A numerically efficient and conservative model for a Riesz space-fractional Klein-Gordon-Zakharov system. (English) Zbl 1464.65084
Commun. Nonlinear Sci. Numer. Simul. 71, 22-37 (2019); corrigendum ibid. 83, Article ID 105109, 4 p. (2020).
Summary: 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.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
| 35L53 | Initial-boundary value problems for second-order hyperbolic systems |
| 35L70 | Second-order nonlinear hyperbolic equations |
Keywords:
fractional Klein-Gordon-Zakharov equations; conservation of energy; Riesz space-fractional equations; energy-conserving method; fractional-order centered differences; numerical efficiency analysisReferences:
| [1] | Texier, B., Derivation of the Zakharov equations, Arch Ration Mech Anal, 184, 1, 121-183 (2007) · Zbl 1370.35249 |
| [2] | Zakharov, V. E., Collapse of Langmuir waves, Soviet Phys JETP, 35, 5, 908-914 (1972) |
| [3] | Garcia, L.; Haas, F.; De Oliveira, L.; Goedert, J., Modified Zakharov equations for plasmas with a quantum correction, Phys Plasmas, 12, 1, 012302 (2005) |
| [4] | Marklund, M., Classical and quantum kinetics of the Zakharov system, Phys Plasmas, 12, 8, 082110 (2005) |
| [5] | Thornhill, S. G.; Ter Haar, D., Langmuir turbulence and modulational instability, Phys Rep, 43, 2, 43-99 (1978) |
| [6] | Bridges, T. J.; Reich, S., Multi-symplectic spectral discretizations for the Zakharov-Kuznetsov and shallow water equations, Physica D, 152, 491-504 (2001) · Zbl 1032.76053 |
| [7] | Colin, T.; Métivier, G., Instabilities in Zakharov equations for laser propagation in a plasma, Phase space analysis of partial differential equations, 63-81 (2006), Springer · Zbl 1133.35303 |
| [8] | Tsutaya, K., Global existence of small amplitude solutions for the Klein-Gordon-Zakharov equations, Nonlinear evolution equations and infinite dimensional dynamical systems-proceedings of the conference, 226 (1997), World Scientific · Zbl 0972.35115 |
| [9] | Devanandhan, S.; Singh, S.; Lakhina, G.; Bharuthram, R., Small amplitude electron acoustic solitary waves in a magnetized superthermal plasma, Commun Nonlinear Sci Numer Simul, 22, 1-3, 1322-1330 (2015) |
| [10] | You, S.; Ning, X., On global generalized solution for a generalized Zakharov equations, Int J Appl Math, 47, 2 (2017) · Zbl 1512.35557 |
| [11] | Sun, C.; Li, L., The global existence and uniqueness of the classical solution with the periodic initial value problem for one-dimension Klein-Gordon-Zakharov equations, Adv Math Phys, 2018 (2018) · Zbl 1404.35279 |
| [12] | You, S.; Ning, X., On global smooth solution for generalized Zakharov equations, Comput Math Appl, 72, 1, 64-75 (2016) · Zbl 1443.35149 |
| [13] | Larkin, N.; Padilha, M., Global regular solutions to one problem of Saut-Temam for the 3D Zakharov-Kuznetsov equation, Appl Math Optim, 77, 2, 253-274 (2018) · Zbl 1391.35342 |
| [14] | Antonova, A. P.; Faminskii, A. V., On the regularity of solutions of the Cauchy problem for the Zakharov-Kuznetsov equation in Hölder norms, Math Notes, 97, 1-2, 12-20 (2015) · Zbl 1319.35213 |
| [15] | Vaibhav, V., Exact solution of the Zakharov-Shabat scattering problem for doubly-truncated multisoliton potentials, Commun Nonlinear Sci Numer Simul, 61, 22-36 (2018) · Zbl 1470.78003 |
