Kolmogorov Petrovskii Piskunov Research Articles

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.

Dynamics of Evolutionary Differential Equations with Several Spatial Variables

The article is devoted to a method for constructing exact and approximate solutions of evolutionary partial differential equations with several spatial variables. The method is based on the theory of completely integrable distributions. Examples of applying this method to calculating exact solutions of the generalized Kolmogorov–Petrovskii–Piskunov–Fishev equations with two space variables are given.

Barycentric rational interpolation method for solving KPP equation

&lt;abstract&gt;&lt;p&gt;In this paper, we seek to solve the Kolmogorov-Petrovskii-Piskunov (KPP) equation by the linear barycentric rational interpolation method (LBRIM). As there are non-linear parts in the KPP equation, three kinds of linearization schemes, direct linearization, partial linearization, Newton linearization, are presented to change the KPP equation into linear equations. With the help of barycentric rational interpolation basis function, matrix equations of three kinds of linearization schemes are obtained from the discrete KPP equation. Convergence rate of LBRIM for solving the KPP equation is also proved. At last, two examples are given to prove the theoretical analysis.&lt;/p&gt;&lt;/abstract&gt;

Domain Structure Formation in Designing of the Opened Informative Measuring Systems

The opened systems possess an increasing significance and possibilities of applying in designing of measuring devices. Now an essentially nonlinear models are used for such systems. The perturbation approach is not enough for these purposes. Models of new types have solutions in a form of soliton or kink and similar objects. The equation of Fisher–Kolmogorov–Petrovskii–Piskunov is one of such equations. This equation is used for description of convection-reaction-diffusion processes. Such processes are used for studying of a self-organisation and formation of a structure in non-equilibrium opened systems. The aim of this work was to construct of a new solution for the modified equation of Fisher–Kolmogorov– Petrovskii–Piskunov in which a space inhomogeneity is accounted.To solve this problem the direct Hirota method for nonlinear partial differential equation is applied.Some modifications into this method were introduced.The new topologically non-trivial solution of the modified Fisher–Kolmogorov–Petrovskii–Piskunov equation is constructed explicitly. This solution has a kink-like form. Some arguments on the stability of such solution are considered.A possibility of domain structure formation in the systems which describe by the Fisher–Kolmogorov– Petrovskii–Piskunov equation is demonstrated.

A Chebyshev Collocation Approach to Solve Fractional Fisher–Kolmogorov–Petrovskii–Piskunov Equation with Nonlocal Condition

We provide a detailed description of a numerical approach that makes use of the shifted Chebyshev polynomials of the sixth kind to approximate the solution of some fractional order differential equations. Specifically, we choose the fractional Fisher–Kolmogorov–Petrovskii–Piskunov equation (FFKPPE) to describe this method. We write our approximate solution in the product form, which consists of unknown coefficients and shifted Chebyshev polynomials. To compute the numerical values of coefficients, we use the initial and boundary conditions and the collocation technique to create a system of equations whose number matches the unknowns. We test the applicability and accuracy of this numerical approach using two examples.

(2+1)-boyutlu Broer-Kaup-Kupershmidt denklemi ve Kolmogorov-Petrovskii-Piskunov denklemine modifiye edilmiş deneme denklem metodu

Many methods have been developed by scientists to find solutions for nonlinear problems. In this paper, the general structure of the modified trial equation method (MTEM) is introduced, and MTEM is used to find some exact solutions of (2+1)-dimensional Broer-Kaup-Kupershmidt (BKK), Kolmogorov-Petrovskii-Piskunov (KPP) equations. Firstly, an algebraic equation system is obtained by reducing the nonlinear partial differential equation (NLPDE) to the ordinary differential equation under the travelling wave transformation. Travelling wave solutions are found by solving the obtained algebraic equation systems. By using Mathematica 9 program, three and two dimensional graphs for suitable parameters were plotted to analyze the physical behavior of wave solutions. MTEM is of great importance in finding exact solutions of some partial differential equations.

