Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms

References

[1] O.P. Agrawal, Solution for a fractional diffusion-wave equation defined a bounded domain. Nonlinear Dynam. 29 (2002), 145–155.10.1023/A:1016539022492Suche in Google Scholar

[2] F. Alabau-Boussouira, Control of Partial Differential Equations: On Some Recent Advances on Stabilization for Hyperbolic Equations. Springer, 2010.10.1007/978-0-387-30440-3_97Suche in Google Scholar

[3] A. Alsaedi, J.J. Nieto, V. Venktesh, Fractional electrical circuits. Adv. in Mechanical Engineering7 (2015), 1–7.10.1177/1687814015618127Suche in Google Scholar

[4] C.N. Angstmann, B.I. Henry, A.V. McGann, A fractional-order infectivity SIR model. Physica A - Statistical Mechanics and Its Applications452 (2016), 86–93.10.1016/j.physa.2016.02.029Suche in Google Scholar

[5] I. Area, H. Batarfi, J. Losada, J.J. Nieto, W. Shammakh, Á . Torres, On a fractional order Ebola epidemic model. Adv. Diff. Equa. 2015 (2015), Art. # 278.10.1186/s13662-015-0613-5Suche in Google Scholar

[6] D.A. Benson, S.W. Wheatcraft, M.M. Meerschaert, Application of a fractional advection-dispersion equation. Water Resour. Res. 36 (2000), 1403–1412.10.1029/2000WR900031Suche in Google Scholar

[7] M. Cajić, D. Karličić, M. Lazarević, Damped vibration of a nonlocal nanobeam resting on viscoelastic foundation: fractional derivative model with two retardation times and fractional parameters. Meccanica52 (2017), 363–382.10.1007/s11012-016-0417-zSuche in Google Scholar

[8] L. Cesbron, A. Mellet, K. Trivisa, Anomalous transport of particles in plasma physics. Appl. Math. Lett. 25 (2012), 2344–2348.10.1016/j.aml.2012.06.029Suche in Google Scholar

[9] S.-Y.A. Chang, M.D.M. González, Fractional Laplacian in conformal geometry. Adv. Math. 226 (2011), 1410–1432.10.1016/j.aim.2010.07.016Suche in Google Scholar

[10] D. Chatterjee, A.P. Misra, Nonlinear Landau damping of wave envelopes in a quantum plasma. Physics of Plasmas23 (2016), Art. # 102114-1-10.10.1063/1.4964910Suche in Google Scholar

[11] J.S. Chen, C.W. Liu, Generalized analytical solution for advection-dispersion equation in finite spatial domain with arbitrary time-dependent inlet boundary condition. Hydrol. Earth Syst. Sci. 15 (2011), 2471–2479.10.5194/hess-15-2471-2011Suche in Google Scholar

[12] I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping. J. Diff. Equa. 252 (2012), 1229–1262.10.1016/j.jde.2011.08.022Suche in Google Scholar

[13] X.L. Ding, Y.L. Jiang, Semilinear fractional differential equations based on a new integral operator approach. Commun. Nonlinear Sci. Numer. Simulat. 17 (2012), 5143–5150.10.1016/j.cnsns.2012.03.036Suche in Google Scholar

[14] X.L. Ding, Y.L. Jiang, Analytical solutions for the multi-term time-space fractional advection-diffusion equations with mixed boundary conditions. Nonlinear Anal. RWA14 (2013), 1026–1033.10.1016/j.nonrwa.2012.08.014Suche in Google Scholar

[15] X.L. Ding, J.J. Nieto, Analytical solutions for the multi-term time-space fractional reaction-diffusion equations on an infinite domain. Fract. Calc. Appl. Anal. 18, No 3 (2015), 697–716; 10.1515/fca-2015-0043; https://www.degruyter.com/view/j/fca.2015.18.issue-3/issue-files/fca.2015.18.issue-3.xml.Suche in Google Scholar

[16] X.L. Ding, J.J. Nieto, Analytical solutions for coupling fractional partial differential equations with Dirichlet boundary conditions. Commun. Nonlinear Sci. Numer. Simulat. 52 (2017), 165–176.10.1016/j.cnsns.2017.04.020Suche in Google Scholar

[17] X.L. Ding, J.J. Nieto, Analytical solutions for multi-time scale fractional stochastic differential equations driven by fractional brownian motion and their applications. Entropy63 (2018); 10.3390/e20010063.Suche in Google Scholar

