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In this note, we propose a mathematical model for the efficiency of a silicon solar cell. The model is rigorously derived from continuity and transport assumptions, along with the Poisson equations to describe the behavior of charge carriers in a semiconductor. The photovoltaic phenomenon was thoroughly studied to develop the model, and some parameters that play a crucial role in its efficiency have been identified. Those parameters include the diffusion, the mobility and the absorption coefficients, the permittivity, the regions width and the minority carrier lifetime. The mathematical model obtained herein is a quasi-linear second-order elliptic-parabolic partial differential equation system with strong coupling and interface. The model is defined over a spatial domain composed of three regions with different electrical properties. The fundamental physical laws of the photovoltaic effect are employed here to develop the mathematical model and its boundary conditions.

Highlights • Mathematical considerations of a SEIQR model to describe the propagation of COVID-19 are proposed. • We derive analytically the reproductive number and the equilibria with and without COVID-19. • Necessary and sufficient conditions for the stability of the equilibria are mathematically established. • An efficient nonstandard method to solve the continuous problem is proposed and analyzed. • The numerical simulations confirm the analytical and numerical results derived in this work., In this manuscript, we develop a mathematical model to describe the spreading of an epidemic disease in a human population. The emphasis in this work will be on the study of the propagation of the coronavirus disease (COVID-19). Various epidemiologically relevant assumptions will be imposed upon the problem, and a coupled system of first-order ordinary differential equations will be obtained. The model adopts the form of a nonlinear susceptible-exposed-infected-quarantined-recovered system, and we investigate it both analytically and numerically. Analytically, we obtain the equilibrium points in the presence and absence of the coronavirus. We also calculate the reproduction number and provide conditions that guarantee the local and global asymptotic stability of the equilibria. To that end, various tools from analysis will be employed, including Volterra-type Lyapunov functions, LaSalle’s invariance principle and the Routh–Hurwitz criterion. To simulate computationally the dynamics of propagation of the disease, we propose a nonstandard finite-difference scheme to approximate the solutions of the mathematical model. A thorough analysis of the discrete model is provided in this work, including the consistency and the stability analyses, along with the capability of the discrete model to preserve the equilibria of the continuous system. Among other interesting results, our numerical simulations confirm the stability properties of the equilibrium points.

In this work, we investigate the Lagrangians and potentials for two coupled damped Duffing oscillators both directionally and bi-directionally. We show that, although it is not always possible to define a potential in dissipative systems, the potential of our model can be defined if the damping coefficient has a logarithmic derivative form. It is possible to apply these results to the analysis of the dynamics of complex networks based on three-node motif configurations. As an example, we study numerically the dynamics for one of the thirteen different possible configurations. Here, the phenomenon of synchronization is observed in terms of the coupling parameter.

In this manuscript, we will consider a parabolic partial differential equation that models the spatial-temporal dynamics in predator–prey systems with Michaelis–Menten-type functional response. We study numerically an extension of those systems by employing finite differences. The extension includes the presence of inertial times with extended reaction components in a diffusive system consisting of two populations. The spatial domain is a closed and bounded rectangle, and initial-boundary conditions are imposed on the model. An easy-to-implement fully discrete finite-difference methodology is designed here. We establish analytically the existence and uniqueness of discrete solutions as a consequence of the explicit character of the scheme, along with the second-order consistency. To prove the stability and the second-order convergence of the technique, the energy method will be applied conveniently in a discrete manner. From the point of view of the applications, we provide various numerical simulations that show the appearance of complex patterns in the parabolic case. The results will resemble qualitatively those of some papers already published in the specialized literature. Our simulations also show that different patterns are present in the hyperbolic scenario. We also perform a numerical analysis of the convergence of the method.

In this work, we introduce and theoretically analyze a relatively simple numerical algorithm to solve a double-fractional condensate model. The mathematical system is a generalization of the famous Gross–Pitaevskii equation, which is a model consisting of two nonlinear complex-valued diffusive differential equations. The continuous model studied in this manuscript is a multidimensional system that includes Riesz-type spatial fractional derivatives. We prove here the relevant features of the numerical algorithm, and illustrative simulations will be shown to verify the quadratic order of convergence in both the space and time variables.

