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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 review various nonlinear systems with fractional derivatives of the Riesz type in space. Concretely, we consider partial differential equations with fractional Laplacians, which consider potentials of the Fermi–Pasta–Ulam, sine-Gordon, Klein–Gordon and double sine-Gordon. These regimes have the important feature that some energy functional is available in the fractional scenario. For each of these cases, we propose numerical techniques to approximate their solutions, and some discrete functionals are provided in order to approximate the energy dynamics of the systems. The methodologies are energy-preserving (conservative) techniques which are employed then to investigate the presence of nonlinear supratransmission in those systems. It is well known that such process is present in continuous Fermi–Pasta–Ulam, sine-Gordon, Klein–Gordon and double sine-Gordon regimes when the derivatives are of integer order. As one of the most important outcomes of this survey, the computer simulations confirm that this process is also present when the derivatives are of fractional order.

In this work, we investigate numerically a nonlinear wave equation with fractional derivatives of the Riesz type in space. As opposed to previously published papers which employed fractional centered differences, the present approach is based on the use of weighted and shifted Grunwald difference operators. The mathematical model has an associated energy function which is preserved under suitable parameter conditions. In this manuscript, we propose a discrete energy function that estimates the continuous counterpart and which is preserved under the same conditions. As some of the main result of this work, we show that the method is stable and second-order convergent. Moreover, we establish that the technique is quadratically consistent, and we prove the existence and uniqueness of solutions of the numerical model for arbitrary initial conditions. Some numerical results are provided in order to confirm the quadratic order of convergence of the method.

In this paper, we include simultaneously additive and multiplicative noise to the Pais–Uhlenbeck oscillator (PUO). We construct an integral of motion of the PUO with a time-dependent coefficient. Viewing the PUO as two coupled harmonic oscillators, we add noise to the corresponding frequencies. The systems are solved with the fourth-order stochastic Runge–Kutta method. Some graphics of the solutions and integrals of motion are presented, and the average deviations are calculated in order to quantify the noise influence.

In this work, we introduce two novel methodologies to compute the envelope of superficial electromyography signals. Our methods are based on the detection of activation and deactivation patterns using a change-point approach on the variances of the sample. More concretely, an iterative algorithms is proposed to select the change-points between two segments of the signal based on some local statistics introduced in this work. The signal is split up into two segments, and a new search for change-points is recursively conducted in each subsequence. The change-points make possible to calculate local envelopes which reflect the shape of the signal without ignoring the activation and deactivation landmarks. Two methods are proposed in this work, and the improvements with respect to methodologies available in the literature are shown using both synthetic and real data. A thorough analysis of the techniques is performed to that end.

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 investigate numerically a nonlinear hyperbolic partial differential equation with space fractional derivatives of the Riesz type. The model under consideration generalizes various nonlinear wave equations, including the sine-Gordon and the nonlinear Klein–Gordon models. The system considered in this work is conservative when homogeneous Dirichlet boundary conditions are imposed. Motivated by this fact, we propose a finite-difference method based on fractional centered differences that is capable of preserving the discrete energy of the system. The method under consideration is a nonlinear implicit scheme which has various numerical properties. Among the most interesting numerical features, we show that the methodology is consistent of second order in time and fourth order in space. Moreover, we show that the technique is stable and convergent. Some numerical simulations show that the method is capable of preserving the energy of the discrete system. This characteristic of the technique is in obvious agreement with the properties of its continuous counterpart.

In this work, we investigate a general nonlinear wave equation with Riesz space-fractional derivatives that generalizes various classical hyperbolic models, including the sine-Gordon and the Klein–Gordon equations from relativistic quantum mechanics. A finite-difference discretization of the model is provided using fractional centered differences. The method is a technique that is capable of preserving an energy-like quantity at each iteration. Some computational comparisons against solutions available in the literature are performed in order to assess the capability of the method to preserve the invariant. Our experiments confirm that the technique yields good approximations to the solutions considered. As an application of our scheme, we provide simulations that confirm, for the first time in the literature, the presence of the phenomenon of nonlinear supratransmission in Riesz space-fractional Klein–Gordon equations driven by a harmonic perturbation at the boundary.

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...

