A comparative study of analytical solutions of space-time fractional hyperbolic-like equations with two reliable methods

References

• Agrawal, O. P. (2002). Solution for a fractional diffusionwave equation defined in a bounded domain. Nonlin. Dyn, 29, 145–155.
   Web of Science ®Google Scholar
• Andrezei, H. (2002). Multi-dimensional solutions of space-time-fractional diffusion equations. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci, 458(2002), 429–450.
   Google Scholar
• Arshad, M., Lu, D., & Wang, J. (2017). (N + 1)-dimensional fractional reduced differential transform method for fractional order partial differential equations. Communications in Nonlinear Science and Numerical Simulation, 48, 509–519.
   Web of Science ®Google Scholar
• Atangana, A., & Alabaraoye, E. (2013). Exact Solutions Fractional Heat-Like and Wave-Like Equations with Variable Coefficients. Scientific Reports, 2(2), 663–667. doi:10.4172/scientificreports.633
   Google Scholar
• Atangana, A., & Baleanu, D. (2016). New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. Thermal Science, 20(2), 763–769.
   Web of Science ®Google Scholar
• Baleanu, D., Machado, J. A. T., & Luo, A. C. (2012). Fractional dynamics and control. Springer.
   Google Scholar
• Carpinteri, A., & Mainardi, F. (1997). Fractals and fractional calculus in continuum mechanics. Springer Verlag, Wien, New York.
   Google Scholar
• El-Sayed, A. M. A. (1996). Fractional - order diffusionwave equation. International Journal of Theoretical Physics, 35(2), 311–322.
   Web of Science ®Google Scholar
• Fu, Z.-J., Chen, W., & Yang, H.-T. (2013). Boundary particle method for Laplace transformed time fractional diffusion equations. Journal of Computational Physics, 235, 52–66.
   Web of Science ®Google Scholar
• Fu, Z.-J., Chen, W., & Ling, L. (2015). Method of approximate particular solutions for constant- and variable-order fractional diffusion models. Engineering Analysis with Boundary Elements, 57, 37–46.
   Web of Science ®Google Scholar
• Gupta, V. G., & Kumar, P. (2012). Applications of laplace homotopy analysis method for solving fractional heat- and wave-like equations. Global Journal of Science Frontier Research Mathematics and Decision Sciences, 12(6), 39–48.
   Google Scholar
• Hanyga, A. (2002). Multidimensional solutions of time - fractional diffusion wave equations. Proc. R. Soc. Lond. A, 458, 933–957.
   Web of Science ®Google Scholar
• He, J. H. (1998). Nonlinear oscillation with fractional derivative and its applications. Int. Conf. Vibrating Engg’98, Dalian, 288–291.
   Google Scholar
• He, J. H. (1999). Homotopy perturbation technique. Comput. Methods Appl. Mechanics Engg, 178(3-4), 257–262.
   Web of Science ®Google Scholar
• Herzallah, M. A. E., A.M.A, E.-S., & Baleanu, D. (2010). On the fractional- order diffusion - wave process, Rom. Journ. Phys. 55, 274–284.
   Google Scholar
• Holliday, J. R., Rundle, J. B., Tiampo, K. F., Klein, W., & Donnellan, A. (2006). Modification of the pattern informatics method for forecasting large earthquake events using complex eigen factors. Tectonophysics, 413(1-2), 87–91.
   Web of Science ®Google Scholar
• Jafari, H., & Momani, S. (2007). Solving fractional diffusion and wave equations by modified homotopy perturbation method. Physics Letters A, 370(5-6), 388–396.
   Web of Science ®Google Scholar
• Jesus, I. S., & Machado, J. A. T. (2008). Fractional control of heat diffusion systems. Nonlinear Dynamics, 54(3), 263–282.
   Web of Science ®Google Scholar
• Kashuri, A., & Fundo, A. (2013). A new integral transform. Adv Theoretical Appl Math, 8(1), 27–43.
   Google Scholar
• Kashuri, A., Fundo, A., & Kreku, M. (2013). Mixture of a new integral transform and homotopy perturbation method for solving nonlinear partial differential equations. Advances in Pure Mathematics, 03(03), 3, 317–23.
   Google Scholar
• Khader, M., Kumar, S., & Abbasbandy, S. (2016). Fractional homotopy analysis transforms method for solving a fractional heat - like physical model. Walailak J Sci & Tech, 13(5), 337–353.
   Google Scholar
• Klafter, J., Blumen, A., & Zumofen, G. (1984). Fractal behavior in trapping and reaction: a random walk study. Journal of Statistical Physics, 36(5-6), 561–578.
   Web of Science ®Google Scholar
• Kunjan, S., Twinkle, S., & Kilicman, A. (2016). Combination of integral and projected differential transform methods for time-fractional gas dynamics equations. Ain Shams Eng J, 2017 doi:10.1016/j.asej.2016.09.012
   Google Scholar
• Magin, R. L. (2006). Fractional Calculus in Bioengineering, Begell House Publisher, Inc. Connecticut.
   Google Scholar
• Mainardi, F. (1971). Fractional calculus: Some basic problems in continuum and statistical mechanics, in: A. carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, SpringerVerlag, New York, 291–348.
   Google Scholar
• Mainardi, F. (1996). Fractional relaxation oscillation and fractional diffusion wave phenomena. Chaos, Solitons and Fractals, 7(9), 1461–1477.
   Web of Science ®Google Scholar
• Manolis, G. D., & Rangelov, T. V. (2006). Non-homogeneous elastic waves in solid: notes on the vector decomposition technique. Soil Dyn Earthq Eng, 26(10), 952–959.
