Galerkin finite element approximation for semilinear stochastic time-tempered fractional wave equations with multiplicative Gaussian noise and additive fractional Gaussian noise. (English) Zbl 1524.35704
Summary: To model wave propagation in inhomogeneous media with frequency dependent power-law attenuation, it is needed to use the fractional powers of symmetric coercive elliptic operators in space and the Caputo tempered fractional derivative in time. The model studied in this paper is semilinear stochastic space-time fractional wave equations driven by infinite dimensional multiplicative Gaussian noise and additive fractional Gaussian noise, because of the potential fluctuations of the external sources. The purpose of this work is to discuss the Galerkin finite element approximation for the semilinear stochastic fractional wave equation. First, the space-time multiplicative Gaussian noise and additive fractional Gaussian noise are discretized, which results in a regularized stochastic fractional wave equation while introducing a modeling error in the mean-square sense. We further present a complete regularity theory for the regularized equation. A standard finite element approximation is used for the spatial operator, and a mean-square priori estimates for the modeling error and the approximation error to the solution of the regularized problem are established. Finally, numerical experiments are performed to confirm the theoretical analysis.
MSC:
| 35R11 | Fractional partial differential equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 60G22 | Fractional processes, including fractional Brownian motion |
Keywords:
Galerkin finite element method; semilinear stochastic time-tempered fractional wave equation; fractional Laplacian; multiplicative Gaussian noise; additive fractional Gaussian noiseReferences:
| [1] | R. ANTON, D. COHEN, S. LARSSON, AND X. J. WANG, Full discretization of semilin-ear stochastic wave equations driven by multiplicative noise, SIAM J. Numer. Anal. 54(2) (2016), 1093-1119. · Zbl 1336.65008 |
| [2] | D. BAINOV AND P. SIMEONOV, Integral Inequalities and Applications, Kluwer Academic Publishers, 1992. · Zbl 0759.26012 |
| [3] | R. M. BALAN AND C. A. TUDOR, The stochastic wave equation with fractional noise: A ran-dom field approach, Stochastic Process. Appl. 120 (2010), 2468-2494. · Zbl 1202.60095 |
| [4] | L. BOYADJIEV AND Y. LUCHKO, Multi-dimensional α-fractional diffusion-wave equation and some properties of its fundamental solution, Comput. Math. Appl. 73 (2017), 2561-2572. · Zbl 1386.35427 |
| [5] | Y. Z. CAO, J. L. HONG, AND Z. H. LIU, Approximating stochastic evolution equations with additive white and rough noises, SIAM J. Numer. Anal. 55(4) (2017), 1958-1981. · Zbl 1373.60124 |
| [6] | Y. Z. CAO, J. L. HONG, AND Z. H. LIU, Finite element approximations for second-order stochastic differential equation driven by fractional Brownian motion, IMA J. Numer. Anal. 38 (2018), 184-197. · Zbl 1406.65106 |
| [7] | W. CHEN AND S. HOLM, Fractional Laplacian time-space models for linear and nonlin-ear lossy media exhibiting arbitrary frequency power-law dependency, J. Acoust. Soc. Am. 115(4) (2004), 1424-1430. |
| [8] | D. COHEN, S. LARSSON, AND M. SIGG, A trigonometric method for the linear stochastic wave equation, SIAM J. Numer. Anal. 51 (2013), 204-222. · Zbl 1273.65010 |
| [9] | E. CUESTA, C. LUBICH, AND C. PALENCIA, Convolution quadrature time discretization of fractional diffusion-wave equations, Math. Comp. 75 (2006), 673-696. · Zbl 1090.65147 |
| [10] | W. H. DENG, R. HOU, W. L. WANG, AND P. B. XU, Modeling Anomalous Diffusion: From Statistics to Mathematics, World Scientific, 2020. · Zbl 1453.35001 |
| [11] | W. H. DENG, B. Y. LI, W. Y. TIAN, AND P. W. ZHANG, Boundary problems for the fractional and tempered fractional operators, Multiscale Model. Simul. 16(1) (2018), 125-149. · Zbl 1391.60104 |
| [12] | W. H. DENG AND Z. J. ZHANG, High Accuracy Algorithm for the Differential Equations Governing Anomalous Diffusion, World Scientific, 2019. · Zbl 1450.65001 |
