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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.

The present work introduces a numerical scheme that preserves the dissipation of energy of the Higgs boson equation in the de Sitter space-time. More precisely, the model considered in this work is a mathematical generalization of Higgs' model which includes a general time-dependent diffusion coefficient and a generalized potential. The mathematical system is dissipative, and we propose an implicit discrete method which approximates consistently the radially symmetric solutions of the continuous system. At the same time, a discrete energy functional is presented, and we prove that, as its continuous counterpart, the numerical technique dissipates the energy of the discrete system. The properties of consistency, stability and convergence of the numerical model are proved rigorously. To confirm the theoretical results, we approximate some radially symmetric solutions of the classical Higgs boson equation in the de Sitter space-time. In particular, the numerical results confirm the stability and the formation of bubble-like solutions.

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 work, we consider a fractional extension of the Klein–Gordon–Zakharov system which describes the propagation of strong turbulences on the Langmuir wave in a high-frequency plasma. Both components consider space-fractional derivatives of the Riesz type, and initial-boundary conditions are imposed on a closed and bounded interval of the real numbers. In a first stage, we show that the total energy of the system is conserved, and that the global solutions of the system are bounded. Motivated by these results, we propose a finite-difference scheme to approximate the solutions, and a discrete form of the energy functional. The advantage of the discretization proposed in this work lies in that the difference equations to solve the component equations are decoupled. This implies that the numerical schemes can be solved separately at each temporal step. We establish rigorously the existence of solutions, as well as the capability of the scheme to conserve the discrete energy. The method has a second-order consistency in both space and time. Moreover, using a discrete form of the energy method, we establish mathematically that the finite-difference scheme is stable and quadratically convergent. We provide some simulations to show that the proposed methodology is quadratically convergent and that it preserves the total energy of the system.

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 the article Martinez et al. (2019) [7] , the authors assumed the triviality of the existence of solutions of a finite-difference model for a fractional Klein–Gordon–Zakharov equation. It turns out that property is not trivial at all. In this short communication, we provide a proof of that result and, in the way, we mention some wrong proofs of similar results available in the literature. We believe that the report on the demonstration of this result will be a useful tool for researchers working on the numerical analysis of partial differential equations.

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.

Departing from an initial-boundary-value problem governed by a Klein–Gordon–Zakarov system with fractional derivatives in the spatial variable, we provide an explicit finite-difference scheme to approximate its solutions. In agreement with the continuous system, the method proposed in this work is also capable of preserving the energy of the system, and the energy quantities are nonnegative under flexible parameter conditions. The boundedness, the consistency, the stability and the convergence of the technique are also established rigorously. The method is easy to implement, and the computer simulations confirm the main analytical and numerical properties of the new model.

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 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.

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.

In this work, we investigate numerically some relevant solutions of a diffusion problem with a general reaction term. The system under investigation is governed by a partial differential equation that generalizes the classical Fisher's equation and the Hodgkin–Huxley model. In a first stage, we provide an equivalent presentation of the problem, and recall an explicit logarithmic numerical model to solve it. As one of the most important outcomes of this work, we prove that logarithmic scheme is a structure-preserving model under suitable conditions. Concretely, we show that the numerical model is capable of preserving the positivity, the boundedness and the monotonicity of the approximations. This feature of the numerical integrator is in agreement with the fact that the continuous model has solutions with the same characteristics. Also, for the first time in the literature, we show that the logarithmic models are consistent schemes which are stable and convergent. Some numerical simulations confirm that the numerical model is a convergent technique, with first order of convergence in time and second order in space.

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.

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.

In this work, we design a numerically efficient finite-difference 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 Riesz fractional derivatives of orders in ( 0 , 1 ) ∪ ( 1 , 2 ] , which considers a generalized potential and a time-dependent diffusion factor. An energy integral for the mathematical model is readily available, and we propose an explicit and consistent numerical technique based on fractional-order centered differences with similar Hamiltonian properties as the continuous model. 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 ( 3 + 1 ) -dimensional de Sitter space–time, and find results qualitatively similar to those available in the literature. The present work is the first paper to report on a Hamiltonian discretization of the Higgs boson equation (both fractional and non-fractional) in the de Sitter space–time and its numerical analysis. More precisely, the present manuscript is the first paper of the literature in which a dissipation-preserving scheme to solve the multi-dimensional (fractional) Higgs boson equation in the de Sitter space–time is proposed and thoroughly analyzed. Indeed, it is worth pointing out that previous efforts used techniques based on the Runge–Kutta method or discretizations that did not preserve the dissipation nor were rigorously analyzed.

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.

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.

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 consider multidimensional diffusion–reaction equations with time-fractional partial derivatives of the Caputo type and orders of differentiation in ( 0 , 1 ) . The models are extensions of various well-known equations from mathematical physics, biology, and chemistry. In the present manuscript, we will impose initial–boundary data on a closed and bounded spatial multidimensional domain. Single-term and multi-term fractional systems are considered in this work. In the first stage, we show that the fractional models possess energy-like functionals which are dissipated in L 2 ( Ω ) with respect to time. The systems are investigated rigorously from the analytical point of view, and dissipative numerical models to approximate their solutions are proposed and rigorously analyzed. Our discretizations will make use of the uniform L 1 approximation scheme to estimate the time-fractional derivatives, and the usual central-difference operators to approximate the spatial Laplacian. To that end, various results of the literature will be crucial, including some useful discrete forms of Paley–Wiener inequalities. Some numerical examples are included to show the asymptotic behavior of the numerical methods and, ultimately, their dissipative character.

