Fisher Kolmogorov Petrovsky Piscounov Research Articles

An Unconditionally Stable Difference Scheme for the Two-Dimensional Modified Fisher–Kolmogorov–Petrovsky–Piscounov Equation

In this article, we develop an unconditionally stable numerical scheme for the modified Fisher–Kolmogorov–Petrovsky–Piscounov (Fisher–KPP) equation modeling population dynamics in two-dimensional space. The Fisher–KPP equation models the process of interaction between reaction and diffusion. The new solution algorithm is based on an alternating direction implicit (ADI) method and an interpolation method so that it is unconditionally stable. The proposed finite difference method is second-order accurate in time and space variables. Therefore, the main purpose of this study is to propose the novel Fisher–KPP equation with a nonlinear growth term and develop an unconditionally stable second-order numerical scheme. The novelty of our method is that it is a numerical method with second-order accuracy using interpolation and ADI methods in two dimensions. We demonstrate the performance of the proposed scheme through computational tests such as convergence and stability tests and the effects of model parameters and initial conditions.

An analytical solution of Balitsky–Kovchegov equation using homotopy perturbation method

An approximate analytical solution of the Balitsky–Kovchegov (BK) equation using the homotopy perturbation method (HPM) is suggested in this work. We carried out our work in perturbative Quantum Chromodynamics (QCD) (pQCD) dipole picture of deep inelastic scattering (DIS). The BK equation in momentum space with some change of variables and truncation of the Balitsky–Fadin–Kuraev–Lipatov (BFKL) kernel can be reduced to Fisher–Kolmogorov–Petrovsky–Piscounov (FKPP) equation. The observed geometric scaling phenomena are similar to the travelling wave solution of the FKPP equation. We solved the BK equation using the HPM. The obtained solution in this work also suggests the travelling wave nature of the measured scattering amplitude [Formula: see text] plotted at various rapidities. We also extracted the saturation momentum, [Formula: see text], from the obtained solution and plotted it against different rapidities. The result obtained in this work can be helpful for various phenomenological studies in high-density QCD.

Computational modeling of the Balitsky–Kovchegov equation and its numerical solution using hybrid B-spline collocation technique

The dynamics of high-energy scattering of hadrons are governed by Quantum Chromodynamics (QCD). In the parton model of QCD, scattering can be represented as a kind of reaction–diffusion process and consequently can be included in the category of the Fisher–Kolmogorov–Petrovsky–Piscounov (FKPP) equation. We perform a numerical study in the mean-field approximation of QCD evolution given by the Balitsky–Kovchegov (BK) equation, using the hybrid B-spline (HB-spline) collocation technique. The Rubin–Graves type linearization process is used to linearize the nonlinear terms. The accuracy and efficiency of the method are checked by taking numerical examples. It is noticed that the present method gives more accurate results than existing methods. The von Neumann stability analysis is also discussed for the discretized system of the BK equation.

An algorithm for numerical solution of some nonlinear multi-dimensional parabolic partial differential equations

In this research paper, a numerical method, named the three-step Ultraspherical wavelet collocation method, is presented for solving some nonlinear multi-dimensional parabolic partial differential equations. The method is third-order accurate in time. In this method, the three-step Taylor method is used to get the time derivative, while the Ultraspherical wavelet collocation method is used to get the space derivatives. Ultraspherical wavelets have good properties which make useful to carry out this aim. The presented method is developed for Burgers’ equation, Fisher–Kolmogorov–Petrovsky–Piscounov (Fisher–KPP) equation, and quasilinear parabolic equation. Three illustrative numerical problems are solved to demonstrate the efficiency, simplicity, and reliability of the presented method.

Fast-diffusion limit for reaction-diffusion equations with multiplicative noise

In this paper we consider a class of a stochastic partial differential equations with multiplicative noise and consider the limit of fast diffusion or other fast elliptic operator. We show that one can approximate solutions of the stochastic partial differential equations by the solution of a suitable stochastic ordinary differential equations on the kernel of the linear operator. We focus on equations with polynomial nonlinearities and give applications to Fisher–Kolmogorov–Petrovsky–Piscounov equation and the Ginzburg–Landau equation.

Particle-number distribution in large fluctuations at the tip of branching random walks.