| [13] | Q. DU AND T. Y. ZHANG, Numerical approximation of some linear stochastic partial differ-ential equations driven by special additive noises, SIAM J. Numer. Anal. 40 (2002), 1421-1445. · Zbl 1030.65002 |
| [14] | W. P. FAN, F. W. LIU, X. Y. JIANG, AND I. TURNER, A novel unstructured mesh finite element method for solving the time-space fractional wave equation on a two-dimensional irregular convex domain, Fract. Calc. Appl. Anal. 20 (2017), 352-383. · Zbl 1364.65162 |
| [15] | J. GAJDA AND M. MAGDZIARZ, Fractional Fokker-Planck equation with tempered α-stable waiting times: Langevin picture and computer simulation Phys. Rev. E 82(3) (2010), 011117. |
| [16] | D. HENRY, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, 1981. · Zbl 0456.35001 |
| [17] | Y. KIAN AND M. YAMAMOTO, On existence and uniqueness of solutions for semilinear frac-tional wave equations, Fract. Calc. Appl. Anal. 20 (2017), 117-138. · Zbl 1362.35323 |
| [18] | A. A. KILBAS, H. M. SRIVASTAVA, AND J. J. TRUJILLO, Theory and Applications of Frac-tional Differential Equations, in: North-Holland Mathematics Studies, Vol. 204, Elsevier Science B.V., 2006. · Zbl 1092.45003 |
| [19] | P. E. KLOEDEN AND E. PLATEN, Numerical solution of stochastic differential equations, Springer-Verlag, 1992. · Zbl 0752.60043 |
| [20] | A. W. C. LAU AND T. C. LUBENSKY, State-dependent diffusion: Thermodynamic consistency and its path integral formulation, Phys. Rev. E 76 (2007), 011123. |
| [21] | Y. J. LI, Y. J. WANG, AND W. H. DENG, Galerkin finite element approximations for stochas-tic space-time fractional wave equations, SIAM J. Numer. Anal. 55(6) (2017), 3173-3202. · Zbl 1380.65017 |
| [22] | Y. J. LI, Y. J. WANG, W. H. DENG, AND D. X. NIE, Existence and regularity results for semilinear stochastic time-tempered fractional wave equations with multiplicative Gaussian noise and additive fractional Gaussian noise, printed. · Zbl 1527.35112 |
| [23] | M. M. MEERSCHAERT, F. SABZIKAR, M. S. PHANIKUMAR, AND A. ZELEKE, Tempered frac-tional time series model for turbulence in geophysical flows, J. Stat. Mech. Theory Exp. 2014(9) (2014), P09023. |
| [24] | M. M. MEERSCHAERT, R. L. SCHILLING, AND A. SIKORSKII, Stochastic solutions for frac-tional wave equations, Nonlinear Dynam. 80(4) (2015), 1685-1695. · Zbl 1345.35134 |
| [25] | Y. S. MISHURA, Stocastic Calculus for Fractional Brownian Motion and Related Processes, Springer-Verlag, 2008. · Zbl 1138.60006 |
| [26] | R. H. NOCHETTO, E. OTÁROLA, AND A. J. SALGADO, A PDE approach to fractional diffusion in general domains: A priori error analysis, Found. Comput. Math. 15 (2015), 733-791. · Zbl 1347.65178 |
| [27] | I. PODLUBNY, Fractional Differential Equations, Mathematics in Science and Engineering, Vol. 198, Academic Press, 1999. · Zbl 0924.34008 |
| [28] | L. QUER-SARDANYONS AND S. TINDEL, The 1-d stochastic wave equation driven by a frac-tional Brownian sheet, Stochastic Process. Appl. 117 (2007), 1448-1472. · Zbl 1123.60049 |
| [29] | M. STOJANOVIC, Wave equation driven by fractional generalized stochastic processes, Mediterr. J. Math. 10 (2013), 1813-1831. · Zbl 1288.26006 |
| [30] | Z. Z. SUN AND X. N. WU, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56 (2006), 193-209. · Zbl 1094.65083 |
| [31] | T. L. SZABO, Time domain wave equations for lossy media obeying a frequency power law, J. Acoust. Soc. Am. 96(1) (1994), 491-500. |
| [32] | X. J. WANG, S. Q. GAN, AND J. T. TANG, Higher order strong approximations of semilinear stochastic wave equation with additive space-time white noise, SIAM J. Sci. Comput. 36 (2014), A2611-A2632. · Zbl 1322.65023 |
| [33] | L. L. WEI, Analysis of a new finite difference/local discontinuous Galerkin method for the fractional diffusion-wave equation, Appl. Math. Comput. 304 (2017), 180-189. · Zbl 1411.65135 |
| [34] | M. ZAYERNOURI, M. AINSWORTH, AND G. E. KARNIADAKIS, Tempered fractional Sturm-Liouville eigenproblems, SIAM J. Sci. Comput. 37 (2015), A1777-A1800. · Zbl 1323.34012 |
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