Elsevier

Applied Numerical Mathematics

Volume 169, November 2021, Pages 179-200
Applied Numerical Mathematics

An implicit semi-linear discretization of a bi-fractional Klein–Gordon–Zakharov system which conserves the total energy

https://doi.org/10.1016/j.apnum.2021.06.014Get rights and content

Abstract

In this work, we propose an implicit finite-difference scheme to approximate the solutions of a generalization of the well-known Klein–Gordon–Zakharov system. More precisely, the system considered in this work is an extension to the spatially fractional case of the classical Klein–Gordon–Zakharov model, considering two different orders of differentiation and fractional derivatives of the Riesz type. The numerical model proposed in this work considers fractional-order centered differences to approximate the spatial fractional derivatives. The energy associated to this discrete system is a non-negative invariant, in agreement with the properties of the continuous fractional model. We establish rigorously the existence of solutions using fixed-point arguments and complex matrix properties. To that end, we use the fact that the two difference equations of the discretization are decoupled, which means that the computational implementation is easier than for other numerical models available in the literature. We prove that the method has square consistency in both time and space. In addition, we prove rigorously the stability and the quadratic convergence of the numerical model. As a corollary of stability, we are able to prove the uniqueness of numerical solutions. Finally, we provide some illustrative simulations with a computer implementation of our scheme.

Introduction

The field of fractional calculus has witnessed a vertiginous development in recent years, partly due to the vast amount of potential applications in the sciences [17]. It is worth pointing out that many different fractional derivatives and integrals have been proposed. For example, some of the first fractional derivatives introduced historically in mathematics were the Riemann–Liouville fractional operators [37], which generalized the classical integer-order derivatives with respect to some specific analytical properties [44]. The fractional derivatives in the senses of Caputo, Riesz and Grünwald–Letnikov are also extensions of the traditional derivatives of integer order. These fractional operators are nonequivalent in general, and various applications of all of them have been proposed to science and engineering [8], [52]. For instance, some reports have provided theoretical foundations for the application of fractional calculus to the theory of viscoelasticity [3], while others have proposed possible applications of fractional calculus to dynamic problems of solid mechanics [45], continuous-time financial economics [13], [47], Earth system dynamics [65], mathematical modeling of biological phenomena [22] and the modeling of two-phase gas/liquid flow systems [38], just to mention some potential applications. However, it is important to recall that Riesz-type derivatives may be the only fractional operators which have known real physical applications [54].

From the mathematical point of view, many interesting avenues of investigation have been opened by the progress in fractional calculus. Indeed, the different fractional derivatives have found discrete analogues which have been used extensively in the literature. As examples, Riesz fractional derivatives have been discretized consistently using fractional-order centered differences [40] and weighted-shifted Grünwald differences [19], [24]. Obviously, those discrete approaches have been studied to determine their analytical properties, and they have been used extensively to provide discrete models to solve Riesz space-fractional conservative/dissipative wave equations [20], a β-Fermi–Pasta–Ulam chains with different ranges of interactions [28], an energy-preserving double fractional Klein–Gordon–Zakharov system [31], and even a Riesz space-fractional generalization of the Higgs boson equation in the de Sitter space-time with generalized time-dependent diffusion coefficient and potential [27] among other complex systems. On the other hand, Caputo fractional derivatives have been discretized consistently using various criteria. For instance, some high-order L2-compact difference approaches have been utilized to that end [60], as well as the standard L1 formulas [12], [39] and L1-2 numerical methodologies [16], [66].

From the analytical point of view, the literature offers a wide range of reports which focus on the extension of integer-order methods and results to the fractional case. For example, there are various articles which tackle the existence, uniqueness, regularity and asymptotic behavior of the solution for the fractional porous medium equation [10], nonlinear fractional diffusion equations [59], nonlinear fractional heat equations [6], the Fisher–Kolmogorov–Petrovskii–Piscounov equation with nonlinear fractional diffusion [51], fractional thin-film equations [48] and the fractional Schrödinger equation with general non-negative potentials [11]. From a more particular point of view, the fractional generalization of the classical vector calculus operators (that is, the gradient, divergence, curl and Laplacian operators) has been also an active topic of research which has been developed from different approaches. Some of the first attempts to extend those operators to the fractional scenario were proposed in [1], [2] using the Nishimoto fractional derivative. These operators were used later on in [36] to provide a physical interpretation for the fractional advection-dispersion equation for flow in heterogeneous porous media (see [55] and references therein for a historic account of the efforts to formulate a fractional form of vector calculus). More recently, an extension of the Helmholtz decomposition theorem for fractional time and space was proposed in [42] using discrete fractional derivatives.