We study the phenomenon of supratransmission (that is, the sudden transmission of energy above a sharp threshold amplitude of periodic boundary excitations) on an N -particle one-dimensional β -Fermi–Pasta–Ulam (FPU) lattice, under different ranges of interactions. When the system is at rest, we set the first particle in sinusoidal motion at a frequency that lies in the forbidden band-gap of the linear nearest-neighbor FPU chain, and use reliable computational techniques to extend earlier results of energy supratransmission to FPU chains with varying ranges of particle interactions than simply nearest neighbors. We thus discover that supratranmission arises at similar amplitude thresholds as the nearest neighbor case, for a much wider class of short-range interaction systems. However, as the interaction range becomes longer and longer, the threshold amplitude for supratransmission recedes to higher and higher values. It would be highly desirable to verify these results analytically as well as experimentally in superconducting Josephson systems, or arrays of coupled optical waveguides.

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.

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 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.

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.

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.

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.

In this work, we investigate numerically Higgs’ boson equation in the de Sitter space-time, which is an important ( 3 + 1 ) -dimensional model arising in particle physics. Associated with this model, there exists a functional of energy which is dissipated with respect to time. Moreover, some relevant solutions of this model present radial symmetry in three spatial dimensions. In this work, we consider a general Hamiltonian form of this model and determine its expression in radial coordinates. The energy functional and the rate of change of the energy are also expressed in radial coordinates. Using a finite-difference approach, we propose a numerical method to approximate the solutions of the continuous model. At the same time, a discrete energy functional is proposed and we show that, as the total energy of the continuous model, the energy of the discrete system is dissipated with respect to time. We analyze the finite-difference scheme theoretically, and we prove that the discrete model and the discrete energy density are consistent discretizations of their continuous counterparts. We prove that the numerical model is stable and convergent, and provide some estimates of the numerical error. Some simulations on the formation of bubble-like solutions are provided for illustration purposes, and we provide comparisons against numerical schemes available in the literature.

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 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.

It is well known that the simulation of fractional systems is a difficult task from all points of view. In particular, the computer implementation of numerical algorithms to simulate fractional systems of partial differential equations in three dimensions is a hard task which has no been solved satisfactorily. In this work, we propose a numerical method to solve systems of hyperbolic (fractional o non-fractional) partial differential equations that generalize various known models from physics, chemistry and biology. The scheme is an explicit technique which has the advantage of being easy to implement for any scientist with minimal knowledge on scientific programming. We propose a computer implementation which exploits the advantages of the efficient matrix algebra already available in Fortran and other languages. The algorithm is presented mathematically as well as in pseudo-code, and a raw Fortran implementation of the computer algorithm is provided in the appendix. This code is susceptible to be compiled in parallel using OpenMP, whence it follows that the computer time can be substantially reduced. As application, we provide some illustrative simulations on the formation of Turing patterns in a three-dimensional system of inhibitor–activator substances in physics. The graphs were obtained using functions of Matlab with the numerical outputs generated by our Fortran code.

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.

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

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

In this work, we design a numerical method to approximate some solutions of a generalization of a nonlinear diffusion-reaction model which appears in the context of population dynamics. The existence of traveling-wave solutions for the equation under consideration is a well-known fact. Some of such solutions are positive, bounded from above, and monotone in both space and time. Motivated by these facts, we propose an explicit, nonlinear, finite-difference methodology to approximate consistently the solutions of the model under investigation. In the linear regime, the method is consistent of first order in time and of second order in space. Under certain, flexible parameter conditions, the method is capable of preserving the positivity, the boundedness, and the spatial and the temporal monotonicity of the traveling-wave solutions. Moreover, we establish analytically and numerically that the nonlinear method is conditionally stable. A computational implementation of our technique shows that the method preserves in practice the mathematical features of interest of the exact solutions considered.

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

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

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 ) ∪ ( 1 , 2 ] . We impose initial conditions on a closed and bounded rectangle, and a finite-difference methodology based on the use of fractional centered differences is proposed. Among the most important results of this work, we prove the existence and the uniqueness of the solutions of the numerical method, and 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 presence of Turing patterns and wave instabilities, in agreement with some reports found in the literature on superdiffusive hyperbolic activator-inhibitor systems. We show numerically that the presence of Turing patterns is independent of the size of the spatial domain. As a new application, we show that Turing patterns are also present in subdiffusive scenarios.

In this work, we investigate a numerical 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. Finally, an extensive set of numerical simulations is provided in order to illustrate the performance of the scheme. The results verify that the logarithmic scheme converges to the exact solution of the continuous problem, with first order of convergence in time and second order in space. Moreover, we provide some comparisons on the efficiency of the implicit method against an explicit logarithmic scheme of the literature. The results show that the present discretization is a more efficient technique.