In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work.

We investigate a generalization of the equation curlw→=g→ to an arbitrary number n of dimensions, which is based on the well-known Moisil–Teodorescu differential operator. Explicit solutions are derived for a particular problem in bounded domains of Rn using classical operators from Clifford analysis. In the physically significant case n=3, two explicit solutions to the div-curl system in exterior domains of R3 are obtained following different constructions of hyper-conjugate harmonic pairs. One of the constructions hinges on the use of a radial integral operator introduced recently in the literature. An exterior Neumann boundary-value problem is considered for the div-curl system. That system is conveniently reduced to a Neumann boundary-value problem for the Laplace equation in exterior domains. Some results on its uniqueness and regularity are derived. Finally, some applications to the construction of solutions of the inhomogeneous Lamé–Navier equation in bounded and unbounded domains are discussed.

This manuscript studies a double fractional extended p-dimensional coupled Gross–Pitaevskii-type system. This system consists of two parabolic partial differential equations with equal interaction constants, coupling terms, and spatial derivatives of the Riesz type. Associated with the mathematical model, there are energy and non-negative mass functions which are conserved throughout time. Motivated by this fact, we propose a finite-difference discretization of the double fractional Gross–Pitaevskii system which inherits the energy and mass conservation properties. As the continuous model, the mass is a non-negative constant and the solutions are bounded under suitable numerical parameter assumptions. We prove rigorously the existence of solutions for any set of initial conditions. As in the continuous system, the discretization has a discrete Hamiltonian associated. The method is implicit, multi-consistent, stable and quadratically convergent. Finally, we implemented the scheme computationally to confirm the validity of the mass and energy conservation properties, obtaining satisfactory results.

This article proposes two conformal Solow models (with and without migration), accompanied by simulations for six Organisation for Economic Co-operation and Development economies. The models are proposed by employing suitable Inada conditions on the Cobb–Douglas function and making use of the truncated M-derivative for the Mittag–Leffler function. In the exact solutions derived in this manuscript, two new parameters play an important role in the convergence towards, or the divergence from, the steady state of capital and per capita product. The economical dynamics of these nations are influenced by the intensity of the capital and labor factors, as well as the level of depreciation, the labor force rate and the level of saving.

In this paper, we propose a high-performance implementation of a space-fractional FitzHugh–Nagumo model. Our implementation is based on a positivity- and boundedness-preserving finite-difference model to approximate the solutions of a Riesz space-fractional reaction-diffusion equation. The model generalizes the FitzHugh–Nagumo model. The stability and convergence of the difference scheme are thoroughly discussed. Moreover, we prove the existence and uniqueness of numerical solutions, positivity, boundedness and consistency of the model. The scheme is based on weighted and shifted Grunwald differences. The conjugate gradient method is used then to solve the sparse matrix system. The MPI and PETSc libraries are used for the computational implementation. We investigate the influence of some computer factors on the performance of our implementation and scalability. More precisely, we consider the number of cores, the size of the computation mesh and the orders of the fractional derivatives. Tests are evaluated on a ccNUMA architecture with two CPUs.

The present work departs from an extended form of the classical multi-dimensional Gross–Pitaevskii equation, which considers fractional derivatives of the Riesz type in space, a generalized potential function and angular momentum rotation. It is well known that the classical system possesses functionals which are preserved throughout time. It is easy to check that the generalized fractional model considered in this work also possesses conserved quantities, whence the development of conservative and efficient numerical schemes is pragmatically justified. Motivated by these facts, we propose a finite-difference method based on weighted-shifted Grünwald differences to approximate the solutions of the generalized Gross–Pitaevskii system. We provide here a discrete extension of the uniform Sobolev inequality to multiple dimensions, and show that the proposed method is capable of preserving discrete forms of the mass and the energy of the model. Moreover, we establish thoroughly the stability and the convergence of the technique, and provide some illustrative simulations to show that the method is capable of preserving the total mass and the total energy of the generalized system.