| [16] | Tracinà, R., On the nonlinear self-adjointness of the Zakharov-Kuznetsov equation, Commun Nonlinear Sci Numer Simul, 19, 2, 377-382 (2014) · Zbl 1344.35128 |
| [17] | Bejenaru, I.; Guo, Z.; Herr, S.; Nakanishi, K., Well-posedness and scattering for the Zakharov system in four dimensions, Anal PDE, 8, 8, 2029-2055 (2015) · Zbl 1331.35093 |
| [18] | Kato, T., Well-posedness for the generalized Zakharov-Kuznetsov equation on modulation spaces, J Fourier Anal Appl, 23, 3, 612-655 (2017) · Zbl 1372.35276 |
| [19] | Fonseca, G.; Pachón, M., Well-posedness for the two dimensional generalized Zakharov-Kuznetsov equation in anisotropic weighted Sobolev spaces, J Math Anal Appl, 443, 1, 566-584 (2016) · Zbl 1342.35074 |
| [20] | Fang, Y.-F.; Shih, H.-W.; Wang, K.-H., Local well-posedness for the quantum Zakharov system in one spatial dimension, J Hyperbolic Differ Equ, 14, 01, 157-192 (2017) · Zbl 1364.35294 |
| [21] | Liang, X.; Khaliq, A. Q.M., An efficient Fourier spectral exponential time differencing method for the space-fractional nonlinear Schrödinger equations, Comput Math Appl, 75, 12, 4438-4457 (2018) · Zbl 1419.65083 |
| [22] | Khaliq, A. Q.M.; Biala, T.; Alzahrani, S.; Furati, K., Linearly implicit predictor-corrector methods for space-fractional reaction-diffusion equations with non-smooth initial data, Comput Math Appl, 75, 8, 2629-2657 (2018) · Zbl 1415.65193 |
| [23] | Furati, K. M.; Yousuf, M.; Khaliq, A. Q.M., Fourth-order methods for space fractional reaction-diffusion equations with non-smooth data, Int J Comput Math, 95, 6-7, 1240-1256 (2018) · Zbl 1499.35637 |
| [24] | Sahoo, S.; Ray, S. S., Improved fractional sub-equation method for (3+1)-dimensional generalized fractional KdV-Zakharov-Kuznetsov equations, Comput Math Appl, 70, 2, 158-166 (2015) · Zbl 1443.35173 |
| [25] | Korkmaz, A., Exact solutions to (3+1) conformable time fractional Jimbo-Miwa, Zakharov-Kuznetsov and modified Zakharov-Kuznetsov equations, Commun Theor Phys, 67, 5, 479 (2017) · Zbl 1365.35208 |
| [26] | Baskonus, H. M.; Yel, G.; Bulut, H., Novel wave surfaces to the fractional Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation, AIP conference proceedings, 1863, 560084 (2017), AIP Publishing |
| [27] | Ray, S. S.; Sahoo, S., Comparison of two reliable analytical methods based on the solutions of fractional coupled Klein-Gordon-Zakharov equations in plasma physics, Comput Math Math Phys, 56, 7, 1319-1335 (2016) · Zbl 1432.35230 |
| [28] | Pinto, L.; Sousa, E., Numerical solution of a time-space fractional Fokker Planck equation with variable force field and diffusion, Commun Nonlinear Sci Numer Simul, 50, 211-228 (2017) · Zbl 1510.82031 |
| [29] | Zhu, L.; Fan, Q., Numerical solution of nonlinear fractional-order Volterra integro-differential equations by SCW, Commun Nonlinear Sci Numer Simul, 18, 5, 1203-1213 (2013) · Zbl 1261.35152 |
| [30] | Bhrawy, A.; Zaky, M., Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations, Comput Math Appl, 73, 6, 1100-1117 (2017) · Zbl 1412.65162 |
| [31] | Khalil, H.; Khan, R. A., A new method based on legendre polynomials for solutions of the fractional two-dimensional heat conduction equation, Comput Math Appl, 67, 10, 1938-1953 (2014) · Zbl 1366.74084 |
| [32] | Wang, T.; Chen, J.; Zhang, L., Conservative difference methods for the Klein-Gordon-Zakharov equations, J Comput Appl Math, 205, 1, 430-452 (2007) · Zbl 1123.65091 |