We investigate properties of the particle distribution near the tip of one-dimensional branching random walks at large times t, focusing on unusual realizations in which the rightmost lead particle is very far ahead of its expected position, but still within a distance smaller than the diffusion radius ∼sqrt[t]. Our approach consists in a study of the generating function G_{Δx}(λ)=∑_{n}λ^{n}p_{n}(Δx) for the probabilities p_{n}(Δx) of observing n particles in an interval of given size Δx from the lead particle to its left, fixing the position of the latter. This generating function can be expressed with the help of functions solving the Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) equation with suitable initial conditions. In the infinite-time and large-Δx limits, we find that the mean number of particles in the interval grows exponentially with Δx, and that the generating function obeys a nontrivial scaling law, depending on Δx and λ through the combined variable [Δx-f(λ)]^{3}/Δx^{2}, where f(λ)≡-ln(1-λ)-ln[-ln(1-λ)]. From this property, one may conjecture that the growth of the typical particle number with the size of the interval is slower than exponential, but, surprisingly enough, only by a subleading factor at large Δx. The scaling we argue is consistent with results from a numerical integration of the FKPP equation.

An efficient wavelet collocation method for nonlinear two-space dimensional Fisher–Kolmogorov–Petrovsky–Piscounov equation and two-space dimensional extended Fisher–Kolmogorov equation

In this paper, we consider two-space dimensional nonlinear Fisher–Kolmogorov–Petrovsky–Piscounov (Fisher–KPP) equation and two-space dimensional nonlinear fourth-order extended Fisher–Kolmogorov (EFK) equation which have a lot of applications in different branches of science, especially in mathematical biology. We present a wavelet collocation method based on Chebyshev wavelets combined with two different time discretization schemes. To generate more accurate results for Fisher–KPP equation, in time variable discretization, forward Euler timestepping scheme along with Taylor series expansion is used. Resultant time-discretized scheme has second-order accuracy and includes second-order time derivative which is evaluated from governing equation. On the other hand for EFK equation forward Euler timestepping scheme with first-order accuracy is used. Then space variables situated in the semi-discrete schemes are discretized with Chebyshev wavelet series expansion. In this way a full discrete scheme is obtained. By this approach obtainment of numerical solution of considered partial differential equations is turned into an operation of finding solution of an algebraic system of equations. Actually the solution of the algebraic system of equations is wavelet coefficients in wavelet series expansion. Putting these coefficients into Chebyshev wavelet series expansion the numerical solution of considered partial differential equations can be obtained successively. The main objective of this paper is to demonstrate that Chebyshev wavelet-based method is accurate, efficient, and reliable for nonlinear two-space dimensional high-order partial differential equations. Six test problems are considered and $$L_2$$, $$L_{\infty }$$ error norms are calculated for comparison of our numerical results with exact results whenever they are available. Also numerical results are plotted for comparison with reference solutions. The obtained results certify the applicability and efficiency of the suggested method for Fisher–KPP and EFK equations.

Fisher–Kolmogorov–Petrovsky– Piscounov Reaction and n-Diffusion Cattaneo Telegraph Equation

This paper presents research for a class of recombination reaction and diffusion problems in which the Cattaneo relaxation, n-diffusion flux, and p-Fisher–Kolmogorov–Petrovsky–Piscounov (KPP) reaction are considered. Approximate analytical solutions are obtained by Adomian decomposition method (ADM) and shown graphically. Some interesting results for spatial variable and temporal variable evolution are obtained. For specified spatial variable, the temperature profiles decrease with respect to the increase of relaxation parameter and power-law index n but decrease with respect to Fisher–KPP reaction parameter p. For specified temporal variable, the temperature profiles are seem oscillating with values of the relaxation parameter and power-law index n.

Numerical Study of One Dimensional Fishers KPP Equation with Finite Difference Schemes

In this paper, we originate results with finite difference schemes to approximate the solution of the classical Fisher Kolmogorov Petrovsky Piscounov (KPP) equation from population dynamics. Fisher’s equation describes a balance between linear diffusion and nonlinear reaction. Numerical example illustrates the efficiency of the proposed schemes, also the Neumann stability analysis reveals that our schemes are indeed stable under certain choices of the model and numerical parameters. Numerical comparisons with analytical solution are also discussed. Numerical results show that Crank Nicolson and Richardson extrapolation are very efficient and reliably numerical schemes for solving one dimension fisher’s KPP equation.

