In this manuscript, we consider an initial-boundary-value problem governed by a (1+1)-dimensional hyperbolic partial differential equation with constant damping that generalizes many nonlinear wave ...equations from mathematical physics. The model considers the presence of a spatial Laplacian of fractional order which is defined in terms of Riesz fractional derivatives, as well as the inclusion of a generic continuously differentiable potential. It is known that the undamped regime has an associated positive energy functional, and we show here that it is preserved throughout time under suitable boundary conditions. To approximate the solutions of this model, we propose a finite-difference discretization based on fractional centered differences. Some discrete quantities are proposed in this work to estimate the energy functional, and we show that the numerical method is capable of conserving the discrete energy under the same boundary conditions for which the continuous model is conservative. Moreover, we establish suitable computational constraints under which the discrete energy of the system is positive. The method is consistent of second order, and is both stable and convergent. The numerical simulations shown here illustrate the most important features of our numerical methodology.
•The existence of positive energy invariants of a damped nonlinear wave equation with Riesz fractional Laplacian is proved.•A dissipation-preserving technique satisfying the same properties of its continuous counterpart is proposed.•The method is consistent, stable, convergent and preserves the positivity of the energy.•Numerical simulations illustrate the capability of the method to dissipate/conserve the energy.
•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.
•The dynamics of a Riesz space-fractional sine-Gordon equation is investigated numerically.•The system is perturbed harmonically at one end at a frequency in the forbidden band gap.•A sudden increase ...in the amplitude of wave signals absorbed by the system is found above a critical amplitude.•The presence of supratransmission in Riesz space-fractional sine-Gordon systems is established for the first time.
In this work, we consider a (1+1)-dimensional Riesz space-fractional damped sine-Gordon equation defined on a bounded spatial interval. Sinusoidal Dirichlet boundary data are imposed at one end of the interval and homogeneous Neumann conditions at the other. The system is initially at rest in the equilibrium position, and is discretized to simulate its complex dynamics. The method employed in this work is a finite-difference discretization of the mathematical model of interest. Our scheme is throughly validated against simulations on the dynamics of the classical and the space-fractional sine-Gordon equations, which are available in the literature. As the main result of this manuscript, we have found numerical evidence on the presence of the phenomenon of nonlinear supratransmission in Riesz space-fractional sine-Gordon systems. Simulations have been conducted in order to predict its occurrence for some values of the fractional order of the spatial derivative, and a wide range of values of the frequency of the sinusoidal perturbation at the boundary. As far as the author knows, this may be one of the first numerical reports on the existence of nonlinear supratransmission in sine-Gordon systems of Riesz space-fractional order.
•The existence of energy invariants for a multidimensional nonlinear wave equation with Riesz fractional derivatives is established.•An explicit dissipation-preserving technique satisfying the same ...properties of its continuous counterpart is proposed.•The method is a second-order consistent, stable and quadratically convergent technique.•The discrete energy operators also provide consistent approximations of the continuous energy functionals.•Numerical simulations illustrate the capability of the method to dissipate/conserve the energy.
In this work, we investigate numerically a model governed by a multidimensional nonlinear wave equation with damping and fractional diffusion. The governing partial differential equation considers the presence of Riesz space-fractional derivatives of orders in (1, 2, and homogeneous Dirichlet boundary data are imposed on a closed and bounded spatial domain. The model under investigation possesses an energy function which is preserved in the undamped regime. In the damped case, we establish the property of energy dissipation of the model using arguments from functional analysis. Motivated by these results, we propose an explicit finite-difference discretization of our fractional model based on the use of fractional centered differences. Associated to our discrete model, we also propose discretizations of the energy quantities. We establish that the discrete energy is conserved in the undamped regime, and that it dissipates in the damped scenario. Among the most important numerical features of our scheme, we show that the method has a consistency of second order, that it is stable and that it has a quadratic order of convergence. Some one- and two-dimensional simulations are shown in this work to illustrate the fact that the technique is capable of preserving the discrete energy in the undamped regime. For the sake of convenience, we provide a Matlab implementation of our method for the one-dimensional scenario.
•A fractional nonlinear chain consisting of anharmonic oscillators is proposed.•The model is an extension of the well-known α-Fermi–Pasta–Ulam chain with global interactions.•The system is perturbed ...harmonically at one end at a frequency in the forbidden band gap.•The presence of nonlinear supratransmission is established thoroughly through numerical simulations.
In this work, we introduce a spatially discrete model that is a modification of the well-known α-Fermi–Pasta–Ulam chain with damping. The system is perturbed at one end by a harmonic disturbance irradiating at a frequency in the forbidden band-gap of the classical regime, and a nonlocal coupling between the oscillators is considered using discrete Riesz fractional derivatives. We propose fully discrete expressions to approximate an energy functional of the system, and we use them to calculate the total energy of fractional chains over a relatively long period of time Fract. Diff. Appl. 4 (2004) 153–162. The approach is thoroughly tested in the case of local couplings against known qualitative results, including simulations of the process of nonlinear recurrence in the traditional chains of anharmonic oscillators. As an application, we provide evidence that the process of supratransmission is present in spatially discrete Fermi–Pasta–Ulam lattices with Riesz fractional derivatives in space. Moreover, we perform numerical experiments for small and large amplitudes of the harmonic disturbance. In either case, we establish the dependency of the critical amplitude at which supratransmission begins as a function of the driving frequency. Our results are in good agreement with the analytic predictions for the classical Fermi–Pasta–Ulam chain.