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.

We consider a family of 1-dimensional Hamiltonian systems consisting of a large number of particles with on-site potentials and global (long range) interactions. The particles are initially at rest at the equilibrium position, and are perturbed sinusoidally at one end using Dirichlet data, while at the other end we place an absorbing boundary to simulate a semi-infinite medium. Using such a lattice with quadratic particle interactions and Klein-Gordon type on-site potential, we use a parameter $0\leq\alphahigher amplitudes the longer the range of interactions, reaching a maximum at a value $\alpha=\alpha_{max} \lesssim 1.5$ that depends on $\Omega$. Below this $\alpha_{max}$ supratransmission thresholds decrease sharply to values lower than the nearest neighbor $\alpha=\infty$ limit. We give a plausible argument for this phenomenon and conjecture that similar results are present in related systems such as the sine-Gordon, the nonlinear Klein-Gordon and the double sine-Gordon type.

This manuscript introduces a discrete technique to estimate the solution of a double-fractional two-component Bose–Einstein condensate. The system consists of two coupled nonlinear parabolic partial differential equations whose solutions are two complex functions, and the spatial fractional derivatives are interpreted in the Riesz sense. Initial and homogeneous Dirichlet boundary data are imposed on a multidimensional spatial domain. To approximate the solutions, we employ a finite difference methodology. We rigorously establish the existence of numerical solutions along with the main numerical properties. Concretely, we show that the scheme is consistent in both space and time as well as stable and convergent. Numerical simulations in the one-dimensional scenario are presented in order to show the performance of the scheme. For the sake of convenience, A MATLAB code of the numerical model is provided in the appendix at the end of this work.

In this manuscript, we derive integrals of motion for general anharmonic oscillators with damping and power-law forcing. The model under investigation has time-dependent coefficients, and the determination of these physical quantities is carried out using Noether’s theorem. The solutions must satisfy appropriate analytical conditions for the proposed quantities to be true integrals of motion. In turn, these analytical conditions are associated to well known physical systems, including the Milne-Pinney and Ermakov-Lewis models. We provide various numerical solutions of our equations of motion and the associated integrals to verify the theoretical results.

We investigate a model of spatio-temporal spreading of human immunodeficiency virus HIV-1. The mathematical model considers the presence of various components in a human tissue, including the uninfected CD4+T cells density, the density of infected CD4+T cells, and the density of free HIV infection particles in the blood. These three components are nonnegative and bounded variables. By expressing the original model in an equivalent exponential form, we propose a positive and bounded discrete model to estimate the solutions of the continuous system. We establish conditions under which the nonnegative and bounded features of the initial-boundary data are preserved under the scheme. Moreover, we show rigorously that the method is a consistent scheme for the differential model under study, with first and second orders of consistency in time and space, respectively. The scheme is an unconditionally stable and convergent technique which has first and second orders of convergence in time and space, respectively. An application to the spatio-temporal dynamics of HIV-1 is presented in this manuscript. For the sake of reproducibility, we provide a computer implementation of our method at the end of this work.

