Search

In this note, we investigate the existence of exact solutions of a nonlinear partial differential equation with time-dependent coefficients that generalizes the well-known nonlinear wave model with damping. The model under consideration generalizes other classical models from physics, like the nonlinear Klein–Gordon equation, the ( 1 + 1 ) -dimensional ϕ 4 -theory, the Fisher–Kolmogorov equation from population dynamics and the Hodgkin–Huxley model used in the description of the propagation of electric signals through the nervous system. An extension of the trial equation method (also known as the direct integral method) for partial differential equations with non-constant coefficients is used in this work in order to derive traveling-wave solutions in exact form.

In this work, we introduce a nonstandard finite-difference scheme to approximate positive solutions of a modified hyperbolic Fisher–Kolmogorov–Petrovski–Piscounov equation appearing in the investigation of the dynamics of some populations. The technique has a consistency of second order, and it provides nonnegative approximations for nonnegative initial profiles. The stability analysis of the method is carried out in detail, and the scheme is validated against known analytical solutions for some initial–boundary-value problems.

In this work, we investigate a parabolic equation with nonlinear reaction, and fractional diffusion and advection terms of the Riesz type. The model under investigation is a fractional generalization of the well-known Burgers–Fisher and Burgers–Huxley models from population and fluid dynamics, which are equations that admit positive, bounded and monotone solutions, some of them being traveling waves. A variable-step Bhattacharya-type finite-difference scheme based on fractional centered differences is proposed to approximate the solutions of the parabolic partial differential equation. The method is an explicit technique which, under suitable parameter conditions, is capable of preserving the positivity, the boundedness and the monotonicity of the approximations. Moreover, the method preserves the constant solutions of the fractional partial differential equation under investigation. The properties of consistency, stability and convergence of the technique are established thoroughly in this manuscript along with some a priori bounds for the numerical solutions. Some illustrative simulations and numerical comparisons against other numerical techniques are carried out in order to show that the method preserves these features of the approximations.

In this work, we consider a general class of damped wave equations in two spatial dimensions. The model considers the presence of Weyl space-fractional derivatives as well as a generic nonlinear potential. The system has an associated positive energy functional when damping is not present, in which case, the model is capable of preserving the energy throughout time. Meanwhile, the energy of the system is dissipated in the damped scenario. In this work, the Weyl space-fractional derivatives are approximated through second-order accurate fractional centered differences. A high-order compact difference scheme with fourth order accuracy in space and second order in time is proposed. Some associated discrete quantities are introduced to estimate the energy functional. We prove that the numerical method is capable of conserving the discrete variational structure under the same conditions for which the continuous model is conservative. The positivity of the discrete energy of the system is also discussed. The properties of consistency, solvability, stability and convergence of the proposed method are rigorously proved. We provide some numerical simulations that illustrate the agreement between the physical properties of the continuous and the discrete models.

Departing from a two-dimensional parabolic system that describes the spatial dynamics in a predator–prey system with Michaelis–Menten-type functional response, we investigate a general form of that model using a finite-difference approach. The model under investigation is a hyperbolic nonlinear system consisting of two coupled partial differential equations with generalized reaction terms. We impose initial conditions on a closed and bounded rectangle, and a fully discrete finite-difference methodology is proposed. Among the most important results of this work, we establish analytically the existence and uniqueness of discrete solutions, along with the second-order consistency of our scheme. Moreover, a discrete form of the energy method is employed to prove the stability and the quadratic convergence of the technique. Some numerical simulations obtained through our method show the appearance of Turing patterns in the parabolic case, in agreement with some reports found in the literature. Moreover, our simulations also show that Turing patterns are present in the hyperbolic scenario.

In this work, we design a numerically efficient finite-difference technique for the solution of a fractional extension of the Higgs boson equation in the de Sitter space–time. The model under investigation is a multidimensional equation with Riesz fractional derivatives of orders in ( 0 , 1 ) ∪ ( 1 , 2 ] , which considers a generalized potential and a time-dependent diffusion factor. An energy integral for the mathematical model is readily available, and we propose an explicit and consistent numerical technique based on fractional-order centered differences with similar Hamiltonian properties as the continuous model. A fractional energy approach is used then to prove the properties of stability and convergence of the technique. For simulation purposes, we consider both the classical and the fractional Higgs real-valued scalar fields in the ( 3 + 1 ) -dimensional de Sitter space–time, and find results qualitatively similar to those available in the literature. The present work is the first paper to report on a Hamiltonian discretization of the Higgs boson equation (both fractional and non-fractional) in the de Sitter space–time and its numerical analysis. More precisely, the present manuscript is the first paper of the literature in which a dissipation-preserving scheme to solve the multi-dimensional (fractional) Higgs boson equation in the de Sitter space–time is proposed and thoroughly analyzed. Indeed, it is worth pointing out that previous efforts used techniques based on the Runge–Kutta method or discretizations that did not preserve the dissipation nor were rigorously analyzed.