[18] A.M.A. El-Sayed, H.M. Nour, A. Elsaid, A.E. Matouk, A. Elsonbaty, Dynamical behaviors, circuit realization, chaos control, and synchronization of a new fractional order hyperchotic system. Appl. Math. Model. 40 (2016), 3516–3534.10.1016/j.apm.2015.10.010Suche in Google Scholar

[19] M. Ferreira, M. Rodrigues, N. Vieira, Fundamental solution of the multi-dimensional time fractional telegraph equation. Fract. Calc. Appl. Anal. 20, No 4 (2017), 868–894; 10.1515/fca-2017-0046; https://www.degruyter.com/view/j/fca.2017.20.issue-4/issue-files/fca.2017.20.issue-4.xml.Suche in Google Scholar

[20] A. Gamba, M. Grilli, C. Castellani, Renormalization group analysis of the quantum non-linear sigma model with a damping term. Nuclear Physics B556 (1999), 463–484.10.1016/S0550-3213(99)00340-5Suche in Google Scholar

[21] M.G. Hall, T.R. Barrick, From diffusion-weighted MRI to anomalous diffusion imaging. Magn. Reson. Med. 59 (2008), 447–455.10.1002/mrm.21453Suche in Google Scholar PubMed

[22] R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific, Singapore, 2000.10.1142/3779Suche in Google Scholar

[23] M. Ilić, F. Liu, I. Turner, V. Anh, Numerical approximation of a fractional-in-space diffusion equation (II)-with nonhomogeneous boundary condtions. Fract. Calc. Appl. Anal. 9, No 4 (2006), 333–349; available at: http://www.math.bas.bg/complan/fcaa.Suche in Google Scholar

[24] H. Jiang, F. Liu, I. Turner, K. Burrage, Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain. J. Math. Anal. Appl. 389 (2012), 1117–1127.10.1016/j.jmaa.2011.12.055Suche in Google Scholar

[25] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, # 204, Elsevier Science B.V., Amsterdam, 2006.Suche in Google Scholar

[26] A.A. Kilbas, M. Saigo, R.K. Saxena, Solution of Volterra integrodifferential equations with generalized Mittag-Leffler function in the kernels. J. of Integral Equations and Appl. 14, No 4 (2002), 377–396.10.1216/jiea/1181074929Suche in Google Scholar

[27] M. Klimek, A. Malinowska, T. Odzijewicz, Applications of the fractional Sturm-Liouville problem to the space-time fractional diffusion in a finite domain. Fract. Calc. Appl. Anal. 19, No 2 (2016), 516–550; 10.1515/fca-2016-0027; https://www.degruyter.com/view/j/fca.2016.19.issue-2/issue-files/fca.2016.19.issue-2.xml.Suche in Google Scholar

[28] A. Kumar, D.K. Jaiswal, N. Kumar, Analytical solutions of one-dimensional advection-diffusion equation with variable coefficients in a finite domain. J. Earth Syst. Sci. 118 (2009), 539–549.10.1007/s12040-009-0049-ySuche in Google Scholar

[29] M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. Appl. Anal. 20, No 1 (2017), 7–51; 10.1515/fca-2017-0002; https://www.degruyter.com/view/j/fca.2017.20.issue-1/issue-files/fca.2017.20.issue-1.xml.Suche in Google Scholar

[30] T.A.M. Langlands, B.I. Henry, S.L. Wearne, Fractional cable equation models for anomalous electrodiffusion in nerve cells: Finite domain solutions. SIAM J. Appl. Math. 71 (2011), 1168–1203.10.1137/090775920Suche in Google Scholar

[31] F.W. Liu, M.M. Meerschaert, R.J. McGough, P.H. Zhuang, Q.X. Liu, Numerical methods for solving the multi-term time-fractional wave-diffusion equation. Fract. Calc. Appl. Anal. 16, No 1 (2013), 9–25; 10.2478/s13540-013-0002-2; https://www.degruyter.com/view/j/fca.2013.16.issue-1/issue-files/fca.2013.16.issue-1.xml.Suche in Google Scholar PubMed PubMed Central

[32] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, 2010.10.1142/p614Suche in Google Scholar

