A study of a Stefan problem governed with space–time fractional derivatives

[1]   A.S. Chaves, A fractional diffusion equation to describe Lévy flights. Phys. Lett. A, 239, 13–16 (1998).

[2]   D.A. Benson, S.W. Wheatcraft, M. M. Meerschaert, The fractional-order governing equation of Lévy motion, Water Resources Res., 36, 1413–1423 (2000).

[3]   Y. Aoki, M. Sen, S. Paolucci, Approximation of transient temperatures in complex geometries using fractional derivatives, Heat Transfer, 44, 771–777 (2008).

[4]      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, Journal of Mathematical Analysis and Applications, 389, 1117-1127 (2012).

[5]   Zˇ. Tomovskia, T. Sandev, R. Metzler, J. Dubbeldam, Generalized space–time fractional diffusion equation with composite fractional time derivative, Physica A, 391, 2527–42 (2012).

[6]   X.C. Li, M. Y. Xu, S.W. Wang, Analytical solutions to the moving boundary problems with time–space fractional derivatives in drug release devices, J Phys A: Math. Theor., 40, 12131–12141 (2007).

[7]   X.C. Li, M.Y. Xu, S.W. Wang, Scale-invariant solutions to partial differential equations of fractional order with a moving boundary condition,  J Phys A: Math. Theor.,   41, 155202 (2008).

[8]   J. Liu, M. Xu, Some exact solutions to Stefan problems with fractional differential equations, J.  Math Anal. Appl.,  351, 536-542 (2009).

[9]  C. J. Vogl, M. J. Miksis, S. H. Davis, Moving boundary problems governed by anomalous diffusion.Proc. R. Soc. A,468, 3348-3369 (2012).

[10]    S. Das, R. Kumar, P.K. Gupta, Analytical approximate solution of space–time fractional diffusion equation with a moving boundary condition, Z. Naturforsch. A 66 a, 281–288 (2011).

[11]    V.R. Voller, An exact solution of a limit case Stefan problem governed by a fractional diffusion equation, Int. J. Heat Mass Transfer., 53, 5622-5625 (2010).

[12]    Y. Zhou, Y. Wang, W. Bu, Exact solution for a Stefan problem with latent heat a power function of position, International Journal of Heat and Mass Transfer, 69, 451–454 (2014).

[13]    L. Xicheng, M. Xu, X. Jiang, Homotopy perturbation method to time-fractional diffusion equation with a moving boundary condition, Applied Mathematics and Computation, 208, 434–439 (2009).

[14]    Rajeev, M.S. Kushwaha, Homotopy perturbation method for a limit case Stefan problem governed by fractional diffusion equation, Appl. Math Model, 37, 3589-3599 (2013).

[15]    S. Das, Rajeev, Solution of fractional diffusion equation with a moving boundary condition by variational iteration method and Adomian decomposition method, Z Naturforsch . 65a, 793-799 (2010).

[16]    R. Grzymkowski, D. Słota, One-phase inverse Stefan problem solved by Adomian decomposition method, Comput Math Appl., 51, 33-40 (2006).

[17]   Rajeev, M. S. Kushwaha, A. Kumar, An approximate solution to a moving boundary problem with space–time fractional derivative in fluvio-deltaic sedimentation process, Ain Shams Engineering Journal, 4, 889–895 (2013).

[18]   V. Marinca, N. Herisanu, Application of homotopy Asymptotic method for solving non-linear equations arising in heat transfer, Int. Comm. Heat Mass Transfer, 35 , 710–715 (2008).

[19]   N. Herisanu, V. Marinca, Accurate analytical solutions to oscillators with discontinuities and fractional-power restoring force by means of the optimal homotopy asymptotic method. Comput. Math. Appl. 60, 1607–1615 (2010).

[20]   V. Marinca, N. Herisanu, Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method, J. Sound Vib. 329, 1450–1459 (2010).

[21]   S. Iqbal, M. Idrees, A.M. Siddiqui, A.R. Ansari, Some solutions of the linear and nonlinear Klein–Gordon equations using the optimal homotopy asymptotic method, Appl. Math. Comput., 216, 2898–2909 (2010).

[22]   S. Iqbal, A. Javed, Application of optimal homotopy asymptotic method for the analytic solution of singular Lane–Emden type equation, Appl. Math. Comput., 217, 7753–7761 (2011).

[23]   M. S. Hashmi, N. Khan, S. Iqbal, Optimal homotopy asymptotic method for solving nonlinear Fredholm integral equations of second kind, Applied Mathematics and Computation, 218, 10982–10989 (2012).

[24]   M. Ghoreishi ,  A.I.B. Md.  Ismail , A.K.  Alomari ,A.S.Bataineh, The comparison between Homotopy Analysis Method and Optimal Homotopy Asymptotic Method for nonlinear age-structured population Models, Commun Nonlinear Sci. Numer. Simulat., 17, 1163–1177 (2012).

[25]   S. Dinarvand, R. Hosseini, Optimal homotopy asymptotic method for convective–radiative cooling of a lumped system, and convective straight fin with temperature-dependent thermal conductivity, Afrika Matematika., 24, 103-116 (2013).