An efficient difference scheme for the non-Fickian time-fractional diffusion equations with variable coefficient

In this paper, we develop an efficient difference scheme for the non-Fickian time-fractional diffusion equations with variable coefficient. This model may be considered as a generalization of the Kolmogorov–Petrovskii–Piskunov type equation, which is widely used to describe some important phenomena in the fields of chemistry, biology and viscoelastic materials. The stability and convergence of the difference scheme in the maximum norm are proved by the discrete energy method under mild conditions. A numerical example is carried out to verify our theoretical analysis results.

Diverse novel computational wave solutions of the time fractional Kolmogorov—Petrovskii - Piskunov and the (2 + 1)-dimensional Zoomeron equations

The numerical wave solutions of two fractional biomathematical and statistical physics models (the Kolmogorov—Petrovskii - Piskunov (KPP) equation and the (2 + 1)-dimensional Zoomeron (Z) equation) are investigated in this manuscript. Many novel analytical solutions in different mathematical formulations such as trigonometric, hyperbolic, exponential, and so on can be constructed using the generalized Riccati—expansion analytical scheme and the Caputo—Fabrizio fractional derivative. The fractional nonlinear evolution equation is converted into an ordinary differential equation with an integer order using this fractional operator. The obtained solution is used to describe the transmission of a preferred allele and the nonlinear interaction of moving waves, and the relative wave mode’s amplitude dynamic. To illustrate the fractional examined models, several drawings are explained in two dimensions and density plots.

On semi analytical and numerical simulations for a mathematical biological model; the time-fractional nonlinear Kolmogorov–Petrovskii–Piskunov (KPP) equation

Abstract Through five latest numerical schemes (Adomian decomposition (AD), El Kalla (EK), cubic B - spline (CBS), expanded Cubic B-Spline (ECBS), exponential cubic B - spline (ExCBS), this manuscript examines semi-analytical and numerical solutions of the time-fractional nonlinear Kolmogorov–Petrovskii–Piskunov (KPP) equation. Using the Caputo–Fabrizio fractional derivative and expanded Riccati - expansion process in Hamed et al.(2020) [1], developed computational solutions are investigated to determine the sufficient conditions for the implementation of the above-suggested schemes. In combustion theory, mathematical biology, and other study fields, the quasi-linear model is parabolic in simulating specific reaction-diffusion systems. The model’s solution represents the proliferation of a favored gene, and moving waves are pursued by nonlinear interaction. By measuring the absolute error between the exact and numerical solutions, the obtained numerical solutions’ consistency is examined. To explain the correspondence between the exact and numerical solutions, several sketches are given.

Further fresh and general traveling wave solutions to some fractional order nonlinear evolution equations in mathematical physics

PurposeFractional order nonlinear evolution equations (FNLEEs) pertaining to conformable fractional derivative are considered to be revealed for well-furnished analytic solutions due to their importance in the nature of real world. In this article, the autors suggest a productive technique, called the rational fractional (DξαG/G)-expansion method, to unravel the nonlinear space-time fractional potential Kadomtsev–Petviashvili (PKP) equation, the nonlinear space-time fractional Sharma–Tasso–Olver (STO) equation and the nonlinear space-time fractional Kolmogorov–Petrovskii–Piskunov (KPP) equation. A fractional complex transformation technique is used to convert the considered equations into the fractional order ordinary differential equation. Then the method is employed to make available their solutions. The constructed solutions in terms of trigonometric function, hyperbolic function and rational function are claimed to be fresh and further general in closed form. These solutions might play important roles to depict the complex physical phenomena arise in physics, mathematical physics and engineering.Design/methodology/approachThe rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is of the form U(ξ)=∑i=0nai(DξαG/G)i/∑i=0nbi(DξαG/G)i.FindingsAchieved fresh and further abundant closed form traveling wave solutions to analyze the inner mechanisms of complex phenomenon in nature world which will bear a significant role in the of research and will be recorded in the literature.Originality/valueThe rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is newly established and productive.

