References
- S. Abbasbandy, The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys. Lett. A 360 (2006), no. 1, 109-113. https://doi.org/10.1016/j.physleta.2006.07.065
- E. Ahmed, A. M. A. El-Sayed, and H. A. A. El-Saka, Equilibrium points, stability and numerical solutions of fractional order predator-prey and rabies models, J. Math. Anal. Appl. 325 (2007), no. 1, 542-553. https://doi.org/10.1016/j.jmaa.2006.01.087
- O. J. J. Algahtani, Comparing the Atangana-Baleanu and Caputo-Fabrizio derivative with fractional order: Allen Cahn model, Chaos Solitons Fractals 89 (2016), 552-559. https://doi.org/10.1016/j.chaos.2016.03.026
- B. S. T. Alkahtani, Chua's circuit model with Atangana-Baleanu derivative with fractional order, Chaos Solitons Fractals 89 (2016), 547-551. https://doi.org/10.1016/j.chaos.2016.03.020
- M. A. Asiru, Sumudu transform and the solution of integral equations of convolution type, Internat. J. Math. Ed. Sci. Tech. 32 (2001), no. 6, 906-910. https://doi.org/10.1080/002073901317147870
- A. Atangana, On the new fractional derivative and application to nonlinear Fisher's reaction-diffusion equation, Appl. Math. Comput. 273 (2016), 948-956.
-
A. Atangana and E. Alabaraoye, Solving a system of fractional partial differential equations arising in the model of HIV infection of
$CD_4{^+}T$ cells and attractor one-dimensional Keller-Segel equations, Adv. Dierence Equ. 2013 (2013), Article 94, 14 pp. https://doi.org/10.1186/1687-1847-2013-14 - A. Atangana and R. T. Alqahtani, Stability analysis of nonlinear thin viscous fluid sheet flow equation with local fractional variable order derivative, J. Comput. Theor. Nanosci. 13 (2016), no. 5, 2710-2717. https://doi.org/10.1166/jctn.2016.4906
- A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Sci. 20 (2016), no. 2, 763-769. https://doi.org/10.2298/TSCI160111018A
- A. Atangana and A. Kilicman, The use of Sumudu transform for solving certain nonlinear fractional heat-like equations, Abstr. Appl. Anal. 2013 (2013), Art. ID 737481, 12 pp.
- A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fractals 89 (2016), 447-454. https://doi.org/10.1016/j.chaos.2016.02.012
- F. B. M. Belgacem and E. H. N. Al-Shemas, Towards a Sumudu based estimation of large scale disasters environmental fitness changes adversely affecting population dispersal and persistence, AIP Conference Proceedings, 1637, 1442 (2014); doi: 10.1063/1.4907311.
- F. B. M. Belgacem and A. A. Karaballi, Sumudu transform fundamental properties investigations and applications, J. Appl. Math. Stoch. Anal. 2006 (2006), Article ID 91083, 23 pp.
- F. B. M. Belgacem, A. A. Karaballi, and S. L Kalla, Analytical investigations of the Sumudu transform and applications to integral production equations, Math. Probl. Eng. 3 (2003), no. 3-4, 103-118.
- M. Caputo, Elasticita e Dissipazione, Zanichelli, Bologna, 1969.
- M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1 (2015), no. 2, 73-85.
- J. Choi and D. Kumar, Solutions of generalized fractional kinetic equations involving Aleph functions, Math. Commun. 20 (2015), no.1, 113-123.
- A. Choudhary, D. Kumar, and J. Singh, Analytical solution of fractional differential equations arising in fluid mechanics by using Sumudu transform method, Nonlinear Eng. Model. Appl. 3 (2014), no. 3, 133-139.
- R. Darzi, B. Mohammadzade, S. Mousavi, and R. Beheshti, Sumudu transform method for solving fractional differential equations and fractional diffusion-wave equation, J. Math. Comput. Sci. 6 (2013), 79-84. https://doi.org/10.22436/jmcs.06.01.08
- H. Eltayeb and A. Kilcman, On some applications of a new integral transform, Int. J. Math. Anal. 4 (2010), no. 1-4, 123-132.
- Z. Z. Ganji, D. D. Ganji, H. Jafari, and M. Rostamian, Application of the homotopy perturbation method to coupled system of partial differential equations with time fractional derivatives, Topol. Methods Nonlinear Anal. 31 (2008), no. 2, 341-348.
- Q. Ghori, M. Ahmed, and A. M. Siddiqui, Application of homotopy perturbation method to squeezing flow of a Newtonian fluid, Int. J. Nonlinear Sci. 8 (2007), 179-184.
- J.-H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg. 178 (1999), no. 3-4, 257-262. https://doi.org/10.1016/S0045-7825(99)00018-3
- J.-H. He, Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput. 135 (2003), no. 1, 73-79. https://doi.org/10.1016/S0096-3003(01)00312-5
- J.-H. He, New interpretation of homotopy perturbation method, Internat. J. Modern Phys. B 20 (2006), no. 18, 2561-2568. https://doi.org/10.1142/S0217979206034819
- H. Jafari, S. Ghasempoor, and C. M. Khalique, A comparison between Adomian's polynomials and He's polynomials for nonlinear functional equations, Math. Probl. Eng. 2013 (2013), Article ID 943232, 4 pp.
