Approximate solution of two-term fractional-order diffusion, wave-diffusion, and telegraph models arising in mathematical physics using optimal homotopy asymptotic method

References

• Mainardi F. Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri A, Mainardi F, editors. Fractal and fractional calculus in continuum mechanics. New York (NY): Springer-Verlag; 1997. p. 291–348.
   Google Scholar
• Podlubny I. Fractional differential equations. New York (NY): Academic Press; 1999.
   Google Scholar
• Sarwar S, Alkhalaf S, Iqbal S, Zahid MA. A note on optimal homotopy asymptotic method for the solutions of fractional order heat- and wave-like partial differential equations. Comput. Math. Appl. 2015;70:942–953.
   Web of Science ®Google Scholar
• Iqbal S, Sarwar F, Mufti MR, Siddique I. Use of optimal asymptotic method for fractional order nonlinear fredholm integro-differential equations. Sci. Int. (Lahore). 2015;27:3033–3040.
   Google Scholar
• Guner O, Bekir A. Traveling wave solutions for time-dependent coefficient nonlinear evolution equations. Waves Random Complex Media. 2015;25:342–349.
   Web of Science ®Google Scholar
• Inc Mustafa, Kılıç Bülent. Classification of traveling wave solutions for time-fractional fifth-order KdV-like equation. Waves Random Complex Media. 2014;24:393–403.
   Web of Science ®Google Scholar
• Mudaliar S. Diffuse waves in a random medium layer with rough boundaries. Waves Random Media. 2001;11:45–60.
   Web of Science ®Google Scholar
• Younesian D, Askari H, Saadatnia Z, et al. Analytical solution for nonlinear wave propagation in shallow media using the variational iterationmethod. Waves Random Complex Media. 2012;22:133–142.
   Web of Science ®Google Scholar
• Ebadi G, Krishnan EV, Labidi M, et al. Analytical and numerical solutions to the Davey--Stewartson equation with power-law nonlinearity. Waves Random Complex Media. 2011;21:559–590.
   Web of Science ®Google Scholar
• Wilhelmsson H, Lazzaro E. Reaction--diffusion problems in the physics of hot plasmas. Bristol: Institute of Physics Publishing; 2001.
   Google Scholar
• Hundsdorfer W, Verwer JG. Numerical solution of time-dependent advection--diffusion--reaction equations. Berlin: Springer-Verlag; 2003.
   Google Scholar
• Sandev T, Metzler R, Tomovski Z. Fractional diffusion equation with a generalized Riemann-Liouville time fractional derivative. J. Phys. A Math. Theor. 2011;44:21. doi:10.1088/1751-8113/44/25/255203.
   Web of Science ®Google Scholar
• Henry BI, Wearne SL. Existence of turing instabilities in a two-species fractional reaction--diffusion system. SIAM J. Appl. Math. 2002;62:870–887.
   Web of Science ®Google Scholar
• Henry BI, Langlands TAM, Wearne SL. Turing pattern formation in fractional activator-inhibitor systems. Phys. Rev. E. 2005;72: doi:10.1103/PhysRevE.72.026101.
   PubMed Web of Science ®Google Scholar
• Haubold HJ, Mathai AM, Saxena RK. Solutions of the reaction--diffusion equations in terms of the H-functions. Bull. Astron. Soc. India. 2007;35:681–689.
   Web of Science ®Google Scholar
• Haubold AJ, Mathai AM, Saxena RK. Further solutions of reaction-diffusion equations in terms of the H-function. J. Comput. Appl. Math. 2011;235:1311–1316.
   Web of Science ®Google Scholar
• Yang AM, Zhang YZ, Cattani C, et al. Application of local fractional series expansion method to solve Klein-Gordon equations on cantor sets. Abstract Appl. Anal. 2014;2014. Article number 372741.
   Google Scholar
• Kumar S, Kumar D, Abbasbandy S, et al. Analytical solution of fractional Navier-Stokes equation by using modified Laplace decomposition method. Ain Shams Eng. J. 2014;5:569–574.
