Kolmogorov Petrovskii Piskunov Equation Research Articles

The Exact Traveling Wave Solutions of a KPP Equation

We obtain the exact analytical traveling wave solutions of the Kolmogorov–Petrovskii–Piskunov equation, with the reaction term belonging to the class of functions, which includes that of the (generalized) Fisher equation, for the particular values of the wave’s speed. Additionally we obtain the exact analytical traveling wave solutions of the generalized Burgers–Huxley equation.

Dynamics Behaviours of Kink Solitons in Conformable Kolmogorov–Petrovskii–Piskunov Equation

The current study introduces the generalised New Extended Direct Algebraic Method (gNEDAM) for producing and examining propagation of kink soliton solutions within the framework of the Conformable Kolmogorov–Petrovskii–Piskunov Equation (CKPPE), which entails conformable fractional derivatives into account. The primary justification around employing conformable derivatives in this study is their special ability to comply with the chain rule, allowing for in the solution of aimed nonlinear model. The CKPPE is a crucial model for a number of disciplines, such as mathematical biology, reaction-diffusion mechanisms, and population increase. CKPPE is transformed into a Nonlinear Ordinary Differential Equation by the proposed gNEDAM, and many kink soliton solutions are found by applying the series form solution. These kink soliton solutions shed light on propagation mechanisms within the framework of the CKPPE model. Furthermore, our research offers multiple graphical depictions that facilitate the examination and analysis of the propagation patterns of the identified kink soliton solutions. Through the integration of mathematical biology and reaction-diffusion principles, our research broadens our comprehension of intricate occurrences in various academic domains.

Barycentric rational interpolation method for solving KPP equation

<abstract><p>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.</p></abstract>

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.

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.

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).

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.

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.

Influence of the Environment on Pattern Formation in the One-Dimensional Nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov Model

A self-consistent model of the dynamics of a cellular population described by the generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation with nonlocal competitive losses and interaction with the environment is formulated, in which the dynamics is described by the diffusion equation with allowance for the interaction of the population and the environment. With the help of computer modeling, the formation of the population pattern under the influence of the environment is considered. Possible applications of the model and its generalizations are discussed.

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

In this paper, we consider the invariance properties of the multiple-term fractional Kolmogorov–Petrovskii–Piskunov (FKPP) equation. By employing the Lie symmetry analysis method, we explicitly investigate the vector fields and symmetry reductions of the FKPP equation. Moreover, an effective method is proposed to succinctly derive the exact power series solutions with their convergence analysis of the equation. Finally, by using the new conservation theorem, the conservation laws associated with Lie symmetries of the equation are well constructed with a detailed analysis.

Traveling-Wave Solutions of the Kolmogorov–Petrovskii–Piskunov Equation

We consider quasi-stationary solutions of a problem without initial conditions for the Kolmogorov–Petrovskii–Piskunov (KPP) equation, which is a quasilinear parabolic one arising in the modeling of certain reaction–diffusion processes in the theory of combustion, mathematical biology, and other areas of natural sciences. A new efficiently numerically implementable analytical representation is constructed for self-similar plane traveling-wave solutions of the KPP equation with a special right-hand side. Sufficient conditions for an auxiliary function involved in this representation to be analytical for all values of its argument, including the endpoints, are obtained. Numerical results are obtained for model examples.

Approximate Solutions of the One-Dimensional Fisher–Kolmogorov–Petrovskii– Piskunov Equation with Quasilocal Competitive Losses

The modified Fisher–Kolmogorov–Petrovskii–Piskunov equation with quasilocal quadratic competitive losses and variable coefficients in the small nonlocality parameter approximation is reduced to an equation with a nonlinear diffusion coefficient. Within the framework of a perturbation method, equations are obtained for the first terms of an asymptotic expansion of an approximate solution of the reduced equation. Particular solutions in separating variables are considered for the equations determining the first terms of the asymptotic series. The problem is reduced to an elliptic integral and one linear, homogeneous ordinary differential equation.

Many Exact Solutions of the Nonlinear KPP Equation Using the Bäcklund Transformation of the Riccati Equation

The Bcklund transformation of the Riccati equation is applied in this article to construct traveling wave solutions for the nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation. Solitons, trigonometric and rational solutions of this equation are obtained. This transformation is straightforward and concise. It gives much more general results than the well-known results obtaining by other methods. With the aid of Maple, some graphical representations for some results are presented by choosing suitable values of parameters.

Wave propagation in the Kolmogorov–Petrovskii–Piskunov problem with delay

The problem of density wave propagation governed by a logistic equation with delay and diffusion (Fisher–Kolmogorov–Petrovskii–Piskunov equation with delay) was studied. To analyze the qualitative behavior of solutions to this equation with periodic boundary conditions in the case of the diffusion parameter tending to zero, the normal form of the problem, i.e., the Ginzburg–Landau equation was constructed near the unit equilibrium. A numerical analysis of wave propagation showed that, for sufficiently small delays, this equation has solutions close to those of the standard Kolmogorov–Petrovskii–Piskunov equation. As the delay parameter increases, a decaying oscillatory component appears in the spatial distribution of the solution and, then, undamped (in time) and slowly propagating (in space) oscillations close to solutions of the corresponding boundary value problem with periodic boundary conditions are observed near the initial perturbation segment.

The Backlund Transformation of the Generalized Riccati Equation and Its Applications to the Nonlinear KPP Equation

The Backlund transformation of the generalized Riccati equation is applied in this article to construct many new exact traveling wave solutions for the nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation. Solutions, trigonometric and rational solutions of this equation are obtained. This transformation is straightforward and concise. It gives much more general results than the well-known results obtaining by other methods. With the aid of Maple, some graphical representations for some results are presented by choosing suitable values of parameters.

Isolation effects in a system of two mutually communicating identical patches

Starting from the Fisher–Kolmogorov–Petrovskii–Piskunov equation (FKPP) we model the dynamic of a diffusive system with two mutually communicating identical patches and isolated of the remaining matrix. For this system we find the minimal size of each fragment in the explicit form and compare with the explicit results for similar problems found in the literature. From this comparison emerges an unexpected result that for a same set of the parameters, the isolated system studied in this work with size L, can be better or worst than the non isolated systems with the same size L, uniquely depending on the parameter a0 (internal conditions of the patches). Due to the fact that this result is unexpected we propose an experimental verification.