This present manuscript studies a nonlinear hyperbolic model in fractional form which generalizes the nonlinear Klein–Gordon system. The equation under investigation includes the presence of a time-fractional operator of the Caputo type. A space-fractional form of that equation with integer-order temporal derivative has been previously investigated to elucidate the existence of localized wave transmission in relativistic wave equations. Here, we employ numerical techniques to estimate the solution of the fractional equation. The method has consistency of fourth order in space. Meanwhile, the temporal order of consistency is equal to 3 − α . We considered herein a sinusoidal perturbation of the medium. The simulations show the existence of the transmission of localized nonlinear modes in some complex fractional media governed by hyperbolic models. Physically, the present work investigates the phenomenon of nonlinear supratransmission in a continuous generalization of linear chains of harmonic oscillators with memory effects. In particular, the present work corroborates the presence of this nonlinear phenomenon in chains of pendula with memory and arrays of Josephson junctions attached through superconducting wires and memory effects.

In this paper, we propose a stochastic model in which the values of the factors involved in the development of a glioma vary randomly in a biologically congruent range. Stability analysis revealed three fixed points which allude to patients with a growing glioma, with an advanced stage glioma and without glioma, respectively. The graphics of the solutions and the diagrams of asymptotic behavior of some of the parameters are presented. We also show the order of influence of them. The results obtained show a decay in serum glucose levels in the presence of a glioma. Moreover, they indicate that the immune system is an important element in the prevention of the growth of gliomata.

In this work, we consider a two-dimensional mathematical model that describes the growth dynamics of cancer when radiotherapy is considered. The mathematical model for the local density of the tumor is described by a parabolic partial differential equation with variable diffusion coefficient. The nonlinear reaction term considers both the logistic law of proliferation of tumor cells and the effect of a treatment against cancer. Suitable initial-boundary conditions are imposed on a bounded spatial domain, and a theorem on the existence and the uniqueness of solutions for the initial-boundary-value problem is proved. Motivated by this result, we design a finite-difference methodology to approximate the solutions of our mathematical model. The scheme is a linear method that is capable of preserving the positivity and the boundedness of the approximations. Some simulations are presented in order to illustrate the performance of the method. Among other conclusions, the numerical results show that the method is able to preserve the analytical features of the relevant solutions of the model.

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 $$\phi ^4$$ -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 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.

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.

Recently, some of the authors of the present note provided a generalization of Bumby's theorem to the injectivity of modules over some classes of monomorphisms. The question of whether Bumby's criterion could be extended to more general classes of categories was left without an affirmative response, though. However, the present note provides an extension of that theorem to Grothendieck categories. More concretely, we establish that any two injective objects in a Grothendick category are isomorphic when they are isomorphic to subobjects of each other. As a corollary, we prove that two objects have isomorphic injective envelops (if they exist) whenever each of them is isomorphic to a subobject of the other. In the way, it will be indispensable to note that the main results of our previous work are still valid under weaker conditions.Keywords: Grothendiek categories, Bumby's theorem, isomorphism criterion, injective objects, injective envelops

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...

We depart from the well-known one-dimensional Fisher’s equation from population dynamics and consider an extension of this model using Riesz fractional derivatives in space. Positive and bounded initial-boundary data are imposed on a closed and bounded domain, and a fully discrete form of this fractional initial-boundary-value problem is provided next using fractional centered differences. The fully discrete population model is implicit and linear, so a convenient vector representation is readily derived. Under suitable conditions, the matrix representing the implicit problem is an inverse-positive matrix. Using this fact, we establish that the discrete population model is capable of preserving the positivity and the boundedness of the discrete initial-boundary conditions. Moreover, the computational solubility of the discrete model is tackled in the closing remarks.

A discrete monotone iterative method is reported here to solve a space-fractional nonlinear diffusion–reaction equation. More precisely, we propose a Crank–Nicolson discretization of a reaction–diffusion system with fractional spatial derivative of the Riesz type. The finite-difference scheme is based on the use of fractional-order centered differences, and it is solved using a monotone iterative technique. The existence and uniqueness of solutions of the numerical model are analyzed using this approach, along with the technique of upper and lower solutions. This methodology is employed also to prove the main numerical properties of the technique, namely, the consistency, stability, and convergence. As an application, the particular case of the space-fractional Fisher’s equation is theoretically analyzed in full detail. In that case, the monotone iterative method guarantees the preservation of the positivity and the boundedness of the numerical approximations. Various numerical examples are provided to illustrate the validity of the numerical approximations. More precisely, we provide an extensive series of comparisons against other numerical methods available in the literature, we show detailed numerical analyses of convergence in time and in space against fractional and integer-order models, and we provide studies on the robustness and the numerical performance of the discrete monotone method. © 2019, The Author(s). Russian Foundation for Basic Research, RFBR: 19-01-00019 Consejo Nacional de Ciencia y Tecnología, CONACYT: A1-S-45928 The first author would like to acknowledge the financial support of the National Council for Science and Technology of Mexico (CONACYT). The second (and corresponding) author acknowledges financial support from CONACYT through grant A1-S-45928. ASH is financed by RFBR Grant 19-01-00019.