New Exact Solutions of Kolmogorov Petrovskii Piskunov Equation, Fitzhugh Nagumo Equation, and Newell-Whitehead Equation

This work presents the new exact solutions of nonlinear partial differential equations (PDEs). The solutions are acquired by using an effectual approach, the first integral method (FIM). The suggested technique is implemented to obtain the solutions of space-time Kolmogorov Petrovskii Piskunov (KPP) equation and its derived equations, namely, Fitzhugh Nagumo (FHN) equation and Newell-Whitehead (NW) equation. The considered models are significant in biology. The KPP equation describes genetic model for spread of dominant gene through population. The FHN equation is imperative in the study of intercellular trigger waves. Similarly, the NW equation is applied for chemical reactions, Faraday instability, and Rayleigh-Benard convection. The proposed technique FIM can be applied to find the exact solutions of PDEs.

Dynamics of Perturbations under Diffusion in a Porous Medium

We consider the dynamics of finite perturbations of a plane phase transition surface in the problem of evaporation of a fluid inside a low-permeability layer of a porous medium. In the case of a nonwettable porous medium, the problem has two stationary solutions, each containing a discontinuity. These discontinuities correspond to plane stationary phase transition surfaces located inside the low-permeability porous layer. One of these surfaces is unstable with respect to long-wavelength perturbations, while the other is stable. We study the evolution of perturbations of the stable plane phase transition surface. It is known that when two phase transition surfaces are located close enough to each other, the dynamics of a weakly nonlinear and weakly unstable wave packet is described by the Kolmogorov–Petrovskii–Piskunov (KPP) diffusion equation. As traveling wave solutions, this equation has heteroclinic solutions with either oscillating or monotonic structure of the front. The boundary value problem in the full statement, which should be considered if the distance between the stable and unstable plane phase transition surfaces is not small, also has similar solutions. We formulate a sufficient condition for the decrease of finite perturbations of the stable plane phase transition surface. This condition depends on their position with respect to the standing wave type and traveling front type solutions of the model equations in the model description when the KPP equation holds.

Abundant analytical and numerical solutions of the fractional microbiological densities model in bacteria cell as a result of diffusion mechanisms

In this paper, an analytical scheme [the generalized Sinh–Gordon equation method) with a new fractional operator (ABR fractional operator] is employed to find novel computational solutions of the nonlinear fractional Kolmogorov–Petrovskii–Piskunov (FKPP) equation. These solutions are used to evaluate the initial and boundary conditions under the numerical scheme (B-spline schemes) to get the numerical solutions of the same model. This equation describes the behavior of genetic models in the increase of microorganisms. Usually, it is used as a biological model to investigate the microbiological densities in bacteria cells as a result of diffusion mechanisms in terms of space–time. Some novel computational solutions are obtained, and the accuracy of them is investigated by calculating the absolute value of error between the obtained computational and numerical solutions. The comparison between the distinct types of obtained solutions is shown by calculating the absolute value of error.

Kolmogorov – Petrovskii – Piskunov denkleminin analitik çözümleri

In the current study, analytical solutions are constructed by applying (1/G') -expansion method to the Kolmogorov–Petrovskii–Piskunov (KPP) equation. Hyperbolic type exact solutions of the KPP equation are presented with the successfully applied method. 3D, 2D and contour graphics are presented by giving special values to the parameters in the solutions obtained. This article explores the applicability and effectiveness of this method on nonlinear evolution equations (NLEEs).

Computational and numerical simulations for the nonlinear fractional Kolmogorov–Petrovskii–Piskunov (FKPP) equation

This research paper elucidates solitary, compacton, and peakon computational solutions, and numerical solutions of the nonlinear fractional Kolmogorov–Petrovskii–Piskunov (FKPP) equation that belongs to the class of reaction–diffusion equation. This equation describes the behavior of genetic models in the increase of microorganisms. Usually, it is used as a biological model to investigate the microbiological densities in bacteria cells as a result of diffusion mechanisms in terms of space-time. The present framework depends on applying the modified Khater method to the FKPP equation to extract the computational solutions then using these solutions to get necessary boundary conditions to implement the numerical B–spline schemes on the suggested equation. The reliability and accuracy of the computational method and solutions are verified by using numerical simulations. For more explanation of the obtained analytical solutions, some sketches are plotted in different types. Also, the comparison between the distinct types of obtained solutions is shown by calculating the absolute value of error.

