A wavelet approach for the multi-term time fractional diffusion-wave equation

References

• T.M. Atanackovic, S. Pilipovic, and D. Zorica, A diffusion wave equation with two fractional derivatives of different order, J. Phys. A: Math. Theor. 40 (2007), pp. 5319–5333. doi: 10.1088/1751-8113/40/20/006
   Web of Science ®Google Scholar
• E. Babolian and M. Mordad, A numerical method for solving systems of linear and nonlinear integral equations of the second kind by hat basis function, Comput. Math. Appl. 62 (2011), pp. 187–198. doi: 10.1016/j.camwa.2011.04.066
   Web of Science ®Google Scholar
• A.H. Bhrawy, E.H. Doha, D. Baleanu, and S.S. Ezz-Eldien, A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations, J. Comput. Phys. 293 (2015), pp. 142–156. doi: 10.1016/j.jcp.2014.03.039
   Web of Science ®Google Scholar
• C. Canuto, M. Hussaini, A. Quarteroni, and T. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlage, Berlin, 1988.
   Google Scholar
• C.M. Chen, F. Liu, I. Turner, and V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys. 227 (2007), pp. 886–897. doi: 10.1016/j.jcp.2007.05.012
   Web of Science ®Google Scholar
• Y. Chena, Y. Wua, Y. Cuib, Z. Wanga, and D. Jin, Wavelet method for a class of fractional convection–diffusion equation with variable coefficients, J. Comput. Sci. 1 (2010), pp. 146–149. doi: 10.1016/j.jocs.2010.07.001
   Web of Science ®Google Scholar
• M. Dehghan, M. Safarpoor, and M. Abbaszadeh, Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equations, J. Comput. Appl. Math. 290 (2015), pp. 174–195. doi: 10.1016/j.cam.2015.04.037
   Web of Science ®Google Scholar
• X.L. Ding and Y.L. Jiang, Analytical solutions for the multi-term time? Space fractional advection–diffusion equations with mixed boundary conditions, Nonlinear Anal. RWA 14 (2013), pp. 1026–1033. doi: 10.1016/j.nonrwa.2012.08.014
   Web of Science ®Google Scholar
• R. Du, W.R. Cao, and Z.Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Appl. Math. Model. 34 (2010), pp. 2998–3007. doi: 10.1016/j.apm.2010.01.008
   Web of Science ®Google Scholar
• G.H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities (Cambridge Mathematical Library), 2nd edn, Cambridge University Press, Cambridge, 1952.
   Google Scholar
• G. Hariharan and R. Rajaraman, Two reliable wavelet methods to Fitzhugh–Nagumo (FN) and fractional FN equations, J. Math. Chem. 51 (2013), pp. 2432–2454. doi: 10.1007/s10910-013-0220-1
   Web of Science ®Google Scholar
• M.H. Heydari, M.R. Hooshmandasl, C. Cattani, and F.M. Maalek Ghaini, An efficient computational method for solving nonlinear stochastic Itô integral equations: Application for stochastic problems in physics, J. Comput. Phys. 283 (2015), pp. 148–168. doi: 10.1016/j.jcp.2014.11.042
   Web of Science ®Google Scholar
• M.H. Heydari, M.R. Hooshmandasl, F.M.M. Ghaini, and C. Cattani, A computational method for solving stochastic Itô-Volterra integral equations based on stochastic operational matrix for generalized hat basis functions, J. Comput. Phys. 270 (2014), pp. 402–415. doi: 10.1016/j.jcp.2014.03.064
   Web of Science ®Google Scholar
• M.H. Heydari, M.R. Hooshmandasl, F.M. Maalek Ghaini, and F. Fereidouni, Two-dimensional Legendre wavelets for solving fractional poisson equation with Dirichlet boundary conditions, Eng. Anal. Boundary Elem. 37 (2013), pp. 1331–1338. doi: 10.1016/j.enganabound.2013.07.002
   Web of Science ®Google Scholar
• M.H. Heydari, M.R. Hooshmandasl, Gh. Barid Loghmani, and C. Cattani, Wavelets Galerkin method for solving stochastic heat equation, Int. J. Comput. Math. 93 (2015), pp. 1579–1596. doi: 10.1080/00207160.2015.1067311
   Web of Science ®Google Scholar
• M.H. Heydari, M.R. Hooshmandasl, and F. Mohammadi, Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions, Appl. Math. Comput. 234 (2014), pp. 267–276.