Striated populations in disordered environments with advection

Growth in static and controlled environments such as a Petri dish can be used to study the spatial population dynamics of microorganisms. However, natural populations such as marine microbes experience fluid advection and often grow up in heterogeneous environments. We investigate a generalized Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) equation describing single species population subject to a constant flow field and quenched random spatially inhomogeneous growth rates with a fertile overall growth condition. We analytically and numerically demonstrate that the non-equilibrium steady-state population density develops a flow-driven striation pattern. The striations are highly asymmetric with a longitudinal correlation length that diverges linearly with the flow speed and a transverse correlation length that approaches a finite velocity-independent value. Linear response theory is developed to study the statistics of the steady states. Theoretical predictions show excellent agreement with the numerical steady states of the generalized FKPP equation obtained from Lattice Boltzmann simulations. These findings suggest that, although the growth disorder can be spatially uncorrelated, correlated population structures with striations emerge naturally at sufficiently strong advection.

A skew symmetry-preserving computational technique for obtaining the positive and the bounded solutions of a time-delayed advection–diffusion–reaction equation

In this work, we consider a one-dimensional, time-delayed, advective version of the well-known Fisher–Kolmogorov–Petrovsky–Piscounov equation from population dynamics, which extends several models from mathematical physics, including the classical wave equation, the nonlinear Klein–Gordon equation, a FitzHugh–Nagumo equation from electrodynamics, and the Burgers–Huxley equation and the Newell–Whitehead–Segel equation from fluid mechanics. We propose a skew symmetry-preserving, finite-difference scheme for approximating the solutions of the model under investigation, and establish conditions on the model coefficients and the numerical parameters under which the method provides positive or bounded approximations for initial data which are likewise positive or bounded, respectively. The derivation of the conditions under which the positivity and the boundedness of the approximations is guaranteed is based on the properties of the inverses of M-matrices; in fact, the conditions obtained here assure that the iterative method is described in vector form through the multiplication by a matrix of this type. We provide simulations in order to show that the technique is indeed conditionally positivity-preserving and boundedness-preserving.

An explicit positivity-preserving finite-difference scheme for the classical Fisher–Kolmogorov–Petrovsky–Piscounov equation

In this work, we introduce a family of explicit finite-difference schemes to approximate positive solutions of the classical Fisher–Kolmogorov–Petrovsky–Piscounov equation from population dynamics. A Neumann stability analysis reveals that our methods are indeed stable under certain choices of the model and numerical parameters, and numerical comparisons with analytical solutions are carried out in some detail.

Sufficient conditions for the preservation of the boundedness in a numerical method for a physical model with transport memory and nonlinear damping

Departing from a finite-difference scheme to approximate solutions of a nonlinear, hyperbolic partial differential equation which generalizes the Burgers–Huxley equation from fluid dynamics, we investigate conditions on the model coefficients and the computational parameters under which positive and bounded initial data evolve into positive and bounded new approximations. The model under investigation includes nonlinear coefficients of damping and advection, and the reaction term extends the reaction law of the classical Fisher–Kolmogorov–Petrovsky–Piscounov equation. The method can be expressed in vector form in terms of a multiplicative matrix which, under certain parametric conditions, becomes an M-matrix. Using the fact that every M-matrix is non-singular and that the entries of its inverse are positive, real numbers, we establish sufficient conditions under which the method provides new, positive and bounded approximations from previous, positive and bounded data and boundary conditions. The numerical results confirm the fact that the conditions derived here are sufficient for the positivity and the boundedness of the approximations; moreover, computational experiments evidence the fact that the method still preserves these properties for values of the model and the numerical parameters outside of the analytic regions of positivity and boundedness. We point out that our simulations show a good agreement between the numerical approximations computed through our method and the corresponding, analytical solutions.

A boundedness-preserving finite-difference scheme for a damped nonlinear wave equation

In this work, we present a non-standard finite-difference scheme to approximate solutions of a damped, hyperbolic partial differential equation with nonlinear reaction which follows a generalized logistic form. Our mathematical model is a hyperbolic generalization of the classical Fisher–Kolmogorov–Petrovsky–Piscounov equation from population dynamics, where the damping term has a constant coefficient; both zero Dirichlet and Neumann boundary conditions are considered. Under certain conditions of the model and the computational parameters, the method is able to preserve the positivity and the boundedness of the solutions of the mathematical model, as evidenced both theoretically (via techniques of matrix algebra which provide sufficient conditions on the positivity and the boundedness of the technique) and computationally (via numerical simulations). The validity of the method is tested against known exact solutions of the parabolic and the hyperbolic regimes, obtaining an excellent agreement between the analytical and numerical results.

