zbMATH — the first resource for mathematics

The boundary element method applied to the solution of the diffusion-wave problem. (English) Zbl 07228770
Summary: A Boundary Element Method formulation is developed for the solution of the two-dimensional diffusion-wave problem, which is governed by a partial differential equation presenting a time fractional derivative of order \(\alpha\), with \(1.0<\alpha<2.0\). In the proposed formulation, the fractional derivative is transferred to the Laplacian through the Riemann-Liouville integro-differential operator; then, the basic integral equation of the method is obtained through the Weighted Residual Method, with the fundamental solution of the Laplace equation as the weighting function. In the final expression, the presence of additional terms containing the history contribution of the boundary variables constitutes the main difference between the proposed formulation and the standard one. The proposed formulation, however, works well for \(1.5\leq\alpha<2.0\), producing results with good agreement with the analytical solutions and with the Finite Difference ones.
65 Numerical analysis
35 Partial differential equations
Full Text: DOI
[1] Miller, KS; Ross, B., An introduction to the fractional calculus and fractional differential equations (1993), Wiley-Interscience
[2] Ortigueira, MD, Fractional calculus for scientist and engineers, 84 (2011), Springer, Lectures Notes in Electrical Engineering
[3] Mainardi, F.; Paradisi, P., A model of diffusive waves in viscoelasticity, (Proceedings of the 36th conference on decision & control (1997), San Diego, California, USA), 4961-4966
[4] Mainardi, F.; Paradisi, P., Fractional diffusive waves, J Comput Acoust, 9, 1417-1436 (2001) · Zbl 1360.76272
[5] El-Saka H, Ahmed E. A fractional order network model for ZIKA. bioRxiv 2016 doi: 10.1101/039917.
[6] Dalir, M.; Bashour, M., Applications of fractional calculus, Appl Math Sci, 4, 1021-1032 (2010) · Zbl 1195.26011
[7] Sun, H.; Zhang, Y.; Baleanu, D.; Chen, W.; Chen, Y., A new collection of real world applications of fractional calculus in science and engineering, Commun Nonlinear Sci Numer Simul, 64, 213-231 (2018)
[8] Carrer, JAM; Seaid, M.; Trevelyan, J.; Solheid, BS, The boundary element method applied to the solution of the anomalous diffusion, Eng Anal Bound Elem, 109, 129-142 (2019) · Zbl 07127496
[9] Brebbia, CA; Telles, JCF; Wrobel, LC, Boundary element techniques: theory and application in engineering (1984), Springer Verlag
[10] Murillo, JQ; Yuste, SB, An explicit difference method for solving fractional diffusion and diffusion-wave equations in the Caputo form, J Comput Nonlinear Dyn, 6, Article 021014 pp. (2011), 6 pages doi:101115/1.4002687
[11] Yuste, SB; Acedo, L., An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J Numer Anal, 42, 1862-1874 (2005) · Zbl 1119.65379
[12] Murio, DA, Implicit finite difference approximation for time fractional diffusion equations, Comput Math Appl, 56, 1138-1145 (2008) · Zbl 1155.65372
[13] Meerschaert, MM; Tadjeran, C., Finite difference approximations for fractional advection-dispertion flow equations, J Comput Appl Math, 172, 65-77 (2004) · Zbl 1126.76346
[14] Li, W.; Li, C., Second order explicit difference schemes for the space fractional advection diffusion equation, Appl Math Comput, 257, 446-457 (2015) · Zbl 1339.65129
[15] Deng, WH, Finite element method for the space and time fractional Fokker-Planck equations, SIAM J Numer Anal, 47, 204-226 (2008)
[16] Roop, JP, Computational aspects of FEM Approximation of fractional advection dispersion equations on bounded domains in ℜ^2, J Comput Appl Math, 193, 243-268 (2006) · Zbl 1092.65122
[17] Huang, Q.; Huang, G.; Zhan, H., A finite element solution for the fractional advection-dispersion equations, Adv Water Res, 31, 1578-1589 (2008)
[18] Katsikadelis, JT, The BEM for numerical solution of partial fractional differential equations, Comput Math Appl, 62, 891-901 (2011) · Zbl 1228.74103
[19] Dehghan, M.; Safarpoor, M., The dual reciprocity boundary elements method for the linear and nonlinear two-dimensional time-fractional partial differential equations, Math Methods Appl Sci, 39, 3979-3995 (2016) · Zbl 1347.65182
[20] Carrer, JAM; Mansur, WJ; Vanzuit, RJ, Scalar Wave equation by the boundary element method: a D-BEM approach with non-homogeneous initial conditions, Comput Mech, 44, 31-44 (2009) · Zbl 1162.74483
[21] Houbolt, JC, A recurrence matrix solution for the dynamic response of elastic aircraft, J Aeronaut Sci, 17, 540-550 (1950)
[22] Ray, SS, Exact solutions for time-fractional diffusion-wave equations by decomposition method, Phys Scr, 75, 53-61 (2007) · Zbl 1197.35147
[23] Mansur WJ. A Time-Stepping Technique to Solve Wave Propagation Problems Using the Boundary Element Method, Ph.D. Thesis, University of Southampton, 1983.
[24] Carrer, JAM; Costa, VL, Boundary element method formulation for the solution of the Scalar Wave equation in one-dimensional problems, J Braz Soc Mech Sci Eng, 37, 959-971 (2015)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.