Departing from a two-dimensional hyperbolic system that describes the interaction between some activator and inhibitor substances in chemical reactions, we investigate a general form of that model using a finite-difference approach. The model under investigation is a nonlinear system consisting of two coupled partial differential equations with generalized reaction terms. The presence of two-dimensional diffusive terms consisting of fractional operators of the Riesz type is considered here, using spatial differentiation orders in the set (1, 2] . We impose initial conditions on a closed and bounded rectangle, and a finite-difference methodology based on the use of weighted-shifted Grunwald differences is proposed. Among the most important results of this work, we establish analytically the second-order consistency of our scheme. Moreover, the discrete 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 and wave instabilities, in agreement with some reports found in the literature on superdiffusive hyperbolic activator-inhibitor systems.

Background and objective: We propose a nonstandard computational model to approximate the solutions of a stochastic system describing the propagation of an infectious disease. The mathematical model considers the existence of various sub-populations, including humans who are susceptible to the disease, asymptomatic humans, infected humans and recovered or quarantined individuals. Various mechanisms of propagation are considered in order to describe the propagation phenomenon accurately. Methods: We propose a stochastic extension of the deterministic model, considering a random component which follows a Brownian motion. In view of the difficulties to solve the system exactly, we propose a computational model to approximate its solutions following a nonstandard approach. Results: The nonstandard discretization is fully analyzed for positivity, boundedness and stability. It is worth pointing out that these properties are realized in the discrete scenario and that they are thoroughly established herein using rigorous mathematical arguments. We provide some illustrative computational simulations to exhibit the main computational features of this approach. Conclusions: The results show that the nonstandard technique is capable of preserving the distinctive characteristics of the epidemiologically relevant solutions of the model, while other (classical) approaches are not able to do it. For the sake of convenience, a computational code of the nonstandard discrete model may be provided to the readers at their requests.

In this work, we use a computational technique with multiple properties of consistency in order to approximate solutions of a bounded β-Fermi–Pasta–Ulam lattice in two space dimensions subject to harmonic driving in two adjacent boundaries. The processes of nonlinear supratransmission and infratransmission are employed in order to propagate binary signals in the medium by means of a suitable modulation of the driving amplitude.

In this work, for the first time in the literature, we provide solid computational evidence that the phenomenon of nonlinear supratransmission is present in time-fractional sine-Gordon equations with damping. The model under study considers a Caputo-type partial derivative of order α ∈ ( 1 , 2 ] with respect to the temporal variable, and it possesses an associated continuous energy density function which is dissipated throughout time. In the present work, we determine the dispersion relation of the physical model and employ a reliable numerical technique to approximate the solutions. Associated with the finite-difference scheme, we also propose a discrete energy density function. This methodology was implemented computationally to approximate the solution of the model on a closed and bounded interval, considering sinusoidal perturbations on one end, and homogeneous Neumann conditions on the other. As the most relevant physical conclusions of our experiments, we have confirmed the presence of nonlinear supratransmission in nonlinear wave equations with time-fractional derivatives. Additionally, we have found out that nonlinear supratransmission gradually ceases to be present in the system as α tends from 2 to 1. Moreover, we provide some bifurcation diagrams depicting the relation between the driving amplitude at which supratransmission is triggered versus driving frequency. To that end, we follow a systematic approach which is relatively standard nowadays. For the sake of convenience, we provide a Matlab implementation of our code as an appendix at the end of this manuscript.

