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The present work is the first manuscript of the literature in which a numerical model that preserves the dissipation of free energy is proposed to solve a time-space fractional generalized nonlinear parabolic system. More precisely, we investigate an extension of the multidimensional heat equation with nonlinear reaction and fractional derivatives in space and time. The model considers homogeneous Dirichlet boundary conditions and initial data. The temporal fractional derivative is understood in the Caputo sense, and Riesz fractional derivatives are employed in the spatial variables. The system possesses a free energy functional and we prove mathematically that it is a decreasing function of time. Motivated by these facts, we propose a discretization of the space-time-fractional model which dissipates the discrete free energy. The discrete model is obtained using fractional-order centered differences to approximate the Riesz derivatives, and the L 1 scheme to estimate the Caputo derivatives. The numerical model is a consistent approximation of the continuous system with quadratic order in space, and at most second order in time. Simulations confirm the capability of the numerical model to dissipate the free energy of the continuous fractional system.

In this work, we consider a fractional extension of the classical nonlinear wave equation, subjected to initial conditions and homogeneous Dirichlet boundary data. We consider space-fractional derivatives of the Riesz type in a bounded real interval. It is known that the problem has an associated energy which is preserved through time. The mathematical model is presented equivalently using the scalar auxiliary variable (SAV) technique, and the expression of the energy is obtained using the new scalar variable. The new differential system is discretized then following the SAV approach. The proposed scheme is a nonlinear implicit method which has an associated discrete energy, and we prove that the discrete model is also conservative. The present work is the first report in which the SAV method is used to design nonlinear conservative numerical method to solve a Hamiltonian space-fractional wave equations.

In this note, we use an extension of the trial equation method (also called the direct integral method) for partial differential equations with non-constant coefficients to derive exact solutions in the form of nonlinear waves. The model considered generalizes other classical models from physics like the Klein–Gordon equation, the $$(1 + 1)$$ -dimensional $$\phi ^4$$ -theory, the Fisher–Kolmogorov equation from population dynamics, and the Hodgkin–Huxley model which describes the propagation of electrical signals through the nervous system. As a particular example, the cylindrically symmetric cubic nonlinear Klein–Gordon equation is considered herein.

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

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.

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

In this work, we investigate a multidimensional system consisting of a finite (though arbitrary) number of coupled hyperbolic partial differential equations with fractional diffusion, constant damping and inertial times, and nonlinear reaction terms. Under suitable analytical conditions, the model has conserved quantities which are preserved in the absence of damping. We establish rigorously the conservation of the proposed quantities and, assuming that solutions of the model exist, we prove their boundedness. Motivated by these facts, we propose a finite-difference methodology to approximate the solutions of the continuous system. As its continuous counterpart, the discrete model has associated discrete quantities that estimate the Hamiltonian functional. Moreover, these quantities are preserved in the absence of damping, and they are dissipated when damping is present. To prove this feature of our finite-difference scheme, a new approximation form of the nonlinear reaction terms is proposed. This approach allows for the scheme to mimic the properties of the continuous system. The numerical properties of consistency, stability, boundedness and convergence of the scheme are proved rigorously. Some illustrative simulations confirm that the scheme is capable of preserving or dissipating the quantities, in agreement with the analytical results.

In this work, we propose a fractional extension of the one-dimensional nonlinear vibration problem on an elastic string. The fractional problem is governed by a hyperbolic partial differential equation that considers a nonlinear function of spatial derivatives of the Riesz type and constant damping. Initial and homogeneous Dirichlet boundary conditions are imposed on a bounded interval of the real line. We show that the problem can be expressed in variational form and propose a Hamiltonian function associated to the system. We prove that the total energy of the system is constant in the absence of damping, and it is non-increasing otherwise. Some boundedness properties of the solutions are established mathematically. Motivated by these facts, we design a finite-difference discretization of the continuous model based on the use of fractional-order centered differences. The discrete scheme has also a variational structure, and we propose a discrete form of the Hamiltonian function. As the continuous counterpart, we prove rigorously that the discrete total energy is conserved in the absence of damping, and dissipated when the damping coefficient is positive. The scheme is a second-order consistent discretization of the continuous model. Moreover, we prove the stability and quadratic convergence of the numerical model using a discrete form of the energy method. We provide some computer simulations using an implementation of our scheme to illustrate the validity of the conservative/dissipative properties.