In this work, we propose a fractional extension of the multi-dimensional Gross–Pitaevskii system that describes a two-component Bose–Einstein condensate with an internal atomic Josephson junction. The fractional problem is governed by two parabolic partial differential equations that consider fractional spatial derivatives of the Riesz type along with coupling terms. Initial and homogeneous Dirichlet boundary conditions are imposed on a bounded interval of a closed and bounded domain. We show that the problem can be expressed in variational form and propose a Hamiltonian function associated to the system. We prove that the total energy of the system is constant, whence the need to provide energy-conserving schemes to solve the system is pragmatically justified. Motivated by these facts, we design a finite-difference discretization of the continuous model based on the use of fractional-order centered differences. The discrete scheme has also a variational structure, and we propose a discrete form of the Hamiltonian function. As the continuous counterpart, we prove rigorously that the discrete total energy is conserved at each temporal step. The scheme is a second-order consistent discretization of the continuous model. Moreover, we prove the stability and quadratic convergence of the numerical model. We provide some computer simulations using an implementation of our scheme to illustrate the validity of the conservation properties.

In this work, we consider a parabolic partial differential equation with fractional diffusion that generalizes the well-known Fisher’s and Hodgkin–Huxley equations. The spatial fractional derivatives are understood in the sense of Riesz, and initial–boundary conditions on a closed and bounded interval are considered here. The mathematical model is presented in an equivalent form, and a finite-difference discretization based on fractional-order centered differences is proposed. The scheme is the first explicit logarithmic model proposed in the literature to solve fractional diffusion–reaction equations. We establish rigorously the capability of the technique to preserve the positivity and the boundedness of the methodology. Moreover, we propose conditions under which the monotonicity of the numerical model is also preserved. The consistency, the stability and the convergence of the scheme are also proved mathematically, and some a priori bounds for the numerical solutions are proposed. We provide some numerical simulations in order to confirm that the method is capable of preserving the positivity and the boundedness of the approximations, and a numerical study of the convergence of the technique is carried out confirming, thus, the analytical results.

In this work, we consider multidimensional diffusion–reaction equations with time-fractional partial derivatives of the Caputo type and orders of differentiation in ( 0 , 1 ) . The models are extensions of various well-known equations from mathematical physics, biology, and chemistry. In the present manuscript, we will impose initial–boundary data on a closed and bounded spatial multidimensional domain. Single-term and multi-term fractional systems are considered in this work. In the first stage, we show that the fractional models possess energy-like functionals which are dissipated in L 2 ( Ω ) with respect to time. The systems are investigated rigorously from the analytical point of view, and dissipative numerical models to approximate their solutions are proposed and rigorously analyzed. Our discretizations will make use of the uniform L 1 approximation scheme to estimate the time-fractional derivatives, and the usual central-difference operators to approximate the spatial Laplacian. To that end, various results of the literature will be crucial, including some useful discrete forms of Paley–Wiener inequalities. Some numerical examples are included to show the asymptotic behavior of the numerical methods and, ultimately, their dissipative character.

In this work, we investigate both analytically and numerically a spatially two-dimensional advection-diffusion-reaction equation that generalizes the Burgers' and the Fisher's equations. The partial differential equation of interest is a nonlinear model for which the existence and the uniqueness of positive and bounded solutions are analytically established here. At the same time, we propose an exact finite-difference discretization of the Burgers-Fisher model of interest and show that, as the continuous counterpart, the method proposed is capable of preserving the positivity and the boundedness of the numerical approximations as well as the temporal and spatial monotonicity of the discrete initial-boundary conditions. It is shown that the method is convergent with first order in time and second order in space. We provide some simulations that illustrate the fact that the proposed technique preserves the positivity, the boundedness and the monotonicity.

We propose a numerical method to approximate the solutions of generalized forms of two bi-dimensional models of mathematical physics, namely, the Burgers-Fisher and the Burgers-Huxley equations. In one-dimensional form, the literature in the area gives account of the existence of analytical solutions for both models, in the form of traveling-wave fronts bounded within an interval I of the real numbers. Motivated by this fact, we propose a finite-difference methodology that guarantees that, under certain analytical conditions on the model and computer parameters, estimates within I will evolve discretely into new estimates which are likewise bounded within I . Additionally, we establish the preservation in the discrete domain of the skew-symmetry of the solutions of the models under study. Our computational implementation of the method confirms numerically that the properties of positivity and boundedness are preserved under the analytical constraints derived theoretically. Our simulations show a good agreement between the analytical solutions derived in the present work and the corresponding numerical approximations.