   Google Scholar
• Khalil H, Khan RA, Rashidi MM. Brenstien polynomials and applications to fractional differential equations. Comput. Methods Differ. Equ. 2015;3:14–35.
   Google Scholar
• Marinca V, Herisanu N. Application of homotopy Asymptotic method for solving non-linear equations arising in heat transfer. I. Commun. Heat Mass Transfer. 2008;35:710–715.
   Web of Science ®Google Scholar
• Marinca V, Herisanu N. 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. 2010;329:1450–1459.
   Web of Science ®Google Scholar
• Herisanu N, Marinca V. Accurate analytical solutions to oscillators with discontinuities and fractional-power restoring force by means of the optimal homotopy asymptotic method. Comput. Math. Appl. 2010;60:1607–1615.
   Web of Science ®Google Scholar
• Marinca V, Herisanu N. The optimal homotopy asymptotic method for solving Blasius equation. Appl. Math. Comput. 2014;231:134–139.
   Web of Science ®Google Scholar
• Iqbal S, Idrees M, Siddiqui AM, et al. Some solutions of the linear and nonlinear Klein--Gordon equations using the optimal homotopy asymptotic method. Appl. Math. Comput. 2010;216:2898–2909.
   Web of Science ®Google Scholar
• Iqbal S, Javed A. Application of optimal homotopy asymptotic method for the analytic solution of singular Lane-Emden type equation. Appl. Math. Comput. 2011;217:7753–7761.
   Web of Science ®Google Scholar
• Iqbal S, Siddiqui AM, Ansari AR, et al. Use of optimal homotopy asymptotic method and Galerkin’s finite element formulation in the study of heat transfer flow of a third grade fluid between parallel plates. J. Heat Transfer. 2011;133:091702.
   Web of Science ®Google Scholar
• Rashidi MM, Erfani E, Rostami B. Optimal homotopy asymptotic method for solving viscous flow through expanding or contracting gaps with permeable walls. Trans. IoT Cloud Comput. 2014;2:76–100.
   Google Scholar
• Caputo M. Linear models of dissipation whose Q is almost frequency independent. Part II. J. Roy. Astro. Soc. 1967;13:529–539.
   Web of Science ®Google Scholar
• Li C, Zeng F. Numerical method for fractional calculus. Boca Raton: Chapman and Hall/CRC; 2015.
   Google Scholar
• Li CP, Deng WH. Remarks on fractional derivatives. Appl. Math. Comput. 2007;187:777–784.
   Web of Science ®Google Scholar
• Li C, Wang Y. Numerical algorithm based on Adomian decomposition for fractional differential equations. Comput. Math. Appl. 2009;57:1672–1681.
   Web of Science ®Google Scholar
• Li CP, Qian DL, Chen YQ. On Riemann--Liouville and Caputo derivatives. Discrete Dyn. Nat. Soc. 2011;2011;15p. Article ID 562494.
   Google Scholar
• Caputo M, Fabrizio M. A new deffinition of fractional derivative without singular Kernel. Progr. Fract. Differ. Appl. 2015;1:73–85.
   Google Scholar
• Losada J, Nieto JJ. Properties of a new fractional derivative without singular Kernel. Progr. Fract. Differ. Appl. 2015;1:87–92.
   Google Scholar
• Atangana A. On the new fractional derivative and application to nonlinear Fisher’s reaction--diffusion equation. Appl. Math. Comput. 2016;273:948–956.
   Web of Science ®Google Scholar
• Atangana A, Nieto JJ. Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel. Adv. Mech. Eng. 2015;7:1–6.
   Web of Science ®Google Scholar
• Liu F, Meerschaert MM, et al. Numerical methods for solving the multi term time fractional wave diffusion equation. Fract Calc Appl Anal. 2013;16:9–25.
   PubMed Web of Science ®Google Scholar