Fungal mycoses have become an important health and environmental concern due to the numerous deleterious side effects on the well-being of plants and humans. Antifungal therapy is limited, expensive, and unspecific (causes toxic effects), thus, more efficient alternatives need to be developed. In this work, Copper (I) Iodide (CuI) nanomaterials (NMs) were synthesized and fully characterized, aiming to develop efficient antifungal agents. The bioactivity of CuI NMs was evaluated using Sporothrix schenckii and Candida albicans as model organisms. CuI NMs were prepared as powders and as colloidal suspensions by a two-step reaction: first, the CuI2 controlled precipitation, followed by hydrazine reduction. Biopolymers (Arabic gum and chitosan) were used as surfactants to control the size of the CuI materials and to enhance its antifungal activity. The materials (powders and colloids) were characterized by SEM-EDX and AFM. The materials exhibit a hierarchical 3D shell morphology composed of ordered nanostructures. Excellent antifungal activity is shown by the NMs against pathogenic fungal strains, due to the simultaneous and multiple mechanisms of the composites to combat fungi. The minimum inhibitory concentration (MIC) and minimum fungicidal concentration (MFC) of CuI-AG and CuI-Chitosan are below 50 g/mL (with 5 h of exposition). Optical and Atomic Force Microscopy (AFM) analyses demonstrate the capability of the materials to disrupt biofilm formation. AFM also demonstrates the ability of the materials to adhere and penetrate fungal cells, followed by their lysis and death. Following the concept of safe by design, the biocompatibility of the materials was tested. The hemolytic activity of the materials was evaluated using red blood cells. Our results indicate that the materials show an excellent antifungal activity at lower doses of hemolytic disruption.

Reaction-diffusion-advection equations provide precise interpretations for many important phenomena in complex interactions between natural and artificial systems. This paper studies second-order semi-discretizations for the numerical solution of reaction-diffusion-advection equations modeling quenching types of singularities occurring in numerous applications. Our investigations particularly focus at cases where nonuniform spatial grids are utilized. Detailed derivations and analysis are accomplished. Easy-to-use and highly effective second-order schemes are acquired. Computational experiments are presented to illustrate our results as well as to demonstrate the viability and capability of the new methods for solving singular quenching problems on arbitrary grid platforms.

In this work, we propose a time‐delayed reaction–diffusion model to describe the propagation of infectious viral diseases like COVID‐19 The model is a two‐dimensional system of partial differential equations that describes the interactions between disjoint groups of a human population More precisely, we assume that the population is conformed by individuals who are susceptible to the virus, subjects who have been exposed to the virus, members who are infected and show symptoms, asymptomatic infected individuals, and recovered subjects Various realistic assumptions are imposed upon the model, including the consideration of a time‐delay parameter which takes into account the effects of social distancing and lockdown We obtain the equilibrium points of the model and analyze them for stability Moreover, we examine the bifurcation of the system in terms of one of the parameters of the model To simulate numerically this mathematical model, we propose a time‐splitting nonlocal finite‐difference scheme The properties of the model are thoroughly established, including its capability to preserve the positivity of solutions, its consistency, and its stability Some numerical experiments are provided for illustration purposes

In the present work, we investigate the efficiency of a numerical scheme to solve a nonlinear time-fractional heat equation with sufficiently smooth solutions, which was previously reported in the literature [Fract. Calc. Appl. Anal. 16: 892&ndash, 910 (2013)]. In that article, the authors established the stability and consistency of the discrete model using arguments from Fourier analysis. As opposed to that work, in the present work, we use the method of energy inequalities to show that the scheme is stable and converges to the exact solution with order O(&tau, 2&minus, &alpha, +h4), in the case that 0&lt, &lt, 1 satisfies 3&alpha, &ge, 32, which means that 0.369⪅&alpha, &le, 1. The novelty of the present work lies in the derivation of suitable energy estimates, and a discrete fractional Gr&ouml, nwall inequality, which is consistent with the discrete approximation of the Caputo fractional derivative of order 0&lt, 1 used for that scheme at tk+1/2.

A class of one-dimensional time-fractional parabolic differential equations with delay effects of functional type in the time component is numerically investigated in this work. To that end, a compact difference scheme is constructed for the numerical solution of those equations based on the idea of separating the current state and the prehistory function. In these terms, the prehistory function is approximated by means of an appropriate interpolation–extrapolation operator. A discrete form of the fractional Gronwall inequality is employed to provide an optimal error estimate. The existence and uniqueness of the numerical solutions, the order of approximation error for the constructed scheme, the stability and the order of convergence are mathematically investigated in this work. © 2019 Wiley Periodicals, Inc. Russian Foundation for Basic Research, RFBR: 19‐01‐00019 A1‐S‐45928 The authors want to thank the associate editor in charge of handling this manuscript and anonymous reviewers for all their comments and criticisms. Their suggestions contributed decisively to improve the overall quality of this work. The first two authors wish to acknowledge the support of RFBR Grant 19‐01‐00019. Meanwhile, the last author wishes to acknowledge the financial support of the National Council for Science and Technology of Mexico through grant A1‐S‐45928.