In this work, we propose a discrete mathematical system to model the evolution of the thickness of two-dimensional viscous thin films subject to a dewetting process. The continuous model under consideration is a degenerate partial differential equation that generalizes the classical thin film equation, and considers the inclusion of a singular potential. The analytical model is discretized using an exponential method that is capable of preserving the positive character of the approximations. In addition, the explicit nature of our approach results in an economic computer implementation which produces fast simulations. We provide some illustrative examples on the dynamics of the growth of thin films in the presence/absence of a dewetting process. The qualitative results exhibit the appearance of typical patterns obtained in experimental settings. The technique was validated against Bhattacharya’s method and a standard explicit discretization of the mathematical model.

The phenomenon of supratransmission in nonlinear systems with global interactions is predicted analytically for the first time. The model considered is a physically significant extension of the classical β-Fermi–Pasta–Ulam–Tsingou (β-FPUT) chain, with power-law interaction of degree α ⩾ 0. Using a relatively simple analytical theory, we derive the threshold that triggers the supratransmission process. The threshold for the short-range model is a particular case of our predictions. Our theoretical derivation shows that supratransmission is present for any power α > 1. We confirm numerically the validity of these results. Moreover, the numerical simulations also confirm the presence of supratransmission in the case of long-range interactions, in spite that the theoretical arguments are not valid for that case. It is worth pointing out that the present approach may be applied to other systems with global interactions.

The present work is the first paper in the literature to report on a Hamiltonian discretization of the (fractional) Higgs boson equation in the de Sitter space-time, and its theoretical analysis. More precisely, we design herein a numerically efficient finite-difference Hamiltonian 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 generalized potential and Riesz space-fractional derivatives of orders in ( 1 , 2 ] . An energy integral for the model is readily available, and we propose a nonlinear, implicit and consistent numerical technique based on fractional-order centered differences, with similar Hamiltonian properties in the discrete scenario. 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 de Sitter space-time, and find results qualitatively similar to those available in the literature. For the sake of convenience, we provide the Matlab code of an alternative linear discretization of the method presented in this work. This linear implicit approach is thoroughly analyzed also.

Background and objective: Epidemic models are used to describe the dynamics of population densities or population sizes under suitable physical conditions. In view that population densities and sizes cannot take on negative values, the positive character of those quantities is an important feature that must be taken into account both analytically and numerically. In particular, susceptible-infected-recovered (SIR) models must also take into account the positivity of the solutions. Unfortunately, many existing schemes to study SIR models do not take into account this relevant feature. As a consequence, the numerical solutions for these systems may exhibit the presence of negative population values. Nowadays, positivity (and, ultimately, boundedness) is an important characteristic sought for in numerical techniques to solve partial differential equations describing epidemic models. Method: In this work, we will develop and analyze a positivity-preserving nonstandard implicit finite-difference scheme to solve an advection-reaction nonlinear epidemic model. More concretely, this discrete model has been proposed to approximate consistently the solutions of a spatio-temporal nonlinear advective dynamical system arising in many infectious disease phenomena. Results: The proposed scheme is capable of guaranteeing the positivity of the approximations. Moreover, we show that the numerical scheme is consistent, stable and convergent. Additionally, our finite-difference method is capable of preserving the endemic and the disease-free equilibrium points. Moreover, we will establish that our methodology is stable in the sense of von Neumann. Conclusion: Comparisons with existing techniques show that the technique proposed in this work is a reliable and efficient structure-preserving numerical model. In summary, the present approach is a structure-preserving and efficient numerical technique which is easy to implement in any scientific language by any scientist with minimal knowledge on scientific programming.

In this work, the phenomenon of supratransmission in nonlinear sine-Gordon chains with global interactions will be predicted analytically for the first time in the literature. The model considered is an extension of the classical sine-Gordon chain which describes the dynamics of pendula attached through strings. Using a relatively simple analytical theory, we derive the threshold that triggers the supratransmission process for relatively high degrees of interaction. The threshold for the classical model is a particular case of our predictions when the degree of interaction tends to infinity. We confirm numerically the validity of our results using a systematic computational approach. We obtain numerically bifurcation diagrams for the supratransmission threshold and compare them with the theoretical predictions, obtaining a good agreement.