•A space-fractional array that generalizes the Josephson transmission model is proposed.•Simulations show that nonlinear supratransmission and infratransmission are present in the fractional ...array.•Nonlinear hysteresis cycles are calculated for various values of the fractional derivative order.•An application of the nonlinear bistability of the system is proposed.
In this note, we depart from a model describing the transmission of electric currents in Josephson-junction chains, and provide a fractional generalization using Riesz discrete differential operators. The fractional model considered has generalized Hamiltonian- and energy-like functionals. The model and the energy functionals are fully discretized in order to investigate numerically the complex dynamics of the system when a sinusoidal perturbation at one end of the chain is imposed. As one of the most important results in this report, we establish the persistence of the nonlinear phenomena of supratransmission and infratransmission in Riesz fractional chains. Nonlinear hysteresis loops are obtained numerically for some values of the order of the fractional derivative, and numerical simulations of the propagation of monochromatic wave signals through the transmission line are presented using the nonlinear bistability of the system.
•A numerical method to solve multidimensional systems with multiple equations is proposed.•The equations are hyperbolic space-fractional PDEs with coupled nonlinear reactions.•A wide diversity of ...systems is generalized by the mathematical model.•The method is rigorously analyzed and a parallel implementation is provided.•The code is used to exhibit three-dimensional patterns in nonlinear systems.
In this work, we consider a general multidimensional system of hyperbolic partial differential equations with fractional diffusion of the Riesz type, constant damping and coupled nonlinear reaction terms. The system generalizes many particular models from the physical sciences (including inhibitor-activator models in chemistry, diffusive nonlinear systems in population dynamics and relativistic wave equations), and considers the presence of an arbitrary number of both spatial dimensions and dependent variables. Motivated by the wide range of applications, we propose an explicit four-step finite-difference methodology to approximate the solutions of the continuous system. The properties of stability, boundedness and convergence of the scheme are proved rigorously using a discrete form of the fractional energy method. An efficient computational implementation of the scheme is also proposed in this work. It is important to recall that algorithms for space-fractional systems are computationally highly demanding. To alleviate this problem, a parallel implementation of our scheme is proposed using a vector reformulation of the numerical method. We provide some illustrative simulations on the formation of complex patterns in the two-dimensional scenario, and even in the computationally intense three-dimensional case. For the sake of convenience, an algorithmic presentation of our computational model is provided in this manuscript.
•A stochastic model for the propagation of diseases is deduced from epidemiological assumptions.•The reproductive number and equilibria of the deterministic system are calculated.•A nonstandard ...scheme to solve the stochastic system is proposed and theoretically analyzed.•The simulations show that the scheme is epidemiologically more robust than other approaches.
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.
•Nonlinear time-fractional diffusion equations with multiple delays are numerically studied.•An L2−1σ difference method is proposed.•The existence and uniqueness of numerical solutions is thoroughly ...proved.•A novel discrete fractional Gronwall-type inequality is established.•The efficiency analysis is effectively carried out using Gronwall’s inequality.
The theoretical analysis (convergence and stability) of an L2−1σ difference method is presented for nonlinear time-fractional diffusion equations with multiple delays. In this communication, we state and prove a new discrete form of a fundamental fractional Gronwall inequality. For the problems under consideration, that result and the L2−1σ formula are the cornerstones to the establishment of the optimal error estimates of our fully discrete linear difference scheme. Our numerical simulations confirm the validity of the theoretical findings. Moreover, both the analytical and the numerical results demonstrate superior rates of convergence when compared to similar methods of the literature.
•The existence of invariants for a fractional Klein–Gordon–Zakharov equation is established.•A second-order numerical model is proposed to solve the continuous model.•The model has discrete forms of ...the invariants which are also conserved.•The existence, uniqueness and boundedness of the numerical solutions are proved.•We prove stability and convergence using a form of the discrete energy method.
Departing from a fractional extension of the well-known one-dimensional Klein–Gordon–Zakharov system, we propose a numerically efficient model to approximate its solutions. The continuous model under investigation considers fractional derivatives of the Riesz type in space, with orders of differentiation in (1, 2. In analogy with the non-fractional regime, the existence of a positive conserved energy quantity is established in this work. Motivated by this fact, the design of the numerical model focuses on the preservation of the energy. Using fractional-order centered differences to approximate the fractional partial derivatives, we propose a numerical model that preserves a positive discrete form of the energy. The existence and uniqueness of solutions are thoroughly established using fixed-point arguments, and the usual argument with Taylor polynomials is employed to prove the consistency of the numerical model. Some suitable bounds in terms of the energy invariants are found for the solutions of the numerical model. Moreover, using an extension of the energy method for fractional systems, we establish the stability and the convergence properties of the methodology. Finally, we provide some examples to illustrate the accuracy of our numerical implementation, including a numerical study of the convergence rate of the scheme that confirms the validity of the analytical results derived in this work.
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