- H. Jafari, A. Golbabai, S. Seifi, and K. Sayevand, Homotopy analysis method for solving multi-term linear and nonlinear diusion wave equations of fractional order, Comput. Math. Appl. 59 (2010), no. 3, 1337-1344. https://doi.org/10.1016/j.camwa.2009.06.020
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006.
- A. Kilcman and H. Eltayeb, A note on classication of hyperbolic and elliptic equations with polynomial coefficients, Appl. Math. Lett. 21 (2008), no. 11, 1124-1128. https://doi.org/10.1016/j.aml.2007.11.002
- A. Kilcman and H. Eltayeb, A note on integral transforms and partial differential equations, Appl. Math. Sci. 4 (2010), no. 1-4, 109-118.
- S. Kumar, Y. Khan, and A. Yildirim, A mathematical modeling arising in the chemical systems and its approximate numerical solution, Asia-Pac. J. Chem. Eng. 7 (2012), 835-840. https://doi.org/10.1002/apj.647
- S. J. Liao, The Proposed Homotopy Analysis Technique for the Solutions of Nonlinear Problems, PhD thesis, Shanghai Jiao Tong University, 1992.
- S. J. Liao, Homotopy analysis method a new analytical technique for nonlinear problems, Commun. Nonlinear Sci. Numer. Simul. 2 (1997), 95-100. https://doi.org/10.1016/S1007-5704(97)90047-2
- S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall/CRC Press, Boca Raton, 2003.
- S. J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput. 147 (2004), no. 2, 499-513. https://doi.org/10.1016/S0096-3003(02)00790-7
- J. Liu and G. Hou, Numerical solutions of the space- and time-fractional coupled Burgers equations by generalized differential transform method, Appl. Math. Comput. 217 (2011), no. 16, 7001-7008. https://doi.org/10.1016/j.amc.2011.01.111
- J. Losada and J. J. Nieto, Properties of the new fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1 (2015), no. 2, 87-92.
- A. Mastroberardino, Homotopy analysis method applied to electrohydrodynamic flow, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), no. 7, 2730-2736. https://doi.org/10.1016/j.cnsns.2010.10.004
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, Inc., New York, USA, 1993.
- J. C. Nassar, J. F. Revelli, and R. J. Bowman, Application of the homotopy analysis method to the Poisson-Boltzmann equation for semiconductor devices, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), no. 6, 2501-2512. https://doi.org/10.1016/j.cnsns.2010.09.015
- P. K. Pandey, Om P. Singh, and V. K. Baranwal, An analytic algorithm for the space-time fractional advection-dispersion equation, Comput. Phys. Comm. 182 (2011), no. 5, 1134-1144. https://doi.org/10.1016/j.cpc.2011.01.015
- S.-H. Park and J.-H. Kim, Homotopy analysis method for option pricing under stochastic volatility, Appl. Math. Lett. 24 (2011), no. 10, 1740-1744. https://doi.org/10.1016/j.aml.2011.04.034
- I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, USA, 1999.
- Y. M. Qin and D. Q. Zeng, Homotopy perturbation method for the q-diffusion equation with a source term, Commun. Fract. Calc. 3 (2012), 34-37.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Switzerland, 1993.
- J. Singh, D. Kumar, and A. Kilicman, Application of homotopy perturbation Sumudu transform method for solving heat and wave-like equations, Malays. J. Math. Sci. 7 (2013), no. 1, 79-95.
- J. Singh, D. Kumar, and A. Kilicman, Homotopy perturbation method for fractional gas dynamics equation using Sumudu transform, Abstr. Appl. Anal. 2013 (2013), Article ID 934060, 8 pp.
- H. M. Srivastava, A. K. Golmankhaneh, D. Baleanu, and X.-J. Yang, Local fractional Sumudu transform with applications to IVPs on Cantor sets, Abstr. Appl. Anal. 2014 (2014), Article ID 620529, 7 pp.
- G. K. Watugala, Sumudu transform: A new integral transform to solve differential equations and control engineering problems, Internat. J. Math. Ed. Sci. Tech. 24 (1993), no. 1, 35-43. https://doi.org/10.1080/0020739930240105
- G. K. Watugala, Sumudu transform: A new integral transform to solve differential equations and control engineering problems, Math. Engrg. Indust. 6 (1998), no. 4, 319-329.
- A. Yildirim, He's homotopy perturbation method for solving the space- and time- fractional telegraph equations, Int. J. Comput. Math. 87 (2010), no. 13, 2998-3006. https://doi.org/10.1080/00207160902874653
- S. P. Zhu, An exact and explicit solution for the valuation of American put options, Quant. Finance 6 (2006), no. 3, 229-242. https://doi.org/10.1080/14697680600699811
- M. Zurigat, S. Momani, and A. Alawneh, Analytical approximate solutions of systems of fractional algebraic-differential equations by homotopy analysis method, Comput. Math. Appl. 59 (2010), no. 3, 1227-1235. https://doi.org/10.1016/j.camwa.2009.07.002