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In this work, we study some nonlinear left-handed metamaterial lattices subjected to a harmonic driving source. Using the semi-discrete rotating-wave approximation, a nonlinear Schrodinger equation...

We consider a multidimensional system of hyperbolic equations with fractional diffusion, constant damping and nonlinear reactions. The system considers fractional Riesz derivatives, and generalizes...

In this paper, we investigate several modified exponential finite-difference methods to approximate the solution of the one-dimensional viscous Burgers' equation. Burgers' equation admits solutions that are positive and bounded under appropriate conditions. Motivated by these facts, we propose nonsingular exponential methods that are capable of preserving some structural properties of the solutions of Burgers' equation. The fact that some of the techniques preserve structural properties of the solutions is thoroughly established in this work. Rigorous analyses of consistency, stability and numerical convergence of these schemes are presented for the first time in the literature, together with estimates of the numerical solutions. The methods are computationally improved for efficiency using the Pade approximation technique. As a result, the computational cost is substantially reduced in this way. Comparisons of the numerical approximations against the exact solutions of some initial-boundary-value...

In this work, we consider a damped hyperbolic partial differential equation in multiple spatial dimensions with spatial partial derivatives of non-integer order. The equation under investigation is a fractional extension of the well-known sine-Gordon and Klein–Gordon equations from relativistic quantum mechanics. The system has associated an energy functional which is conserved in the undamped regime, and dissipated in the damped case. In this manuscript, we restrict our study to a bounded spatial domain and propose an explicit finite-difference discretization of the problem using fractional centred differences. Together with the scheme, we propose an approximation for the energy functional and show that the energy of the discrete system is conserved/dissipated when the energy of the continuous model is conserved/dissipated. The method guarantees that the energy functionals are positive, in agreement with the continuous counterparts. We show in this work that the method is a uniquely solvable, con...

Departing from a general stochastic differential equation with Brownian diffusion, we establish that the distribution of probability of the stopping time is governed by a parabolic partial differential equation. A particular form of the problem under investigation may be associated to a stochastic generalization of the well-known Paris’ law from structural mechanics, in which case, the solution of the boundary-value problem represents the probability distribution of the hitting time. An implicit, convergent and probability-based discretization to approximate the solution of the boundary-value problem is proposed in this work. Using a convenient vector representation of our scheme, we prove that the method preserves the most relevant properties of a probability distribution function, namely, the non-negativity, the boundedness from above by 1, and the monotonicity. In addition, we establish that our method is a convergent technique, and provide some illustrative comparisons against known exact solu...

In this work, we propose an efficient implementation of a finite-difference method employed to approximate the solutions of a system of partial differential equations that appears in the investigation of the growth of biological films. The associated homogeneous Dirichlet problem is discretized using a linear approach. This discretization yields a positivity- and boundedness-preserving implicit technique which is represented in vector form through the multiplication by a sparse matrix. A straightforward implementation of this methodology would require a substantial amount of computer memory and time, but the problem is conveniently coded using a continual reduction of the zero sub-matrices of the representing matrix. In addition to the conditions that guarantee the positivity and the boundedness of the numerical approximations, we establish some parametric constraints that assure that the same properties for the discrete total mass at each point of the mesh-grid and each discrete time are actually satisfi...

In this note, we establish analytically the convergence of a nonlinear finite-difference discretization of the generalized Burgers–Fisher equation. The existence and uniqueness of positive, bounded and monotone solutions for this scheme was recently established in [J. Diff. Eq. Appl. 19 (2014), pp. 1907–1920]. In the present work, we prove additionally that the method is convergent of order one in time, and of order two in space. Some numerical experiments are conducted in order to assess the validity of the analytical results. We conclude that the methodology under investigation is a fast, nonlinear, explicit, stable, convergent numerical technique that preserves the positivity, the boundedness and the monotonicity of approximations, making it an ideal tool in the study of some travelling-wave solutions of the mathematical model of interest. This note closes proposing new avenues of future research.

This note presents a weak generalization of a time-delayed partial differential equation which, in turn, generalizes the well-known Burger–Fisher and Burgers–Huxley models. In this work, we provide a full discretization which is consistent with the integro-differential equation under consideration. The main analytical result of this note establishes that the discrete temporal rate of change of the discretization yields a consistent approximation to the differential form of the integro-differential equation investigated. Some numerical examples are provided in order to assess the efficiency and effectiveness of our methodology.

In this note, we establish the property of convergence for a finite-difference discretization of a diffusive partial differential equation with generalized Burgers convective law and generalized Hodgkin–Huxley reaction. The numerical method was previously investigated in the literature and, amongst other features of interest, it is a fast and nonlinear technique that is capable of preserving positivity, boundedness and monotonicity. In the present work, we establish that the method is convergent with linear order of convergence in time and quadratic order in space. Some numerical experiments are provided in order to support the analytical results.