   Web of Science ®Google Scholar
• Mentrelli, A., & Pagnini, G. (2015). Front propagation in anomalous diffusive media governed by time - fractional diffusion. Journal of Computational Physics., 293, 427–441.
   Web of Science ®Google Scholar
• Miller, K. S., & Ross, B. (1993). An Introduction to the fractional calculus and fractional differential equations. Wiley, New York, 1993.
   Google Scholar
• Molliq, Y. R., Noorani, M. S. M., & Hashim, I. (2009). Variational iteration method for fractional heat- and wave-like equations. Nonlinear Analysis: Real World Applications, 10(3), 1854–1869.
   Web of Science ®Google Scholar
• Momani, S. M. (2005). Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method. Applied Mathematics and Computation, 165(2), 459–472.
   Web of Science ®Google Scholar
• Özis, T., & Agirseven, D. (2008). He’s homotopy perturbation method for solving heat-like and wave-like equations with variable coefficients. Phys Lett A, 372, 5944–5950.
   Web of Science ®Google Scholar
• Pandey, R. K., & Mishra, H. K. (2017). Semi- analytic numerical method for solution of time-space fractional heat and wave type equations with variable coefficients. Open Physics, 15(1), 74–86.
   Web of Science ®Google Scholar
• Pang, G., Chen, W., & Fu, Z. (2015). Space-fractional advection-dispersion equations by the Kansa method. Journal of Computational Physics, 293, 280–296.
   Web of Science ®Google Scholar
• Secer, A. (2012). Approximate analytic solution of fractional heat-like and wave-like equations with variable coefficients using the differential transforms method. Advances in Difference Equations, 2012(1), 198. doi:10.1186/1687-1847-2012-198.
   Web of Science ®Google Scholar
• Singh, B. K., & Srivastava, V. K. (2015). Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM. R. Soc. Open Sci. 29, 2(4), 140511. doi:10.1098/rsos.140511.
   Web of Science ®Google Scholar
• Singh, B. K., & Kumar, P. (2018). Homotopy perturbation transform method for solving fractional partial differential equations with proportional delay. Sema Journal, 75(1), 111–125.
   Google Scholar
• Singh, B. K., & Kumar, P. (2017). Extended Fractional Reduced Differential Transform for Solving Fractional Partial Differential Equations with Proportional Delay. International Journal of Applied and Computational Mathematics, 3(S1), 631–649.
   Google Scholar
• Singh, B. K., & Kumar, P. (2017). Fractional variation iteration method for solving fractional partial differential equations with proportional delay. Int. J. Differ. Equ., 2017, 5206380. doi:10.1155/2017/5206380.
   Google Scholar
• Singh, B. K., & Kumar, P. (2016). FRDTM for numerical simulation of multi-dimensional, time-fractional model of Navier-Stokes equation. Ain Shams Engineering Journal, doi:10.1016/j.asej.2016.04.009.
   Web of Science ®Google Scholar
• Singh, J., Kumar, D., & Kilicman, A. (2013). Application of homotopy perturbation Sumudu transform method for solving heat and wave-like equations. Malaysian J Math. Sc, 7(1), 79–95.
   Google Scholar
• Shou, D. H., & He, J. H. (2008). Beyond adomian method: the variational iteration method for solving heat-like and wave-like equations with variable coefficients. Phys Lett A, 372(3), 233–237.
   Web of Science ®Google Scholar
• Taghizadeh, N., & Noori, S. R. M. (2017). Reduced differential transform method for solving parabolic-like and hyperbolic-like equations. SeMA Journal, 74(4), 559. doi:10.1007/s40324-016-0101-1.
   Google Scholar
• Tarasov, V. E. (2008). Fractional vector calculus and fractional Maxwell’s equations. Anna. of Phys, 323(11), 2756–2778.
   Web of Science ®Google Scholar
• Tang, B., Wang, X., Wei, L., & Zhang, X. (2014). Exact solutions of fractional heat-like and wave-like equations with variable coefficients. International Journal of Numerical Methods for Heat & Fluid Flow, 24(2), 455–467. doi:10.1108/HFF-05-2012-0106.
   Web of Science ®Google Scholar
• Wazwaz, A. M., & Goruis, A. (2004). Exact solution for heat-like and wave-like equations with variable coefficient. Journal of Applied Mathematics and Computing., 149, 51–59.
   Google Scholar
• Xu, H., & Cang, J. (2008). Analysis of a time fractional wave-like equation with the homotopy analysis method. Phys. Lett. A, 372(8), 1250–1255.
   Web of Science ®Google Scholar
• Yin, F., Song, J., & Cao, X. (2013). A general iteration formula of VIM for fractional heat- and wave-like equations. Journal of Applied Mathematics, 2013, 428079. doi:10.1155/2013/428079.
   Web of Science ®Google Scholar
• Yin, X. B., Kumar, S., & Kumar, D. (2015). A modified homotopy analysis method for solution of fractional wave equations. Advances in Mechanical Engineering, 7(12), 1–8.
   Web of Science ®Google Scholar
• Zhang, X., Zhao, J., Liu, J., & Tang, B. (2014). Homotopy perturbation method for two dimensional time-fractional wave equation. Applied Mathematical Modelling, 38(23), 5545–5552.
   Web of Science ®Google Scholar
• Zhang, S., Zhu, R., & Zhang, L. (2016). Exact solutions of time fractional heat-like and wave-like equations with variable coefficients. THERMAL Science, 20(suppl. 3), 689–S693.
   Web of Science ®Google Scholar