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Article Content:-
Abstract
This article investigates the approximate analytical solutions of the time-fractional diffusion equations using a novel analytical approach, namely the Sumudu transform iterative method. The time-fractional derivatives are considered in the Caputo sense. The analytical solutions are found in closed form, in terms of Mittag-Leffler functions. Furthermore, the findings are shown graphically, and the solution graphs demonstrate a strong relationship between the approximate and exact solutions.
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References
Bairwa, R. K. and Singh, K. (2020). An efficient analytical approach for solving fractional Fokker-Planck equations, Malaya Journal of Matematik, 8(3), 1248-1258.
Bairwa, R.K., Kumar, A. and Singh, K. (2020). Analytical solutions for time-fractional Cauchy reaction-diffusion equations using iterative Laplace transform method, Jñānābha, 50(1), 207-217.
Belgacem, F.B.M. and Karaballi, A. A. (2006). Sumudu transform fundamental properties investigations and applications, International Journal of Stochastic Analysis.
Bhalekar, S. and Daftardar-Gejji, V. (2011). Convergence of the new iterative method, International Journal of Differential Equations, Article ID 989065, 10 pages.
Bhalekar, S. and Daftardar-Gejji, V. (2010). Solving evolution equations using a new iterative method, Numerical Methods for Partial Differential Equations: An International Journal, 26(4), 906-916.
Çetinkaya, A. and Kıymaz, O. (2013). The solution of the time-fractional diffusion equation by the generalized differential transform method, Mathematical and Computer Modelling, 57(9-10), 2349-2354.
Daftardar-Gejji, V. and Bhalekar, S. (2010). Solving fractional boundary value problems with Dirichlet boundary conditions using a new iterative method, Computers & Mathematics with Applica -tions, 59(5), 1801-1809.
Daftardar-Gejji, V. and Jafari, H. (2006). An iterative method for solving nonlinear functional equations, Journal of Mathematical Analysis and Applications, 316(2), 753-763.
Das,S.(2009).Analytical solution of a fractional diffusion equation by variational iteration method ,Computers & Mathematics with Applications, 57(3), 483-487.
Demiray, S.T., Bulut, H. and Belgacem, F. B. M. (2015). Sumudu transform method for analytical solutions of fractional type ordinary differential equations, Mathematical Problems in Engineering, Article ID 131690, 6 pages.
Hilfer, R. (Ed.). (2000). Applications of fractional calculus in physics, World scientific.
Khan, M. and Hussain, M. (2011). Application of Laplace decomposition method on semi-infinite domain, Numer. Algorithms, 56, 211-218.
Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. (2006). Theory and applications of fractional differential equations (Vol. 204), Elsevier.
Kumar, M. and Daftardar-Gejji,V. (2018). Exact solutions of fractional partial differential equations by Sumudu transform iterative method, Fractional Calculus and Fractional Differential Equations, Birkhuser, Singapore, 157-180.
Kumar, S., Yildirim, A., Khan, Y. and Wei, L. (2012). A fractional model of the diffusion equation and its analytical solution using Laplace transform, Scientia Iranica, 19(4), 1117-1123.
Liao, S. (2003). Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall/CRC Press. Boca Raton.
Miller, K. S. and Ross, B. (1993). An introduction to the fractional calculus and fractional differential equations, John Wiley & Sons, New York, USA.
Mittag-Leffler, G.M. (1903). Sur la nouvelle fonction, C. R. Acad. Sci. Paris, No.137, 554-558.
Nakhushev, A. M. (2003). Fractional Calculus and its Application, Fizmatlit, Moscow.
Oldham, K. B. and Spanier, J. (1974). The Fractional Calculus, Academic Press, New York.
Podlubny, I. (1999). Fractional differential equations, mathematics in science and engineering, Academic Press, New York, USA.
Ray, S. S. and Bera, R. K. (2006). Analytical solution of a fractional diffusion equation by Adomian decomposition method. Applied Mathematics and Computation, 174(1), 329-336.
Shah, K., Khalil, H. and Khan, R. A. (2018). Analytical solutions of fractional order diffusion equations by natural transform method. Iranian Journal of Science and Technology, Transactions A: Science, 42(3), 1479-1490.
Shah, R., Khan, H., Mustafa, S., Kumam, P. and Arif, M. (2019). Analytical solutions of fractional-order diffusion equations by natural transform decomposition method. Entropy, 21(6), 557.
Uchaikin, V. V. (2008). Method of fractional derivatives (Artishok, Ul’janovsk, 2008).
Wang, K. and Liu, S. (2016). A new Sumudu transform iterative method for time-fractional Cauchy reaction-diffusion equation, Springer Plus, 5(1), 1-20.
Watugala, G. (1993). Sumudu transform: a new integral transform to solve differential equations and control engineering problems, International Journal of Mathematical Education in Science and Technology, 24(1), 35-43.
Wiman, A. (1905). Uber die Nullstellen der Funktionen Acta Math., 29, 217-234.