In this work, we investigate numerically a diffusion equation with nonlinear reaction, defined spatially over a closed and bounded interval of the real line. The partial differential equation is expressed in an equivalent logarithmic form, and initial and Dirichlet boundary data are imposed upon the problem. An implicit finite-difference discretization of this logarithmic model is proposed then. We show that the numerical scheme is capable of preserving the constant solutions of the continuous model. Moreover, we establish the existence of positive and bounded numerical solutions using analytical arguments. The theoretical analysis of the numerical model is carried out also. In particular, we establish that the method is a consistent technique, we provide a priori bounds for the numerical solutions, and we prove the stability and the convergence of the scheme by applying a suitable Gronwall-type inequality. As one of the consequences of stability, we show not only that the solutions of the numerical model exist, but also that they are unique.

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 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.

The theoretical analysis (convergence and stability) of an L 2 − 1 σ 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 L 2 − 1 σ 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.

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.

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 $$(0 , 1) \cup (1 , 2]$$ . We impose initial conditions on a closed and bounded rectangle, and a four-step fully explicit finite-difference methodology based on the use of fractional-order centered differences is proposed. Among the most important results of this work, we establish analytically the second-order consistency of our scheme. Moreover, a discrete form of the energy method is employed to prove the stability, the boundedness 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.

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.

In this paper, we present and investigate the analytical properties of a new set of orthogonal basis functions derived from the block-pulse functions. Also, we present a numerical method based on this new class of functions to solve nonlinear Volterra–Fredholm integral equations. In particular, an alternative and efficient method based on the formalism of artificial neural networks is discussed. The efficiency of the mentioned approach is theoretically justified and illustrated through several qualitative and quantitative examples.

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.

Background and objective. We present and analyze a nonstandard numerical method to solve an epidemic model with memory that describes the propagation of Ebola-type diseases. The epidemiological system contemplates the presence of sub-populations of susceptible, exposed, infected and recovered individuals, along with nonlinear interactions between the members of those sub-populations. The system possesses disease-free and endemic equilibrium points, whose stability is studied rigorously. Methods. To solve the epidemic model with memory, a nonstandard approach based on Grunwald–Letnikov differences is used to discretize the problem. The discretization is conveniently carried out in order to produce a fully explicit and non-singular scheme. The discrete problem is thus well defined for any set of non-negative initial conditions. Results. The existence and uniqueness of the solutions of the discrete problem for non-negative initial data is thoroughly proved. Moreover, the positivity and the boundedness of the approximations is also theoretically elucidated. Some simulations confirm the validity of these theoretical results. Moreover, the simulations prove that the computational model is capable of preserving the equilibria of the system (both the disease-free and the endemic equilibria) as well as the stability of those points. Conclusions. Both theoretical and numerical results establish that the computational method proposed in this work is capable of preserving distinctive features of an epidemiological model with memory for the propagation of Ebola-type diseases. Among the main characteristics of the numerical integrator, the existence and the uniqueness of solutions, the preservation of both positivity and boundedness, the preservation of the equilibrium points and their stabilities as well as the easiness to implement it computationally are the most important features of the approach proposed in this manuscript.

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

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

We investigate a family of one-dimensional (1D) Hamiltonian semi-infinite particle lattices whose interactions involve exclusively terms of fourth order in the potential. Our aim is to examine their distinct role in the dynamics, in the absence of quadratic (harmonic) interactions, which are typically included in most studies, as they are known to play an important role in many physical phenomena. We also include in our potentials on-site terms of the sine-Gordon type, which are also considered in many studies in connection with localization effects. Our 1D lattices are subjected to sinusoidal perturbation on one end and an absorbing boundary on the other. To simulate a semi-infinite chain, we will consider a relatively long chain with string coupling. Using reliable finite difference discretization schemes, we establish the existence of nonlinear supratransmission for both short-range and long-range interactions, and demonstrate that the presence of quadratic interactions is not necessary for a system to show nonlinear supratransmission. Additionally, we provide diagrams depicting novel relations between the critical amplitude at which supratransmission is triggered versus driving frequency and a parameter measuring the length of the interactions. Our investigation also shows that the presence of on-site potentials is also not crucial for the system to present supratransmission.