In this work, we provide a generalization of the β -Fermi–Pasta–Ulam–Tsingou (FPUT) chains which considers the absolute value of a power-law in a semi-infinite interval of the real line. It is well known that the β -FPUT chain shows the presence of nonlinear supratransmission, but not the α -FPUT model. The purpose of this work is to provide a generalization of the β -FPUT chain which considers potentials with arbitrary real power-laws, in such way that supratransmission is present independently of the value of the power. The systems considered in this work are all Hamiltonian, and we will use a Hamiltonian numerical discretization of the system to produce simulations. In our computational experiments, we will consider finite FPUT arrays which are perturbed sinusoidally on one end, and have a free boundary on the other. The system will be initially at rest, and the particles will be in resting position. For illustration purposes, we will consider various particular cases in order to exhibit the existence of supratransmission in those models, both in the undamped and the damped scenarios. Bifurcation diagrams will be obtained for various values of the power-law orders, and we will elucidate the behavior of the supratransmission threshold as a function of the driving amplitude and the power-law order.

In this work, a method to calculate the threshold value in the detection of activity phases of electromyographic signals is introduced. The criterion used by the method is based on the number of runs produced by the threshold process. The novelty of this technique lies in that the threshold value is theoretically determined employing the statistical properties of the number of runs. This value is adjusted then to the empirical distribution of the data in the signal. We present some experiments with synthetic and experimental signals to demonstrate the performance of the proposed method. Moreover, we compare it against other recent methods available in the literature. Our results show a superior performance with respect to other recent approaches.

We investigate analytically and numerically the conditions for wave instabilities in a hyperbolic activator-inhibitor system with species undergoing anomalous superdiffusion. In the present work, anomalous superdiffusion is modeled using the two-dimensional Weyl fractional operator, with derivative orders $\ensuremath{\alpha}\ensuremath{\in}$ [1,2]. We perform a linear stability analysis and derive the conditions for diffusion-driven wave instabilities. Emphasis is placed on the effect of the superdiffusion exponent $\ensuremath{\alpha}$, the diffusion ratio $d$, and the inertial time $\ensuremath{\tau}$. As the superdiffusive exponent increases, so does the wave number of the Turing instability. Opposite to the requirement for Turing instability, the activator needs to diffuse sufficiently faster than the inhibitor in order for the wave instability to occur. The critical wave number for wave instability decreases with the superdiffusive exponent and increases with the inertial time. The maximum value of the inertial time for a wave instability to occur in the system is ${\ensuremath{\tau}}_{max}=3.6$. As one of the main results of this work, we conclude that both anomalous diffusion and inertial time influence strongly the conditions for wave instabilities in hyperbolic fractional reaction-diffusion systems. Some numerical simulations are conducted as evidence of the analytical predictions derived in this work.

In this note, we provide a generalization of a well-known result of module theory which states that two injective modules are isomorphic when they are isomorphic to submodules of each other. More precisely, we show here that two RD-injective (respectively, pure-injective) modules over an integral domain are isomorphic if they are isomorphic to relatively divisible (respectively, pure) sub-modules of each other. Keywords: Relatively divisible submodules, pure submodules, injective modules, RD- injective modules, pure-injective modules

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...

We present a finite element scheme capable of preserving the nonnegative and bounded solutions of the generalized Burgers–Huxley equation. Proofs of existence and uniqueness of a solution to the continuous problem together with some results concerning the boundedness and the nonnegativity of the solution are given. Under appropriate conditions on the mesh and the initial and boundary data, boundedness and nonnegativity of the finite element approximation are established. An a priori error estimate for the approximation is also derived. Numerical experiments are presented which support the derived theoretical results.

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 manuscript, we follow a nonlinear approach in the problem of numerically estimating solutions of a generic, diffusive partial differential equation with nonpolynomial advection/convection, and reaction laws. The nonlocal, finite-difference methodology presented in this manuscript has the advantage of conditionally preserving the properties of non-negativity, boundedness, and temporal and spatial monotonicity of approximations, which are characteristics that are inherent to some kink-like solutions of particular forms of the mathematical model under investigation. We establish theorems on the stability of the technique, on the existence and uniqueness of non-negative and bounded solutions of our computational model, and on the property of preservation of monotonicity. As corollaries, we show that the method is capable of preserving the spatial and temporal monotonicity of numerical approximations. We provide some illustrative, numerical simulations that support our analytical results. Our simulations show that the method provides good approximations to the exact solutions of the particular problems considered and that the properties of non-negativity, boundedness, and monotonicity (both temporal and spatial) are conditionally preserved at each iteration. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 652–669, 2015

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.