Numerical treatment of the spherically symmetric solutions of a generalized Fisher–Kolmogorov–Petrovsky–Piscounov equation

In the present work, the connection of the generalized Fisher–KPP equation to physical and biological fields is noted. Radially symmetric solutions to the generalized Fisher–KPP equation are considered, and analytical results for the positivity and asymptotic stability of solutions to the corresponding time-independent elliptic differential equation are quoted. An energy analysis of the generalized theory is carried out with further physical applications in mind, and a numerical method that consistently approximates the energy of the system and its rate of change is presented. The method is thoroughly tested against analytical and numerical results on the classical Fisher–KPP equation, the Heaviside equation, and the generalized Fisher–KPP equation with logistic nonlinearity and Heaviside initial profile, obtaining as a result that our method is highly stable and accurate, even in the presence of discontinuities. As an application, we establish numerically that, under the presence of suitable initial conditions, there exists a threshold for the relaxation time with the property that solutions to the problems considered are nonnegative if and only if the relaxation time is below a critical value. An analytical prediction is provided for the Heaviside equation, against which we verify the validity of our computational code, and numerical approximations are provided for several generalized Fisher–KPP problems.

On the maximal noise for stochastic and QCD travelling waves

Using the relation of a set of nonlinear Langevin equations to reaction–diffusion processes, we note the existence of a maximal strength of the noise for the stochastic travelling wave solutions of these equations. Its determination is obtained using the field-theoretical analysis of branching-annihilation random walks near the directed percolation transition. We study its consequence for the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov equation. For the related Langevin equation modeling the quantum chromodynamic nonlinear evolution of gluon density with rapidity, the physical maximal-noise limit may appear before the directed percolation transition, due to a shift in the travelling-wave speed. In this regime, an exact solution is known from a coalescence process. Universality and other open problems and applications are discussed in the outlook.

Solution to the Balitsky–Kovchegov equation in the saturation domain

The solution to the Balitsky–Kovchegov equation is found in the deep saturation domain. The controversy between different approaches regarding the asymptotic behaviour of the scattering amplitude is solved. It is shown that the dipole amplitude behaves as 1 − exp ( − z + ln z ) with z = ln ( r 2 Q s 2 ) ( r-size of the dipole, Q s is the saturation scale) in the deep saturation region. This solution is developed from the scaling solution to the homogeneous Balitsky–Kovchegov equation. The dangers associated with making simplifications in the BFKL kernel, to investigate the asymptotic behaviour of the scattering amplitude, is pointed out. In particular, the fact that the Balitsky–Kovchegov equation belongs to the Fisher–Kolmogorov–Petrovsky–Piscounov type of equation, needs further careful investigation.

A Langevin equation for high-energy evolution with pomeron loops

We show that the Balitsky–JIMWLK equations proposed to describe non-linear evolution in QCD at high energy fail to include the effects of fluctuations in the gluon number, and thus to correctly describe both the low-density regime and the approach towards saturation. On the other hand, these fluctuations are correctly encoded (in the limit where the number of colors is large) in Mueller's color dipole picture, which, however, neglects saturation. By combining the dipole picture at low density with the JIMWLK evolution at high density, we construct a generalization of the Balitsky hierarchy which includes the particle number fluctuations, and thus the pomeron loops. After an additional coarse-graining in impact parameter space, this hierarchy is shown to reduce to a Langevin equation in the universality class of the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov (sFKPP) equation. This equation implies that the non-linear effects in the evolution become important already in the high-momentum regime where the average density is small, which signals the breakdown of the BFKL approximation.

Emergence of pulled fronts in fermionic microscopic particle models.

We study the emergence and dynamics of pulled fronts described by the Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) equation in the microscopic reaction-diffusion process A+A<-->A on the lattice when only a particle is allowed per site. To this end we identify the parameter that controls the strength of internal fluctuations in this model, namely, the number of particles per correlated volume. When internal fluctuations are suppressed, we explictly see the matching between the deterministic FKPP description and the microscopic particle model.

Interacting particles, the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov equation, and duality

The stochastic Fisher–Kolmogorov–Petrovsky–Piscunov equation is ∂ tU(x,t)=D ∂ xxU+γU(1−U)+ε U(1−U) η(x,t) for 0⩽ U⩽1 where η( x, t) is a Gaussian white noise process in space and time. Here D, γ and ε are parameters and the equation is interpreted as the continuum limit of a spatially discretized set of Itô equations. Solutions of this stochastic partial differential equation have an exact connection to the A⇌ A+ A reaction–diffusion system at appropriate values of the rate coefficients and particles’ diffusion constant. This relationship is called “duality” by the probabilists; it is not via some hydrodynamic description of the interacting particle system. In this paper we present a complete derivation of the duality relationship and use it to deduce some properties of solutions to the stochastic Fisher–Kolmogorov–Petrovsky–Piscunov equation.