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.

For the first time in literature, semi-implicit spectral approximations for nonlinear Caputo time- and Riesz space-fractional diffusion equations with both smooth and non-smooth solutions are proposed. More precisely, the governing partial differential equation generalizes the Hodgkin–Huxley, the Allen–Cahn and the Fisher–Kolmogorov–Petrovskii–Piscounov equations. The schemes employ a Legendre-based Galerkin spectral method for the Riesz space-fractional derivative, and L1-type approximations with both uniform and graded meshes for the Caputo time-fractional derivative. More importantly, by using fractional Gronwall inequalities and their associated discrete forms, sharp error estimates are proved which show an enhancement in the convergence rate compared with the standard L1 approximation on uniform meshes. This analysis encompasses both uniform meshes as well as meshes that are graded in time, and guarantees the unconditional stability. The numerical results that accompany our analysis confirm our theoretical error estimates, and give significant insights into the convergence behavior of our schemes for problems with smooth and non-smooth solutions.

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.

This work is motivated by an extension of both the Burgers–Fisher and the Burgers–Huxley equations in multiple dimensions, considering Riesz fractional diffusion. Initial–boundary conditions which are positive and bounded are imposed on a closed and bounded set, and a finite-difference method is proposed to approximate the solutions of the fractional model. The methodology is a linear and implicit technique which is based on fractional centered differences. We show in this manuscript that the method can be expressed in vector form using a Minkowski matrix under suitable conditions. The main properties of Minkowski matrices are used then to establish the existence and the uniqueness of the solutions of the finite-difference method, as well as the capability of the technique to preserve the positivity and the boundedness. Additionally we show that the method is a consistent technique which is stable and convergent, with first order of convergence in time and second order in space. Some illustrative simulations show that the scheme is capable of preserving the positivity and the boundedness of the numerical approximations.

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.

This work is motivated by the investigation of a fractional extension of a general nonlinear multidimensional wave equation with damping. The model under study considers partial derivatives of orders in $$(0, 1) \cup (1, 2]$$ with respect to the spatial variables. The undamped one-dimensional version of the model was previously used in the literature to investigate the presence of the nonlinear phenomenon of supratransmission in fractional relativistic wave equations. In agreement with the continuous counterpart, the capability of the method to preserve the total energy of the system was demonstrated then. In the present work, we show that the modified methodology is capable of preserving the dissipation of energy of the extended continuous model. We note that the discrete energy functional is positive, a fact which is in agreement with the positivity of the continuous energy. Moreover, the unique solubility of the method is established, and we show that our technique is second-order consistent, stable and quadratically convergent. Some bounds for the numerical solution are computed in the way, and various simulations are performed in order to assess the capability of the method to preserve or dissipate the energy of the system.

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 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.

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 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.

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.

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.

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.

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 provide some discretizations of a partial differential equation that generalizes the well-known Fisher’s equation from population dynamics. The mathematical model of interest is a nonlinear diffusion-reaction equation that appears in the investigation of the proliferation and motility effect of transforming growth factor β on cancer cells. Only positive and bounded solutions are physically relevant in this context, and the discretizations that we provide in this manuscript are able to preserve both properties. One of the techniques is an implicit linear method that is motivated by previous approaches of the author. On the other hand, the second method is a novel explicit exponential technique which has the advantage of requiring less computational resources and less computer time. Similar qualitative results are obtained with both methods, but the latter one is able to handle finer grid meshes. Some qualitative and quantitative comparisons are carried out in support of the advantages of the exponential scheme. It is worthwhile to note that the explicit technique used in the present manuscript has the advantage over other exponential methodologies that it yields no singularities. In addition, the preservation of the properties of non-negativity and boundedness of both the solution and the total mass are distinctive features which are established analytically in this work. The numerical simulations on cancer growth obtained with the exponential method are found to be in good agreement with the experimental results available in the literature.

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 the article [Commun Nonlinear Sci Numer Simul 2019;71:22–37], the authors provided an erroneous proof for the existence of solutions of a finite-difference model for a fractional Klein–Gordon–Zakharov equation. This letter is intended to provide a correct proof of that theorem.

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.