Departing from a system of parabolic partial differential equations that describes the interaction of a microbial colony and a substrate of nutrients, we propose a finite-difference discretization to approximate the bounded and non-negative solutions of the model. The literature establishes the existence and uniqueness of bounded and non-negative solutions of the continuous problem under suitable, analytical conditions; however, the exact determination of such solutions for arbitrary initial-boundary-value problems is a difficult task, whence the need of designing numerical techniques to approximate them is pragmatically justified. The numerical properties of existence and uniqueness of non-negative and bounded solutions are established using the theory of M-matrices. We provide some illustrative simulations to evince the fact that the method preserves the properties of non-negativity and boundedness in the practice.

Departing from a generalized Burgers–Huxley partial differential equation, we provide a Mickens-type, nonlinear, finite-difference discretization of this model. The continuous system is a nonlinear regime for which the existence of travelling-wave solutions has been established previously in the literature. We prove that the method proposed also preserves many of the relevant characteristics of these solutions, such as the positivity, the boundedness and the spatial and the temporal monotonicity. The main results provide conditions that guarantee the existence and the uniqueness of monotone and bounded solutions of our scheme. The technique was implemented and tested computationally, and the results confirm both a good agreement with respect to the travelling-wave solutions reported in the literature and the preservation of the mathematical features of interest.

We present a nonlinear method to approximate solutions of a Burgers–Huxley equation with generalized advection factor and logistic reaction. The equation under investigation possesses travelling-wave solutions that are temporally and spatially monotone functions; the travelling-wave fronts considered are bounded and connect asymptotically the stationary solutions of the model. For the linear regime, the method is consistent of first order in time and second order in space. In the nonlinear scenario, we investigate conditions under which bounded initial profiles evolve into bounded new approximations. The main results report on parametric conditions that guarantee the boundedness, the positivity and the monotonicity preservation of the method. As a consequence, our recursive method is capable of preserving the temporal and the spatial monotonicity of the solutions. We provide simulations that show that, indeed, our technique preserves the positivity, the boundedness and the temporal and spatial monotonicit...

In this work, we present a computational method to approximate the occurrence of the change-points in a temporal series consisting of independent and normally distributed observations, with equal mean and two possible variance values. This type of temporal series occurs in the investigation of electric signals associated to rhythmic activity patterns of nerves and muscles of animals, in which the change-points represent the actual moments when the electrical activity passes from a phase of silence to one of activity, or vice versa. We confront the hypothesis that there is no change-point in the temporal series, against the alternative hypothesis that there exists at least one change-point, employing the corresponding likelihood ratio as the test statistic; a computational implementation of the technique of quadratic penalization is employed in order to approximate the quotient of the logarithmic likelihood associated to the set of hypotheses. When the null hypothesis is rejected, the method provides estim...

Departing from a method to approximate the solutions of a two-dimensional generalization of the well-known Fisher's equation from population dynamics, we extend this computational technique to calculate the solutions of a FitzHugh–Nagumo model and derive conditions under which its positive and bounded analytic solutions are estimated consistently by positive and bounded numerical approximations. The constraints are relatively flexible, and they are provided exclusively in terms of the model coefficients and the computational parameters. The proofs are established with the help of the theory of M -matrices, using the facts that such matrices are non-singular, and that the entries of their inverses are positive numbers. Some numerical experiments are performed in order to show that our method is capable of preserving the positivity and the boundedness of the numerical solutions. The simulations evince a good agreement between the numerical estimations and the corresponding exact solutions derived in this work.

We introduce non-standard, finite-difference schemes to approximate nonnegative solutions of a weakly hyperbolic (that is, a hyperbolic partial differential equation in which the second-order time-derivative is multiplied by a relatively small positive constant), nonlinear partial differential equation that generalizes the well-known equation of Fisher-KPP from mathematical biology. The methods are consistent of order O(Δ t+(Δ x)2). As a means to verify the validity of the techniques, we compare our numerical simulations with known exact solutions of particular cases of our model. The results show that there is an excellent agreement between the theory and the computational outcomes.

In this work, we present a finite-difference scheme that preserves the non-negativity and the boundedness of some solutions of a FitzHugh-Nagumo equation. The method is explicit, and it approximates the solutions of the nonlinear, parabolic partial differential equation under study with a consistency of order O (Δ t+(Δ x)2) in the Dirichlet regime investigated. We give sufficient conditions in terms of the computational and the model parameters, in order to guarantee the non-negativity and the boundedness of the approximations. We also provide analyses of consistency, linear stability and convergence of the method. Our simulations establish that the properties of non-negativity and boundedness are actually preserved by the scheme when the proposed constraints are satisfied. Finally, a comparison against some second-order accurate methods reveals that our technique is easier to implement computationally, and it is better at preserving the properties of non-negativity and boundedness of the solutions of the FitzHugh-Nagumo equation under study.