| [33] | Macías-Díaz, J. E., An explicit dissipation-preserving method for Riesz space-fractional nonlinear wave equations in multiple dimensions, Commun Nonlinear Sci Numer Simul, 59, 67-87 (2018) · Zbl 1510.65200 |
| [34] | Macías-Díaz, J. E., A structure-preserving method for a class of nonlinear dissipative wave equations with Riesz space-fractional derivatives, J Comput Phys, 351, 40-58 (2017) · Zbl 1380.65164 |
| [35] | Macías-Díaz, J. E.; Hendy, A. S.; De Staelen, R., A pseudo energy-invariant method for relativistic wave equations with Riesz space-fractional derivatives, Comput Phys Commun, 224, 98-107 (2018) · Zbl 07694296 |
| [36] | Macías-Díaz, J. E., Numerical study of the process of nonlinear supratransmission in Riesz space-fractional sine-Gordon equations, Commun Nonlinear Sci Numer Simul, 46, 89-102 (2017) · Zbl 1485.35400 |
| [37] | Macías-Díaz, J. E., Numerical simulation of the nonlinear dynamics of harmonically driven Riesz-fractional extensions of the Fermi-Pasta-Ulam chains, Commun Nonlinear Sci Numer Simul, 55, 248-264 (2018) · Zbl 1510.82018 |
| [38] | Macías-Díaz, J. E., Persistence of nonlinear hysteresis in fractional models of Josephson transmission lines, Commun Nonlinear Sci Numer Simul, 53, 31-43 (2017) · Zbl 1510.94055 |
| [39] | Macías-Díaz, J. E.; Bountis, A., Supratransmission in \(β\)-Fermi-Pasta-Ulam chains with different ranges of interactions, Commun Nonlinear Sci Numer Simul, 63, 307-321 (2018) · Zbl 1528.82024 |
| [40] | Ortigueira, M. D., Riesz potential operators and inverses via fractional centred derivatives, Int J Math Math Sci, 2006, ID48391 (2006) · Zbl 1122.26007 |
| [41] | Byrd, P. F.; Friedman, M. D., Handbook of elliptic integrals for scientists and engineers, Grundlehren der mathematische wissenschaften, LXVII (1971), Springer-Verlag: Springer-Verlag Berlin · Zbl 0213.16602 |
| [42] | De Oliveira, E. C.; Tenreiro Machado, J. A., A review of definitions for fractional derivatives and integral, Math Probl Eng, 2014 (2014) · Zbl 1407.26013 |
| [43] | Friedman, A., Foundations of modern analysis (1970), Courier Corporation: Courier Corporation New York · Zbl 0198.07601 |
| [44] | Nirenberg, L., On elliptic partial differential equations, Il principio di minimo e sue applicazioni alle equazioni funzionali, 1-48 (2011), Springer · Zbl 0108.10001 |
| [45] | Wang, X.; Liu, F.; Chen, X., Novel second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations, Adv Math Phys, 2015, 590435 (2015) · Zbl 1380.65188 |
| [46] | Wang, P.; Huang, C.; Zhao, L., Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation, J Comput Appl Math, 306, 231-247 (2016) · Zbl 1382.65260 |
| [47] | Zhou, Y., Application of discrete functional analysis to the finite difference methods (1990) |
| [48] | Glassey, R., Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension, Math Comput, 58, 197, 83-102 (1992) · Zbl 0746.65066 |
| [49] | Liu, S.; Fu, Z.; Liu, S.; Zhao, Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys Lett A, 289, 1-2, 69-74 (2001) · Zbl 0972.35062 |
| [50] | Liu, S.; Fu, Z.; Liu, S.; Wang, Z., The periodic solutions for a class of coupled nonlinear Klein-Gordon equations, Phys Lett A, 323, 5-6, 415-420 (2004) · Zbl 1118.81399 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