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.

This work is motivated by a generalization of the well-known Burgers–Fisher and Burgers–Huxley equations in multiple dimensions, considering Riesz fractional diffusion and convection. Initial–boundary conditions which are positive and bounded are imposed on a closed and bounded rectangular domain. In this manuscript we propose a finite-difference method to approximate the positive and bounded solutions of the fractional model. The methodology is a linear three-steps Crank–Nicolson 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 and the boundedness of the approximations. We show in this work that the method is capable of preserving some of the constant solutions of the continuous model. Additionally, we prove that our technique is a second-order consistent, stable and quadratically convergent scheme. Suitable bounds for the numerical solutions are also derived in this work. Finally, some illustrative simulations show that the method is able to preserve the positivity and the boundedness of the numerical approximations, in agreement with the analytic results proved in this work. Numerical comparisons provided in this work confirm the rate of convergence of the numerical technique.

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 work, we consider a model for the first hitting time of a moving boundary problem associated to some stochastic processes. In addition to a Gaussian component, the model investigated in this manuscript includes a general deterministic source. In either case, the coefficients of the deterministic and the stochastic terms are general functions with suitable analytical properties that guarantee that the process under investigation possesses continuous paths. As the cornerstone of this manuscript, we establish that the cumulative distribution of probability of the first hitting time is governed by a deterministic advection–diffusion partial differential equation subject to initial–boundary data, for which the exact solution is known only in certain specific scenarios. Motivated by these limitations, we propose a Crank–Nicolson discretization of the deterministic model which is capable of preserving the main structural features of probability distributions, namely, the non-negativity, the boundedness from above by 1 and the monotonicity. To guarantee the preservation of those properties, relatively flexible conditions on the parameters need to be imposed. Additionally, we also prove that our scheme is a consistent, and unconditionally stable and convergent technique which has second order of convergence. Some illustrative simulations demonstrate that the order of convergence is indeed quadratic, and the comparisons against the known exact solutions establish that the method preserves the same structural properties of the relevant solutions.

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.

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.

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.

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.

Departing from a general stochastic model for a moving boundary problem, we consider the density function of probability for the first passing time. It is well known that the distribution of this random variable satisfies a problem ruled by an advection&ndash, diffusion system for which very few solutions are known in exact form. The model considers also a deterministic source, and the coefficients of this equation are functions with sufficient regularity. A numerical scheme is designed to estimate the solutions of the initial-boundary-value problem. We prove rigorously that the numerical model is capable of preserving the main characteristics of the solutions of the stochastic model, that is, positivity, boundedness and monotonicity. The scheme has spatial symmetry, and it is theoretically analyzed for consistency, stability and convergence. Some numerical simulations are carried out in this work to assess the capability of the discrete model to preserve the main structural features of the solutions of the model. Moreover, a numerical study confirms the efficiency of the scheme, in agreement with the mathematical results obtained in this work.

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.

For the first time in the literature, a dissipation-preserving computational technique to approximate the solutions of a dissipative generalization of the Higgs boson equation in the de Sitter space–time is proposed. The model is a multidimensional system which considers a generalized potential and a general time-dependent diffusion coefficient. The system has an associated energy functional which is dissipated in time. Motivated by this fact, we propose a finite-difference methodology to approximate the solutions of the mathematical model. In addition to the numerical approximation for the solutions of the system, we propose also a discrete energy functional which is dissipated with respect to the discrete time. Using a computer implementation in two spatial dimensions, we provide some simulations that confirm the presence of bubble-like solutions, in agreement with the theory available in the literature.

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 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.