   Web of Science ®Google Scholar
• M.H. Heydari, M.R. Hooshmandasl, and F. Mohammadi, Two-dimensional Legendre wavelets for solving time-fractional telegraph equation, Adv. Appl. Math. Mech. 6(2) (2014), pp. 247–260. doi: 10.4208/aamm.12-m12132
   Web of Science ®Google Scholar
• I. Aziz, A.S. Al-Fhaid, and A. Shah, A numerical assessment of parabolic partial differential equations using Haar and Legendre wavelets, Appl. Math. Model. 37(23) (2013), pp. 9455–9481. doi: 10.1016/j.apm.2013.04.014
   Web of Science ®Google Scholar
• H. Jiang, F. Liu, I. Turner, and K. Burrage, Analytical solutions for the multi-term time fractional diffusion-wave/diffusion equations in a finite domain, Comput. Math. Appl. 64 (2012), pp. 3377–3388. doi: 10.1016/j.camwa.2012.02.042
   Web of Science ®Google Scholar
• T.A.M. Langlands and B.I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys. 205 (2005), pp. 719–736. doi: 10.1016/j.jcp.2004.11.025
   Web of Science ®Google Scholar
• Ü. Lepik, Solving PDEs with the aid of two-dimensional Haar wavelets, Comput. Math. Appl. 61 (2011), pp. 1873–1879. doi: 10.1016/j.camwa.2011.02.016
   Web of Science ®Google Scholar
• F. Liu, V. Anh, and I. Turner, Numerical solution of the space fractional Fokker–Planck equation, J. Comput. Appl. Math. 166 (2004), pp. 209–219. doi: 10.1016/j.cam.2003.09.028
   Web of Science ®Google Scholar
• F. Liu, C. Yang, and K. Burrage, Numerical method and analytical technique of the modified anomalous sub-diffusion equation with a nonlinear source term, J. Comput. Appl. Math. 231 (2009), pp. 160–176. doi: 10.1016/j.cam.2009.02.013
   Web of Science ®Google Scholar
• F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, Stability and convergence of the difference methods for the space-time fractional advection–diffusion equation, Appl. Math. Comp. 191 (2007), pp. 12–20. doi: 10.1016/j.amc.2006.08.162
   Web of Science ®Google Scholar
• M. Mahalakshmi, G. Hariharan, and K. Kannan, The wavelet methods to linear and nonlinear reaction–diffusion model arising in mathematical chemistry, J. Math. Chem. 51 (2013), pp. 2361–2385. doi: 10.1007/s10910-013-0216-x
   Web of Science ®Google Scholar
• J.C. Mason and D.C. Handscomb, Chebyshev Polynomials, CRC Press, Boca Raton, 2003.
   Google Scholar
• K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993.
   Google Scholar
• R.R. Nigmatullin, To the theoretical explanation of the universal response, Phys. Status B Basic Res. 123 (1984), pp. 739–745. doi: 10.1002/pssb.2221230241
   Web of Science ®Google Scholar
• R.R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Status B Basic Res. 133 (1986), pp. 425–430. doi: 10.1002/pssb.2221330150
   Web of Science ®Google Scholar
• K.B. Oldham and J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order, Academic Press, New York, 1974.
   Google Scholar
• A. Pedas and E. Tamme, Spline collocation methods for linear multi-term fractional differential equations, J. Comput. Appl. Math. 236 (2011), pp. 167–176. doi: 10.1016/j.cam.2011.06.015
   Web of Science ®Google Scholar
• I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
   Google Scholar
• M. Polezzi, On the weighted mean value theorem for integrals, Int. J. Math. Educ. Sci. Technol. 37 (2006), pp. 868–870. doi: 10.1080/00207390600741744
   Google Scholar
• S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Amsterdam, 1993.
   Google Scholar
• R. Scherer, S.L. Kalla, L. Boyadjiev, and B. Al-Saqabi, Numerical treatment of fractional heat equations, Appl. Numer. Math. 58 (2008), pp. 1212–1223. doi: 10.1016/j.apnum.2007.06.003
   Web of Science ®Google Scholar
• H. Schiessel, R. Metzler, A. Blumen, and T.F. Nonnenmacher, Generalized viscoelastic models: Their fractional equations with solutions, J. Phys. A: Math. 28 (1995), pp. 6567–6584. doi: 10.1088/0305-4470/28/23/012
   Google Scholar
• V. Srivastava and K.N. Rai, A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues, Math. Comput. Model. 51 (2010), pp. 616–624. doi: 10.1016/j.mcm.2009.11.002
   Google Scholar
• C. Tadjeran, M.M. Meerschaert, and H.-P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys. 213 (2006), pp. 205–213. doi: 10.1016/j.jcp.2005.08.008
   Web of Science ®Google Scholar
• M.P. Tripathi, V.K. Baranwal, R.K. Pandey, and O.P. Singh, A new numerical algorithm to solve fractional differential equations based on operational matrix of generalized hat functions, Commun. Nonlinear Sci. Numer. Simul. 18 (2013), pp. 1327–1340. doi: 10.1016/j.cnsns.2012.10.014
   Web of Science ®Google Scholar
• V.A.H. Ye, F. Liu, and I. Turner, Maximum principle and numerical method for the multi-term time-space Riesz? Caputo fractional differential equations, Appl. Math. Comput. 227 (2014), pp. 531–540.
   Web of Science ®Google Scholar
• S. Yuste, Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys. 216 (2006), pp. 264–274. doi: 10.1016/j.jcp.2005.12.006
   Web of Science ®Google Scholar
• S.B. Yuste and L. Acedo, An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal. 42 (2005), pp. 1862–1874. doi: 10.1137/030602666
   Web of Science ®Google Scholar
• P. Zhuang, F. Liu, V. Anh, and I. Turner, New solution and analytical techniques of the implicit numerical methods for the anomalous sub-diffusion equation, SIAM J. Numer. Anal. 46 (2008), pp. 1079–1095. doi: 10.1137/060673114
   Web of Science ®Google Scholar