The Traveling Wave Solutions of Space-Time Fractional Partial Differential Equations by Modified Kudryashov Method

In this paper, the modified Kudryashov method is employed to find the traveling wave solutions of two well-known space-time fractional partial differential equations, namely the Zakharov Kuznetshov Benjamin Bona Mahony equation and Kolmogorov Petrovskii Piskunov equation, and as a helping tool, the sense of modified Riemann-Liouville derivative is also used. The propagation properties of obtained solutions are investigated where the graphical representations and justifications of the results are done by mathematical software Maple.

Kink soliton solutions and bifurcation for a nonlinear space-fractional Kolmogorov–Petrovskii–Piskunov equation in circuitry, chemistry or biology

Circuitry and chemistry are applied in such fields as communication engineering and automatic control, environmental protection and material/medicine sciences, respectively. Biology works as the basis of agriculture and medicine. Studied in this paper is a nonlinear space-fractional Kolmogorov–Petrovskii–Piskunov equation for the electronic circuitry, chemical kinetics, population dynamics, neurophysiology, population genetics, mutant gene propagation, nerve impulses transmission or molecular crossbridge property in living muscles. Kink soliton solutions are obtained via the fractional sub-equation method. Change of the fractional order does not affect the amplitudes of the kink solitons. Via the traveling transformation, the original equation is transformed into the ordinary differential equation, while we obtain two equivalent two-dimensional planar dynamic systems of that ordinary differential equation. According to the bifurcation and qualitative considerations of the planar dynamic systems, we display the corresponding phase portraits when the traveling-wave velocity is nonzero or zero. Nonlinear periodic waves of the original equation are obtained when the traveling-wave velocity is zero.

An Efficient Numerical Technique for the Nonlinear Fractional Kolmogorov–Petrovskii–Piskunov Equation

The q -homotopy analysis transform method ( q -HATM) is employed to find the solution for the fractional Kolmogorov–Petrovskii–Piskunov (FKPP) equation in the present frame work. To ensure the applicability and efficiency of the proposed algorithm, we consider three distinct initial conditions with two of them having Jacobi elliptic functions. The numerical simulations have been conducted to verify that the proposed scheme is reliable and accurate. Moreover, the uniqueness and convergence analysis for the projected problem is also presented. The obtained results elucidate that the proposed technique is easy to implement and very effective to analyze the complex problems arising in science and technology.

Approximate Solutions and Symmetry of a Two-Component Nonlocal Reaction-Diffusion Population Model of the Fisher–KPP Type

We propose an approximate analytical approach to a ( 1 + 1 ) dimensional two-component system consisting of a nonlocal generalization of the well-known Fisher–Kolmogorov–Petrovskii– Piskunov (KPP) population equation and a diffusion equation for the density of the active substance solution surrounding the population. Both equations of the system have terms that describe the interaction effects between the population and the active substance. The first order perturbation theory is applied to the system assuming that the interaction parameter is small. The Wentzel–Kramers–Brillouin (WKB)–Maslov semiclassical approximation is applied to the generalized nonlocal Fisher–KPP equation with the diffusion parameter assumed to be small, which corresponds to population dynamics under certain conditions. In the framework of the approach proposed, we consider symmetry operators which can be used to construct families of special approximate solutions to the system of model equations, and the procedure for constructing the solutions is illustrated by an example. The approximate solutions are discussed in the context of the released activity effect variously debated in the literature.

Lie symmetry analysis, explicit solutions and conservation laws for the time fractional Kolmogorov–Petrovskii–Piskunov equation

ABSTRACT In this paper, Lie symmetry method is applied to investigate the invariance properties of the time fractional Kolmogorov–Petrovskii–Piskunov equation with Riemann–Liouville derivative. In view of point symmetry, the vector fields for the governing equation are well constructed. And then the equation can be reduced to a fractional ODE with the help of Erdélyi–Kober operator. Moreover, a kind of explicit power series solutions of the equation is derived by virtue of the power series theory. Lastly, based on Ibragimov's method, two kinds of conservation laws for the equation are established.