The present work is motivated by the Klein–Gordon–Zakharov system, which is a model of utmost interest in plasma physics. It consists of two coupled hyperbolic partial differential equations which describe the interaction between the Langmuir waves and ion-acoustic waves in a high-frequency plasma [56]. This system is a physical and mathematical extension of the Zakharov model discussed in [62]. In fact, V. E. Zakharov analyzed the interaction of the fast-time scale component of an electric field with the deviation of ion density from equilibrium. In spite that the Klein–Gordon–Zakharov system is more general than the Zakharov model [30], the latter is considered as one of the best models to describe simultaneously various physical phenomena. In addition, another important feature of this system is the strong relationship with the nonlinear Schrödinger equation [34]. In fact, under suitable circumstances, it is possible to obtain the Schrödinger and the Klein–Gordon equations from Zakharov's system [35]. As a matter of fact, some exact solutions for this system and their behaviors have been studied in [43], [5], [49], using normal forms, the a priori estimate method and the hyperbolic functions method, respectively. However, we must point out that there are many other methods to obtain traveling waves and solitons solutions for the Klein–Gordon–Zakharov equations, as well as reports which study the qualitative behavior of its solutions [21], [63], [64].

Evidently, a natural extension in the mathematical investigation of systems like the Klein–Gordon–Zakharov model hinges on the introduction of fractional-order partial derivatives. It is worth recalling here that fractional derivatives have been used to describe phenomena more realistically [15], [23]. From the analytical point of view, fractional extensions of classical systems provide a more complicated theoretical analysis [50]. In general, fractional calculus has been employed recently in many areas of science outside from mathematics, like in physics, chemistry, engineering, finances and social sciences [25]. Motivated by those efforts, we consider herein an extension of the Klein–Gordon–Zakharov system to the fractional case with space-fractional derivatives of the Riesz type [9]. In fact, we will consider two parameters to represent the orders of differentiation, and we will assume that the values are not necessarily equal. The model will be defined on a closed and bounded interval of the real line, and we will impose homogeneous Neumann boundary conditions at the ends of the interval. Like its classical counterpart, the fractional model considered here has associated an energy-like non-negative quantity which is constant with respect to time. As a consequence of the existence of a non-negative energy functional, the solutions of the model are uniformly bounded.

As expected, the determination of exact solutions for the fractional system investigated in this work is a hard task [58]. In light of this fact, the development of numerical methods to approximate the solutions is justified in the practice. In this work, we propose a numerical model based on finite-differences to approximate the solutions of the continuous bi-fractional Klein–Gordon–Zakharov system. Our approach will make use of fractional-order centered differences to approximate the space-fractional derivatives [41]. The numerical model is an implicit scheme which has an associated discrete energy functional. As its continuous counterpart, the energy associated to the discrete model is non-negative and constant with respect to time. In that sense, the present discretization belongs to that family of numerical methods which are called conservative for their capability to preserve the energy [14], [53], [4]. In addition, we use a compact-difference operator to approximate the second derivatives in time for both functions of the system, and we show that the scheme has a consistency of quadratic order. Solutions exist by virtue of the Leray–Schauder fixed-point theorem and they are uniformly bounded. We prove rigorously the remaining numerical properties (that is, stability and convergence), and we provide some illustrative simulations.

Section snippets

Preliminaries

Throughout, we fix a nonempty, open and bounded interval B=(xL,xR) of R. Let T>0, and define the set Ω=B×(0,T). In general, for each SR2, we let S be the closure of S with respect to the standard topology of R2. In particular, this means that Ω=B×[0,T]. Let u:ΩC, m:ΩR, u0,u1:BC and m0,m1:BR be sufficiently smooth functions. In this report, functions defined on Ω will be extended to all R×[0,T] by letting them be equal to zero on (R[xL,xR])×[0,T].