In this manuscript, we propose a simple, two-step, finite-difference scheme to approximate the solutions of an advective Fisher's equation. The method proposed is nonlinear, explicit and, in the linear regime, it approximates the solutions of the equation of interest with a consistency of first order in time and second order in space. We prove that the technique is capable of preserving the positive, the bounded, and the temporally and spatially monotone characters of initial approximations; moreover, we establish that the method is conditionally stable under suitable constraints on the model and numerical parameters. Some simulations are provided to evince the validity of our analytical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

Departing from a finite-difference scheme to approximate solutions of a nonlinear, hyperbolic partial differential equation which generalizes the Burgers–Huxley equation from fluid dynamics, we investigate conditions on the model coefficients and the computational parameters under which positive and bounded initial data evolve into positive and bounded new approximations. The model under investigation includes nonlinear coefficients of damping and advection, and the reaction term extends the reaction law of the classical Fisher–Kolmogorov–Petrovsky–Piscounov equation. The method can be expressed in vector form in terms of a multiplicative matrix which, under certain parametric conditions, becomes an M-matrix. Using the fact that every M-matrix is non-singular and that the entries of its inverse are positive, real numbers, we establish sufficient conditions under which the method provides new, positive and bounded approximations from previous, positive and bounded data and boundary conditions. The numerical results confirm the fact that the conditions derived here are sufficient for the positivity and the boundedness of the approximations; moreover, computational experiments evidence the fact that the method still preserves these properties for values of the model and the numerical parameters outside of the analytic regions of positivity and boundedness. We point out that our simulations show a good agreement between the numerical approximations computed through our method and the corresponding, analytical solutions.

In this article, we propose a non-standard, finite-difference scheme to approximate the solutions of a generalized Burgers–Huxley equation from fluid dynamics. Our numerical method preserves the skew-symmetry of the partial differential equation under study and, under some analytical constraints of the model constants and the computational parameters involved, it is capable of preserving the boundedness and the positivity of the solutions. In the linear regime, the scheme is consistent to first order in time (due partially to the inclusion of a tuning parameter in the approximation of a temporal derivative), and to second order in space. We compare the results of our computational technique against the exact solutions of some particular initial-boundary-value problems. Our simulations indicate that the method presented in this work approximates well the theoretical solutions and, moreover, that the method preserves the boundedness of solutions within the analytical constraints derived here. In the problem of approximating solitary-wave solutions of the model under consideration, we present numerical evidence on the existence of an optimum value of the tuning parameter of our technique, for which a minimum relative error is achieved. Finally, we linearly perturb a steady-state solution of the partial differential equation under investigation, and show that our simulations still converge to the same constant solution, establishing thus robustness of our method in this sense.

In this work, we propose a finite-difference scheme to approximate the solutions of a generalization of the classical, one-dimensional, Newell-Whitehead-Segel equation from fluid mechanics, which is an equation for which the existence of bounded solutions is a well-known fact. The numerical method preserves the skew-symmetry of the problem of interest, and it is a non-standard technique which consistently approximates the solutions of the equation under investigation, with a consistency of the first order in time and of the second order in space. We prove that, under relatively flexible conditions on the computational parameters of the method, our technique yields bounded numerical approximations for every set of bounded initial estimates. Some simulations are provided in order to verify the validity of our analytical results. In turn, the validity of the computational constraints under which the method guarantees the preservation of the boundedness of the approximations, is successfully tested by means of computational experiments in some particular instances.

In this work, we present a numerical method to consistently approximate solutions of a spatially discrete, double sine-Gordon chain which considers the presence of external damping. In addition to the finite-difference scheme employed to approximate the solution of the difference-differential equations of the model under investigation, our method provides positivity-preserving schemes to approximate the local and the total energy of the system, in such a way that the discrete rate of change of the total energy with respect to time provides a consistent approximation of the corresponding continuous rate of change. Simulations are performed, first of all, to assess the validity of the computational technique against known qualitative solutions of coupled sine-Gordon and coupled double sine-Gordon chains. Secondly, the method is used in the investigation of the phenomenon of nonlinear transmission of energy in double sine-Gordon systems; the qualitative effects of the damping coefficient on the occurrence of the nonlinear process of supratransmission are briefly determined in this work, too.

In this work, we present a non-standard finite-difference scheme to approximate solutions of a damped, hyperbolic partial differential equation with nonlinear reaction which follows a generalized logistic form. Our mathematical model is a hyperbolic generalization of the classical Fisher-Kolmogorov-Petrovsky-Piscounov equation from population dynamics, where the damping term has a constant coefficient; both zero Dirichlet and Neumann boundary conditions are considered. Under certain conditions of the model and the computational parameters, the method is able to preserve the positivity and the boundedness of the solutions of the mathematical model, as evidenced both theoretically (via techniques of matrix algebra which provide sufficient conditions on the positivity and the boundedness of the technique) and computationally (via numerical simulations). The validity of the method is tested against known exact solutions of the parabolic and the hyperbolic regimes, obtaining an excellent agreement between the analytical and numerical results.