Model of Coupled System of Fractional Reaction-Diffusion Within a New Fractional Derivative Without Singular Kernel
In this chapter, the approximate analytical solutions of a new reaction-diffusion fractional time model are studied. For this analysis is used the p-homotopy transform method based on different kernels (power, exponential and Mittag-Leffler). The system nonlinearities are addressed by the Adomian polynomials. The system convergence is studied by determining the interval of the convergence by \(\hbar \)-curves, as well as, searching for the optimal value of \(\hbar \) which minimize the residual error. Therefore, the optimal \(\hbar \) value is calculated to estimate the order \(\beta \) error. At the end of the chapter, we explained the obtained behavior by plotting the solutions in 3D. The results are accurate.
KeywordsFractional calculus Atangana–Baleanu fractional derivative Cubic isothermal auto-catalytic chemical system
José Francisco Gómez Aguilar acknowledges the support provided by CONACyT: Cátedras CONACyT para jóvenes investigadores 2014.
- 4.Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular Kernel. Progr. Fract. Differ. Appl. 1(2), 73–85 (2015)Google Scholar
- 7.Atangana, A., Nieto, J.J.: Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel. Adv. Mech. Eng. 7, 1–7 (2015)Google Scholar
- 14.Abro, K.A., Hussain, M., Baig, M.M.: An analytic study of molybdenum disulfide nanofluids using the modern approach of Atangana-Baleanu fractional derivatives. Eur. Phys. J. Plus 132(10), 1–14 (2017)Google Scholar
- 19.El-Tawil, M.A., Huseen, S.N.: The q-homotopy analysis method (q-ham). Int. J. Appl. Math. Mech. 8(15), 51–75 (2012)Google Scholar
- 22.Liao, S.-J.: The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. thesis, Shanghai Jiao Tong University (1992)Google Scholar
- 30.Khan, Y., Austin, F.: Application of the Laplace decomposition method to nonlinear homogeneous and non-homogenous advection equations. Z. Nat. Sect. A 65, 849–853 (2010)Google Scholar
- 32.Yin, X.B., Kumar, S., Kumar, D.: A modified homotopy analysis method for solution of fractional wave equations. Adv. Fract. Dyn. Mech. Eng. 7(12), 1–8 (2015)Google Scholar
- 35.Abo-Dahab, S.M., Mohamed, M.S., Nofal, T.A.: A one step optimal homotopy analysis method for propagation of harmonic waves in nonlinear generalized magnetothermoelasticity with two relaxation times under influence of rotation. Abstract and Applied Analysis, vol. 1, pp. 1–15. Hindawi Publishing Corporation, Cairo (2013)Google Scholar
- 37.Gondal, M.A., Arife, A.S., Khan, M., Hussain, I.: An efficient numerical method for solving linear and nonlinear partial differential equations by combining homotopy analysis and transform method. World Appl. Sci. J. 14(12), 1786–1791 (2011)Google Scholar
- 40.Singh, J., Kumar, D., Swroop, R., Kumar, S.: An efficient computational approach for time-fractional Rosenau-Hyman equation. Neural Comput. Appl. 45, 192–204 (2017)Google Scholar