Definition 2.1

Podlubny [44]

Let f:RR be a function, and let nN

Discrete model

The first aim of this section is to provide the necessary discrete nomenclature which will be required later on in this work. To give a fresh start, we will focus firstly on the concept of fractional-order centered differences, which will allow us to obtain a discretization of Riesz space-fractional derivatives. We must point out that we opted to use fractional centered differences for computational implementation reasons. However, there are various other alternative approaches to provide such

Structural properties

The present section is devoted to establishing the main structural properties of the discrete model (3.16). More precisely, we will prove herein that the discrete model is solvable, and that the quantities (3.22) are non-negative temporal invariants of the scheme (3.16). Firstly, we prove and state some crucial results in our analysis.

Definition 4.1

If U,VVh, then we define the product of U and V point-wisely. More precisely, UV=(UjVj)jIJ.

Lemma 4.2

Let U=(Un)nIN and M=(Mn)nIN be sequences in Vh. Assume that U is

Numerical properties

The present section will be devoted to prove rigorously the main numerical properties of the scheme (3.16). More precisely, we will show that the numerical approximations are second-order consistent estimates of the solutions of the continuous model (2.5). Moreover, we will establish the stability and the second-order convergence of the scheme using a suitable discrete Gronwall inequality. As a corollary of the stability property of our scheme, we will prove that the solutions of the

Computer implementation

We describe now the computational implementation of the scheme (3.16) and provide some illustrative simulations. About the computational implementation, it is important to point out that the left-hand side of the first recursive equation of (3.16) is a function of the unknown complex vector Un+1. That function considers the presence of |Un+1| which, unfortunately, is not analytic in the variable Un+1. As a consequence, the use of numerical methods to approximate the roots of complex functions

Conclusions

In this work, we investigated numerically a generalization of the well-known Klein–Gordon–Zakharov system which considers two fractional derivatives of the Riesz type. The fractional derivatives are not required to be of the same order, but they belong to the interval (1,2]. It is known that the system considered in this work has a conserved energy-like quantity, and that the solutions are uniformly bounded. Motivated by these facts, we proposed a finite-difference discretization of the system

Funding

The first author (R.M.) was financially supported by the National Council for Science and Technology of Mexico (CONACYT) through a grant to carry out doctoral studies at the Universidad Autónoma de Aguascalientes, Mexico. The third author (Q.S.) was supported in part by a summer research award from the College of Arts and Sciences, Baylor University, USA. Meanwhile, the corresponding author (J.E.M.-D.) wishes to acknowledge the financial support from CONACYT through grant A1-S-45928.

Financial disclosure

None reported.

Declaration of Competing Interest

The authors declare no potential conflict of interests.

Acknowledgements

The authors wish to thank the anonymous reviewers and the associate editor in charge of handling this manuscript for their criticisms. All of their suggestions and comments were taken into account in the revision of this paper.

References (67)

  • A. Houwe et al.

    Complex traveling-wave and solitons solutions to the Klein-Gordon-Zakharov equations

    Results Phys.

    (2020)
  • C. Ionescu et al.

    The role of fractional calculus in modeling biological phenomena: a review

    Commun. Nonlinear Sci. Numer. Simul.

    (2017)
  • J.T. Machado et al.

    Recent history of fractional calculus

    Commun. Nonlinear Sci. Numer. Simul.

    (2011)
  • J.E. Macías-Díaz

    An explicit dissipation-preserving method for Riesz space-fractional nonlinear wave equations in multiple dimensions

    Commun. Nonlinear Sci. Numer. Simul.

    (2018)
  • J.E. Macías-Díaz et al.

    Supratransmission in β-Fermi–Pasta–Ulam chains with different ranges of interactions

    Commun. Nonlinear Sci. Numer. Simul.

    (2018)
  • J. Macías-Díaz et al.

    A boundedness-preserving finite-difference scheme for a damped nonlinear wave equation

    Appl. Numer. Math.

    (2010)
  • R. Martínez et al.

    An energy-preserving and efficient scheme for a double-fractional conservative Klein–Gordon–Zakharov system

    Appl. Numer. Math.

    (2020)
  • R. Martínez et al.

    An energy-preserving and efficient scheme for a double-fractional conservative Klein-Gordon-Zakharov system

    Appl. Numer. Math.

    (2020)
  • N. Masmoudi et al.

    From the Klein-Gordon-Zakharov system to a singular nonlinear Schrödinger system

    Ann. Inst. Henri Poincaré, Anal. Non Linéaire

    (2010)
  • M.M. Meerschaert et al.

