Exact solutions of multi-term fractional diffusion-wave equations with Robin type boundary conditions. (English) Zbl 1284.35455
Summary: General exact solutions in terms of wavelet expansion are obtained for multiterm time-fractional diffusion-wave equations with Robin type boundary conditions. By proposing a new method of integral transform for solving boundary value problems, such fractional partial differential equations are converted into time-fractional ordinary differential equations, which are further reduced to algebraic equations by using the Laplace transform. Then, with a wavelet-based exact formula of Laplace inversion, the resulting exact solutions in the Laplace transform domain are reversed to the time-space domain. Three examples of wave-diffusion problems are given to validate the proposed analytical method.
MSC:
| 35R11 | Fractional partial differential equations |
| 34A08 | Fractional ordinary differential equations |
| 44A10 | Laplace transform |
| 65T60 | Numerical methods for wavelets |
Keywords:
fractional derivative; diffusion-wave equation; Laplace transform; integral transform; exact solution; waveletReferences:
| [1] | Huang, F. H. and Guo, B. L. General solutions to a class of time fractional partial differential equations. Applied Mathematics and Mechanics (English Edition), 31, 815-826 (2010) DOI 10.1007/s10483-010-1316-9 · Zbl 1204.35170 · doi:10.1007/s10483-010-1316-9 |
| [2] | Mainardi, F. Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos, Solitons & Fractals, 7, 1461-1477 (1996) · Zbl 1080.26505 · doi:10.1016/0960-0779(95)00125-5 |
| [3] | Caputo, M. Linear models of dissipation whose Q is almost frequency independent-II. Geophysical Journal of the Royal Astronomical Society, 13, 529-539 (1967) · doi:10.1111/j.1365-246X.1967.tb02303.x |
| [4] | Caputo, M. and Mainardi, F. Linear models of dissipation in anelastic solids. La Rivista del Nuovo Cimento, 1, 161-198 (1971) · doi:10.1007/BF02820620 |
| [5] | Nigmatullin, R. R. The realization of the generalized transfer equation in a medium with fractal geometry. Physica Status Solidi (B), 133, 425-430 (1986) · doi:10.1002/pssb.2221330150 |
| [6] | Nigmatullin, R. R. To the theoretical explanation of the universal response. Physica B, 123, 739-745 (1984) |
| [7] | Agrawal, O. P. Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dynamics, 29, 145-155 (2002) · Zbl 1009.65085 · doi:10.1023/A:1016539022492 |
| [8] | Chen, W. Time-space fabric underlying anomalous diffusion. Chaos, Solitons & Fractals, 28, 923-929 (2006) · Zbl 1098.60078 · doi:10.1016/j.chaos.2005.08.199 |
| [9] | Chen, W., Sun, H., Zhang, X., and Koroak, D. Anomalous diffusion modeling by fractal and fractional derivatives. Computers and Mathematics with Applications, 59, 1754-1758 (2010) · Zbl 1189.35355 · doi:10.1016/j.camwa.2009.08.020 |
| [10] | Chen, W. An intuitive study of fractional derivative modeling and fractional quantum in soft matter. Journal of Vibration and Control, 14, 1651-1657 (2008) · Zbl 1229.74009 · doi:10.1177/1077546307087398 |
| [11] | Chen, W. A speculative study of 2/3-order fractional Laplacian modeling of turbulence: some thoughts and conjectures. Chaos, 16, 023126 (2006) · Zbl 1146.37312 · doi:10.1063/1.2208452 |
| [12] | Chen, W. and Holm, S. Modified Szaboo wave equation models for lossy media obeying frequency power law. Journal of the Acoustical Society of America, 114, 2570-2574 (2003) · doi:10.1121/1.1621392 |
| [13] | Chen, W. and Holm, S. Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency dependency. Journal of the Acoustical Society of America, 115, 1424-1430 (2004) · doi:10.1121/1.1646399 |
| [14] | Li, C., Zhang, F., Kurths, J., and Zeng, F. Equivalent system for a multiple-rational-order fractional differential system. Philosophical Transactions of the Royal Society A, 371, 20120156 (2013) · Zbl 1342.34012 · doi:10.1098/rsta.2012.0156 |
| [15] | Schneider, W. R. and Wyss, W. Fractional diffusion and wave equations. Journal of Mathematical Physics, 30, 134-144 (1989) · Zbl 0692.45004 · doi:10.1063/1.528578 |
| [16] | Mainardi, F. The fundamental solutions for the fractional diffusion-wave equation. Applied Mathematics Letters, 9, 23-28 (1996) · Zbl 0879.35036 · doi:10.1016/0893-9659(96)00089-4 |
| [17] | Daftardar-Gejji, V. and Bhalekar, S. Solving multi-term linear and non-linear diffusion-wave equations of fractional order by Adomian decomposition method. Applied Mathematics and Computation, 202, 113-120 (2008) · Zbl 1147.65106 · doi:10.1016/j.amc.2008.01.027 |
| [18] | Jafari, H. and Seifi, S. Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation. Communications in Nonlinear Science and Numerical Simulation, 14, 2009-2012 (2009) · Zbl 1221.65278 |