[33] M. Mamchuev, Solutions of the main boundary value problems for the time-fractional telegraph equation by the green function method. Fract. Calc. Appl. Anal. 20, No 1 (2017), 190–211; 10.1515/fca-2017-0010; https://www.degruyter.com/view/j/fca.2017.20.issue-1/issue-files/fca.2017.20.issue-1.xml.Suche in Google Scholar

[34] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1–77.10.1016/S0370-1573(00)00070-3Suche in Google Scholar

[35] S. Momani, Analytic and approximate solutions of the space- and time-fractional telegraph equations. Appl. Math. Comput. 170 (2005), 1126–1134.10.1016/j.amc.2005.01.009Suche in Google Scholar

[36] Myong-Ha Kim, Guk-Chol Ri, Hyong-Chol O, Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives. Fract. Calc. Appl. Anal. 17, No 1 (2014), 79–95; 10.2478/s13540-014-0156-6; https://www.degruyter.com/view/j/fca.2014.17.issue-1/issue-files/fca.2014.17.issue-1.xml.Suche in Google Scholar

[37] O. Nevanlinna, Convergence of Iterations for Linear Equations. Berlin, 1983.Suche in Google Scholar

[38] B.W. Philippa, R.D. White, R.E. Robson, Analytic solution of the fractional advection-diffusion equation for the time-of-flight experiment in a finite geometry. Phys. Rev. E84 (2011), 1–9.10.1103/PhysRevE.84.041138Suche in Google Scholar PubMed

[39] C.M.A. Pinto, A.R.M. Carvalho, The role of synaptic transmission in a HIV model with memory. Appl. Math. Comput. 292 (2017), 76–95.10.1016/j.amc.2016.07.031Suche in Google Scholar

[40] I. Podlubny, Fractional Differential Equations. Academic Press, New York, 1999.Suche in Google Scholar

[41] Y.Z. Povstenko, Analytical solution of the advection-diffusion equation for a ground-level finite area source. Atmos. Environ. 42 (2008), 9063–9069.10.1016/j.atmosenv.2008.09.019Suche in Google Scholar

[42] A.K. Shukla, J.C. Prajapati, On a generalization of Mittag-Leffler function and its properties. J. Math. Anal. Appl. 336 (2007), 797–811.10.1016/j.jmaa.2007.03.018Suche in Google Scholar

[43] E. Sousa, Finite difference approximations for a fractional advection diffusion problem. J. Comput. Phys. 11 (2009), 4038–4054.10.1016/j.jcp.2009.02.011Suche in Google Scholar

[44] R. Stern, F. Effenberger, H. Fichtner, T. Schäfer, The space-fractional diffusion-advection equation: analytical solutions and critical assessment of numerical solutions. Fract. Calc. Appl. Anal. 17, No 1 (2014), 171–190; 10.2478/s13540-014-0161-9; https://www.degruyter.com/view/j/fca.2014.17.issue-1/issue-files/fca.2014.17.issue-1.xml.Suche in Google Scholar

[45] Ž. Tomovski, R. Garra, Analytic solutions of fractional integro-differential equations of Volterra type with variable coefficients. Fract. Calc. Appl. Anal. 17, No 1 (2014), 38–60; 10.2478/s13540-014-0154-8; https://www.degruyter.com/view/j/fca.2014.17.issue-1/issue-files/fca.2014.17.issue-1.xml.Suche in Google Scholar

[46] D. Valerio, J. Sa da Costa, Introduction to single-input, single-output fractional control. IET Control Theory Appl. 8 (2011), 1033–1057.10.1049/iet-cta.2010.0332Suche in Google Scholar

[47] O. Vasilyeva, F. Lutscher, Competition of three species in an advective environment. Nonlinear Anal. RWA13 (2012), 1730–1748.10.1016/j.nonrwa.2011.12.004Suche in Google Scholar

[48] F.F. Zhang, X.Y. Jiang, Analytical solutions for a time-fractional axisymmetric diffusion-wave equation with a source term. Nonlinear Anal. RWA12 (2011), 1841–1849.10.1016/j.nonrwa.2010.11.015Suche in Google Scholar

[49] P. Zhang, Y.T. Gu, F. Liu, I. Turner, P.K.D.V. Yarlagadda, Time-dependent fractional advection-diffusion equations by an implicit MLS meshless method. Int. J. Numer. Meth. Engng. 13 (2011), 1346–1362.10.1002/nme.3223Suche in Google Scholar