    Fractional vector calculus for fractional advection–dispersion

    Phys. A, Stat. Mech. Appl.

    (2006)
  • M.D. Ortigueira

    Fractional central differences and derivatives

    IFAC Proc. Vol.

    (2006)
  • M.D. Ortigueira et al.

    From a generalised Helmholtz decomposition theorem to fractional Maxwell equations

    Commun. Nonlinear Sci. Numer. Simul.

    (2015)
  • T. Ozawa et al.

    Normal form and global solutions for the Klein-Gordon-Zakharov equations

    Ann. Inst. Henri Poincaré, Anal. Non Linéaire

    (1995)
  • E. Scalas et al.

    Fractional calculus and continuous-time finance

    Phys. A, Stat. Mech. Appl.

    (2000)
  • Y. Shang et al.

    New exact traveling wave solutions for the Klein-Gordon-Zakharov equations

    Comput. Math. Appl.

    (2008)
  • H. Sun et al.

    A new collection of real world applications of fractional calculus in science and engineering

    Commun. Nonlinear Sci. Numer. Simul.

    (2018)
  • Y.-F. Tang et al.

    Symplectic methods for the nonlinear Schrödinger equation

    Comput. Math. Appl.

    (1996)
  • V.E. Tarasov

    Fractional vector calculus and fractional Maxwell's equations

    Ann. Phys.

    (2008)
  • Y.-M. Wang et al.

    A high-order L2-compact difference method for Caputo-type time-fractional sub-diffusion equations with variable coefficients

    Appl. Math. Comput.

    (2019)
  • Y. Zhang et al.

    A review of applications of fractional calculus in Earth system dynamics

    Chaos Solitons Fractals

    (2017)
  • F.B. Adda

    Geometric interpretation of the differentiability and gradient of real order

    C. R. Acad. Sci., Ser. 1 Math.

    (1998)
  • R.L. Bagley et al.

    A theoretical basis for the application of fractional calculus to viscoelasticity

    J. Rheol.

    (1983)
  • G. Boling et al.

    Global smooth solution for the Klein–Gordon–Zakharov equations

    J. Math. Phys.

    (1995)
  • Cited by (7)

    • A simulation expressivity of the quenching phenomenon in a two-sided space-fractional diffusion equation

      2023, Applied Mathematics and Computation
      Citation Excerpt :

      Xu focused at fractional diffusion operators and their quenching applications [9]. The many heuristic discussions on fractional ordered differential equations have quickly become an intensive research field attracting more and more attentions for in-depth theory and simulations [21,26]. Both theoretical and numerical explorations will be carried out for the stability, positivity and monotonicity of the semiadaptive scheme proposed.

    • Analysis of a scheme which preserves the dissipation and positivity of Gibbs' energy for a nonlinear parabolic equation with variable diffusion

      2023, Applied Numerical Mathematics
      Citation Excerpt :

      The design and theoretical analysis of a dissipation-preserving numerical model to solve that fractional problem would be an interesting avenue of research in that case. It is worth pointing out that the authors of this manuscript have devoted some efforts to design some energy-preserving techniques to solve fractional extensions of the Klein–Gordon–Zakharov equations [16,18] and the Gross–Pitaevskii system [15]. A priori, the authors of the present manuscript believe that such investigation is feasible, though substantial efforts need to be carried out to that end.

    • Stable and efficient time second-order difference schemes for fractional Klein–Gordon–Zakharov system

      2022, Journal of Computational Science
      Citation Excerpt :

      To the authors’ knowledge, a few high-order numerical schemes that preserve some physical and intrinsic properties have been developed for simulating the propagation of the fractional KGZ system, such as, second-order energy conservative linear difference scheme [40], second-order conservative explicit difference scheme [41], second-order energy conservative nonlinear difference scheme [42]. More recently, R. Martínez, J.E. Macías-Díaz and Q. Sheng developed and analyzed several efficient energy-preserving second-order finite difference schemes for solving multi-fractional extension of the KGZ system [1–3] and Zakharov system [43]. Therefore, it is significance to develop high-order stable efficient numerical schemes with energy conservation for simulating the propagation of fractional KGZ system (1)–(4) over long time.

    View all citing articles on Scopus
    View full text