In this work, we consider a stochastic differential equation that generalizes the well known Paris' equation from fracture of materials. The model describes the propagation of cracks on solids, and it includes a deterministic summand and a stochastic component in terms of a Brownian motion. The use of Ito's stochastic integral gives an equivalent stochastic integral equation that is further generalized here. We note that the probability distribution of the stopping time of the general model satisfies a deterministic diffusion-advection partial differential equation for which the solution is known only in a reduced number of particular cases. Motivated by these analytical results, we develop a fast finite-difference method to approximate the distribution of the stopping time. The method is an explicit exponential-like technique that preserves the main features of a probability distribution, namely, the non-negativity, the boundedness from above by 1 as well as the spatial monotonicity. Moreover, the method is a monotone technique that is also capable of preserving the temporal monotonicity of the approximations. These properties of the proposed methodology are thoroughly established in the present manuscript. A continuity condition of the numerical solutions in terms of the initial conditions and the temporal computational parameter is established also, together with a limiting property of the methodology when the free parameter tends to infinity. For comparison purposes, we are providing an implicit and stable discretization of the mathematical model which has a second order of convergence but for which conditions that guarantee the positivity, the boundedness and the monotonicity of approximations are not available. The numerical simulations obtained with implementations of our techniques show that the explicit method is an efficient scheme that preserves the characteristics of interest (non-negativity, boundedness from above by 1 and monotonicity), and that the numerical approximations are in good agreement with the known exact solutions.

In this work, we investigate numerically the classical Burgers-Fisher equation using a modified Bhattacharya method. The partial differential equation under investigation possesses nonnegative and bounded solutions and, under some suitable parameter conditions, these solutions are traveling waves. It is well known that the use of the Bhattacharya approach leads to the design of numerical techniques that are sensitive to zero solutions. However, in this manuscript, we provide a correction of that technique in order to approximate solutions of the Burgers-Fisher equation that are bounded in 0 , 1 . The proposed methodology is explicit, and we establish thoroughly the capability of the technique to preserve the non-negativity, the boundedness and the monotonicity of the numerical approximations, as well as the constant solutions of the continuous model. The new class of methods introduced in this work considers the presence of a free parameter, and we show that this family tends to an explicit and standard discretization of the Burgers-Fisher equation when the free parameter tends to infinity. Some simulations illustrate the main features of the method.

In this work, we propose a discrete system to model the dynamics of individual opinions when the agents of a population have three equally likely choices. The social network consists of a finite number of agents with pairwise interactions at discrete times, and the opinion space is identified as a triangle in the plane. After a suitable homotopic transformation, one may convert the opinion space into the classical circle S 1 of the Cartesian plane. The opinion of each agent is updated following a general nonlinear law which considers individual parameters of the members. We establish conditions that guarantee the existence of attracting points (or strong consensus), and infer the existence of attracting intervals (identified here as weak consensus). Moreover, we notice that the conditions that lead to global consensuses are independent of the weight matrix and the number of agents in the network. The simulations obtained in this work confirm the validity of the analytical results.

Departing from a numerical method designed to solve ordinary differential equations, in this manuscript we extend such approach to solve problems involving fuzzy partial differential equations. The method proposed in this work is a non-recursive technique that combines differential quadrature rules and a Picard-like scheme in order to obtain general solutions of systems of partial differential equations derived from a general fuzzy partial differential model. The property of stability and a bound for the Hausdorff distance are established under suitable conditions. Several numerical examples are provided in order to show the effectiveness of the proposed technique.

This work is motivated by a generalization of the well-known one-dimensional Burgers–Fisher equation, considering a Riesz anomalous diffusion. Initial–boundary conditions which are positive and bounded are imposed on a closed and bounded spatial domain. In this manuscript we propose a finite-difference method to approximate the positive and bounded solutions of the anomalous model. The methodology is an exponential and explicit technique which is based on the use of fractional centered differences. The properties of fractional centered differences are employed to establish the existence and the uniqueness of solutions of the finite-difference method, as well as the capability of the technique to preserve the positivity, the boundedness and the monotonicity of the approximations. Additionally, we provide some a priori bounds for the numerical solutions and a consistent analysis of the scheme. Some illustrative simulations show that the method is able to preserve the positivity, the boundedness and the monotonicity of the numerical approximations.

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.

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.

In this paper we study the phenomenon of nonlinear supratransmission in a semi-infinite discrete chain of coupled oscillators described by modified sine-Gordon equations with constant external and internal damping, and subject to harmonic external driving at the end. We develop a consistent and conditionally stable finite-difference scheme in order to analyze the effect of damping in the amount of energy injected in the chain of oscillators; numerical bifurcation analyses to determine the dependence of the amplitude at which supratransmission first occurs with respect to the frequency of the driving oscillator are carried out in order to show the consequences of damping on harmonic phonon quenching and the delay of appearance of critical amplitude.