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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.

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.

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 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 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.