In the present work, we develop a computational technique to approximate the changes of phase in temporal series associated to electric signals of muscles which perform activities at three different levels of intensity. We suppose that the temporal series are samples of independent, normally distributed random variables with mean equal to zero, and variance equal to one of three possible values, each of them associated to a certain degree of electric intensity. For example, these intensity levels may represent a leg muscle at rest, or active during a light activity (walking), or active during a highly demanding performance (jogging). The model is presented as a maximum likelihood problem involving discrete variables. In turn, this problem is transformed into a continuous one via the introduction of continuous variables with penalization parameters, and it is solved recursively through an iterative numerical method. An a posteriori treatment of the results is used in order to avoid the detection of relatively short periods of silence or activity. We perform simulations with synthetic data in order to assess the validity of our technique. Our computational results show that the method approximates well the occurrence of the change points in synthetic temporal series, even in the presence of autocorrelated sequences. In the way, we show that a generalization of a computational technique for the change-point detection of electric signals with two phases of activity (Esquivel-Frausto et al., 2010 [40]), may be inapplicable in cases of temporal series with three levels of intensity. In this sense, the method proposed in the present manuscript improves previous efforts of the authors.

In this work, we investigate numerically a system of partial differential equations that describes the interactions between populations of predators and preys. The system considers the effects of anomalous diffusion and generalized Michaelis&ndash, Menten-type reactions. For the sake of generality, we consider an extended form of that system in various spatial dimensions and propose two finite-difference methods to approximate its solutions. Both methodologies are presented in alternative forms to facilitate their analyses and computer implementations. We show that both schemes are structure-preserving techniques, in the sense that they can keep the positive and bounded character of the computational approximations. This is in agreement with the relevant solutions of the original population model. Moreover, we prove rigorously that the schemes are consistent discretizations of the generalized continuous model and that they are stable and convergent. The methodologies were implemented efficiently using MATLAB. Some computer simulations are provided for illustration purposes. In particular, we use our schemes in the investigation of complex patterns in some two- and three-dimensional predator&ndash, prey systems with anomalous diffusion.

In this work, we numerically investigate a three-dimensional nonlinear reaction-diffusion susceptible-infected-recovered hepatitis B epidemic model. To that end, the stability and bifurcation analyses of the mathematical model are rigorously discussed using the Routh&ndash, Hurwitz condition. Numerically, an efficient structure-preserving nonstandard finite-difference time-splitting method is proposed to approximate the solutions of the hepatitis B model. The dynamical consistency of the splitting method is verified mathematically and graphically. Moreover, we perform a mathematical study of the stability of the proposed scheme. The properties of consistency, stability and convergence of our technique are thoroughly analyzed in this work. Some comparisons are provided against existing standard techniques in order to validate the efficacy of our scheme. Our computational results show a superior performance of the present approach when compared against existing methods available in the literature.

In this work, we investigate numerically a one-dimensional wave equation in generalized form. The system considers the presence of constant damping and functional anomalous diffusion of the Riesz type. Reaction terms are also considered, in such way that the mathematical model can be presented in variational form when damping is not present. As opposed to previous efforts available in the literature, the reaction terms are not only functions of the solution. Instead, we consider the presence of smooth functions that depend on fractional derivatives of the solution function. Using a finite-difference approach, we propose a numerical scheme to approximate the solutions of the fractional wave equation. Along with this integrator, we propose discrete forms of the local and the total energy operators. In a first stage, we show rigorously that the energy properties of the continuous system are mimicked by our discrete methodology. In particular, we prove that the discrete system is dissipative (respectively, conservative) when damping is present (respectively, absent), in agreement with the continuous model. The theoretical numerical analysis of this system is more complicated in light of the presence of the functional form of the anomalous diffusion. To solve this problem, some novel technical lemmas are proved and used to establish the stability and the quadratic convergence of the scheme. Finally, we provide some computer simulations to show the capability of the scheme to conserve/dissipate the energy. Various fractional problems with functional forms of the anomalous diffusion of the solution are considered to that effect.

In this short communication, integrals of motion for the undamped and damped cubic-quintic Duffing oscillators with time-dependent coefficients are obtained for the first time in the literature under appropriate analytical conditions. The integrals of motion are obtained using Noether’s theorem, and the conditions for their existence are directly related to the well-known Milne–Pinney equation, which is associated to the Ermakov–Lewis systems. We perform here some numerical simulations to illustrate the validity of our analytical results.