| [19] | Daftardar-Gejji, V. and Bhalekar, S. Boundary value problems for multi-term fractional differential equations. Journal of Mathematical Analysis and Applications, 345, 754-765 (2008) · Zbl 1151.26004 · doi:10.1016/j.jmaa.2008.04.065 |
| [20] | Welch, S. W. J., Ropper, R. A. L., and Duren, R. G. Application of time-based fractional calculus methods to viscoelastic creep and stress relaxation of materials. Mechanics of Time-Dependent Materials, 3, 279-303 (1999) · doi:10.1023/A:1009834317545 |
| [21] | Ford, N. J., Xiao, J., and Yan, Y. A finite element method for time fractional partial differential equations. Fractional Calculus and Applied Analysis, 14, 454-474 (2011) · Zbl 1273.65142 · doi:10.2478/s13540-011-0028-2 |
| [22] | Esen, A., Ucar, Y., Yagmurlu, N., and Tasbozan, O. A Galerkin finite element method to solve fractional diffusion and fractional diffusion-wave equations. Mathematical Modelling and Analysis, 18, 260-273 (2013) · Zbl 1266.65026 · doi:10.3846/13926292.2013.783884 |
| [23] | Li, C. and Zeng, F. The finite difference methods for fractional ordinary differential equations. Numerical Functional Analysis and Optimization, 34, 149-179 (2013) · Zbl 1267.65094 · doi:10.1080/01630563.2012.706673 |
| [24] | Li, C. and Zeng, F. Finite difference methods for fractional differential equations. International Journal of Bifurcation and Chaos, 22, 1230014 (2012) · Zbl 1258.34018 · doi:10.1142/S0218127412300145 |
| [25] | Zhou, Y. H., Wang, X. M., Wang, J. Z., and Liu, X. J. A wavelet numerical method for solving nonlinear fractional vibration, diffusion and wave equations. Computer Modeling in Engineering and Sciences, 77, 137-160 (2011) · Zbl 1356.65272 |
| [26] | Li, Y. Solving a nonlinear fractional differential equation using Chebyshev wavelets. Communications in Nonlinear Science and Numerical Simulation, 15, 2284-2292 (2010) · Zbl 1222.65087 · doi:10.1016/j.cnsns.2009.09.020 |
| [27] | Wang, J. Z., Zhou, Y. H., and Gao, H. J. Computation of the Laplace inverse transform by application of the wavelet theory. Communications in Numerical Methods in Engineering, 19, 959-975 (2003) · Zbl 1035.65159 · doi:10.1002/cnm.645 |
| [28] | Koziol, P. and Hryniewicz, Z. Analysis of bending waves in beam on viscoelastic random foundation using wavelet technique. International Journal of Solids and Structures, 43, 6965-6977 (2006) · Zbl 1120.74543 · doi:10.1016/j.ijsolstr.2006.02.018 |
| [29] | Koziol, P., Mares, C., and Esat, I. Wavelet approach to vibratory analysis of surface due to a load moving in the layer. International Journal of Solids and Structures, 45, 2140-2159 (2008) · Zbl 1152.74023 · doi:10.1016/j.ijsolstr.2007.11.008 |
| [30] | Koziol, P., Hryniewicz1, Z., and Mares, C. Wavelet analysis of beam-soil structure response for fast moving train. Journal of Physics: Conference Series, 181, 012052 (2009) |
| [31] | Hong, D. P., Kim, Y. M., and Wang, J. Z. A new approach for the analysis solution of dynamic systems containing fractional derivative. Journal of Mechanical Science and Technology, 20, 658-667 (2006) · doi:10.1007/BF02915983 |
| [32] | Wang, J. Z. Fractional stochastic description of hinge motions in single protein molecules. Chinese Science Bulletin, 56, 495-501 (2011) · doi:10.1007/s11434-010-4218-9 |
| [33] | Wei, D. Coiflet-Type Wavelets: Theory, Design, and Applications, Ph. D. dissertation, The University of Texas, Austin (1998) |
| [34] | Donoho, D. L. Interpolating Wavelet Transforms, Report, Stanford University, Stanford (1992) |
| [35] | Xu, C. F., Cai, C., Pi, M. H., Zhu, C. X., and Li, G. K. Interpolating wavelet and its applications. Conference of International Symposium on Multispectral Image Process, 3545, 428-432 (1998) · doi:10.1117/12.323558 |
| [36] | Daubechies, I. Orthonormal bases of compactly supported wavelets. Communications on Pure and Applied Mathematics, 41, 909-996 (1988) · Zbl 0644.42026 · doi:10.1002/cpa.3160410705 |
| [37] | Comincioli, V., Naldi, G., and Scapolla, T. A wavelet-based method for numerical solution of nonlinear evolution equations. Applied Numerical Mathematics, 33, 291-297 (2000) · Zbl 0964.65112 · doi:10.1016/S0168-9274(99)00095-1 |
| [38] | Metzler, R. and Klafter, J. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports, 339, 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3 |
| [39] | Tarasov, V. E. Review of some promising fractional physical models. International Journal of Modern Physics B, 27, 1330005 (2013) · Zbl 1267.34012 · doi:10.1142/S0217979213300053 |
| [40] | Metzler, R. and Klafter, J. Boundary value problems for fractional diffusion equations. Physica A, 278, 107-125 (2000) · doi:10.1016/S0378-4371(99)00503-8 |
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.
