack of emphasis in prevailing works. Primarily, the T1-weighted brain Magnetic Resonance Imaging (MRI) images from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database undergo preprocessing. After that, for spatiotemporal evaluation, region segmentation, tau accumulation analysis, and Feature Extraction (FE), the image is processed. By employing the Fisher-Kolmogorov (FK) model, the essential parameters are extracted in spatio-temporal evaluation. In the meantime, for tissue segmentation, the Knowledge Partitioned Clustering (KPC) is utilized, and for hippocampus segmentation, the Kullback-Leibler Within-layer Regularized UNet (KLW-RU-Net) is employed. Moreover, by binarizing the image, the tau accumulation is analyzed; in addition, Principal Component Analysis (PCA) is utilized for FE. After PCA, the Adaptive Step-sized Gauss Rabbits Optimization (ASGRO) is employed for feature selection. Subsequently, to classify AD, the Dilated Derivative Sigmoid-Weighted DenseNet (DD-SWDN) is employed. Additionally, by employing Type-1 Fuzzy Logic (T1FL), the subtype of AD is classified, thus achieving higher accuracy for AD subtype classification. Therefore, as per these outcomes, the proposed work performs better than the other conventional approaches.

Front stability of infinitely steep travelling waves in population biology

Reaction–diffusion models are often used to describe biological invasion, where populations of individuals that undergo random motility and proliferation lead to moving fronts. Many reaction–diffusion models of biological invasion are extensions of the well–known Fisher–Kolmogorov–Petrovskii–Piskunov model that describes the spatiotemporal evolution of a 1D population density, u(x,t) , as a result of linear diffusion with flux J=−∂u/∂x , and logistic growth source term, S=u(1−u) . In 2020 Fadai introduced a new reaction–diffusion model of biological invasion with a nonlinear degenerate diffusive flux, J=−u∂u/∂x , and the model was formulated as a moving boundary problem on 0<x<L(t) , with u(L(t),t)=0 and dL(t)/dt=−κu∂u/∂x at x=L(t) (Fadai and Simpson 2020 J. Phys. A: Math. Theor. 53 095601). Fadai’s model leads to travelling wave solutions with infinitely steep, well–defined fronts at the moving boundary, and the model has the mathematical advantage of being analytically tractable in certain parameter limits. In this work we consider the stability of the travelling wave solutions presented by Fadai. We provide general insight by first presenting two key extensions of Fadai’s model by considering: (i) generalised nonlinear degenerate diffusion with flux J=−um∂u/∂x for some constant m > 0; and, (ii) solutions describing both biological invasion with dL(t)/dt>0 , and biological recession with dL(t)/dt<0 . After establishing the existence of travelling wave solutions for these two extensions, our main contribution is to consider stability of the travelling wave solutions by introducing a lateral perturbation of the travelling wavefront. Full 2D time–dependent level–set numerical solutions indicate that invasive travelling waves are stable to small amplitude lateral perturbations, whereas receding travelling waves are unstable. These preliminary numerical observations are corroborated through a linear stability analysis that gives more formal insight into short time growth/decay of wavefront perturbation amplitude. Julia–based software, including level–set algorithms, is available on Github to replicate all results in this study.

Solving nonlinear Fisher–Kolmogorov–Petrovsky–Piskunov equation using two meshless methods

Solving nonlinear Fisher–Kolmogorov–Petrovsky–Piskunov equation using two meshless methods

Analytical and numerical solutions to the Klein–Gordon model with cubic nonlinearity

In this paper, the nonlinear Klein–Gordon equation’s exact solutions are obtained through the application of an appropriate transformation based on He’s semi-inverse approach. This equation considered a generalization of other famous models in applied science, such as Phi-4 equation, Duffing equation, Fisher–Kolmogorov model through population dynamics and Hodgkin–Huxley equation that characterizes the propagation of electrical signals via nervous system. The suggested approach is simple, robust, and efficient, and its application in other partial differential equations in applied science seems promising. Numerical solution of the nonlinear Klein–Gordon equation is presented using finite difference method. The method’s accuracy is demonstrated by contrasting it with the exact solution that we obtained. The Von Neumann stability technique is applied to obtain the stability condition and a convergent test is presented Some 2D and 3D graphs matching to chosen solutions are simulated by taking into account appropriate values for the parameters.

Travelling wave solution of fourth order reaction diffusion equation using hybrid quintic hermite splines collocation technique

Fourth order extended Fisher Kolmogorov reaction diffusion equation has been solved numerically using a hybrid technique. The temporal direction has been discretized using Crank Nicolson technique. The space direction has been split into second order equation using twice continuously differentiable function. The space splitting results into a system of equations with linear heat equation and non linear reaction diffusion equation. Quintic Hermite interpolating polynomials have been implemented to discretize the space direction which gives a system of collocation equations to be solved numerically. The hybrid technique ensures the fourth order convergence in space and second order in time direction. Unconditional stability has been obtained by plotting the eigen values of the matrix of iterations. Travelling wave behaviour of dependent variable has been obtained and the computed numerical values are shown by surfaces and curves for analyzing the behaviour of the numerical solution in both space and time directions.

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.

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.

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

&lt;abstract&gt;&lt;p&gt;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.&lt;/p&gt;&lt;/abstract&gt;

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.