### Novel solution methods for initial boundary value problems of fractional order with conformable differentiation

Mehmet Yavuz

#### Abstract

In this work, we develop a formulation for the approximate-analytical solution of fractional partial differential equations (PDEs) by using conformable fractional derivative. Firstly, we redefine the conformable fractional Adomian decomposition method (CFADM) and conformable fractional modified homotopy perturbation method (CFMHPM). Then, we solve some initial boundary value problems (IBVP) by using the proposed methods, which can analytically solve the fractional partial differential equations (FPDE). In order to show the efficiencies of these methods, we have compared the numerical and exact solutions of the IBVP. Also, we have found out that the proposed models are very efficient and powerful techniques in finding approximate solutions for the IBVP of fractional order in the conformable sense.

#### Keywords

Conformable fractional derivative; approximate-analytical solution; Adomian decomposition method; modified homotopy perturbation method

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#### References

Zhang, Y. (2009). A finite difference method for fractional partial differential equation. Applied Mathematics and Computation, 215(2), 524-529.

Ibrahim, R.W. (2011). On holomorphic solutions for nonlinear singular fractional differential equations. Computers & Mathematics with Applications, 62(3), 1084-1090.

Odibat, Z. and Momani, S. (2008). A generalized differential transform method for linear partial differential equations of fractional order. Applied Mathematics Letters, 21(2), 194-199.

Odibat, Z. and Momani, S. (2008). Numerical methods for nonlinear partial differential equations of fractional order. Applied Mathematical Modelling, 32(1), 28-39.

Bildik, N. and Bayramoglu, H. (2005). The solution of two dimensional nonlinear differential equation by the Adomian decomposition method. Applied mathematics and computation, 163(2), 519-524.

Bildik, N., Konuralp, A., Bek, F.O., & Küçükarslan, S. (2006). Solution of different type of the partial differential equation by differential transform method and Adomian’s decomposition method. Applied Mathematics and Computation, 172(1), 551-567.

Daftardar-Gejji, V. and Jafari, H. (2005). Adomian decomposition: A tool for solving a system of fractional differential equations. Journal of Mathematical Analysis and Applications, 301(2), 508-518.

Elbeleze, A.A., Kılıçman, A., & Taib, B.M. (2013). Homotopy perturbation method for fractional Black-Scholes European option pricing equations using sumudu transform. Mathematical Problems in Engineering, 2013.

El-Sayed, A. and Gaber, M. (2006). The adomian decomposition method for solving partial differential equations of fractal order in finite domains. Physics Letters A, 359(3), 175-182.

El-Wakil, S.A., Abdou, M.A., & Elhanbaly, A. (2006). Adomian decomposition method for solving the diffusion–convection–reaction equations. Applied Mathematics and Computation, 177(2), 729-736.

Gülkaç, V. (2010). The homotopy perturbation method for the Black–Scholes equation. Journal of Statistical Computation and Simulation, 80(12), 1349-1354.

Momani, S. and Odibat, Z. (2007). Numerical approach to differential equations of fractional order. Journal of Computational and Applied Mathematics, 207(1), 96-110.

Momani, S. and Odibat, Z. (2007). Homotopy perturbation method for nonlinear partial differential equations of fractional order. Physics Letters A, 365(5), 345-350.

Evirgen, F. and Özdemir, N. (2012). A fractional order dynamical trajectory approach for optimization problem with HPM. In: D. Baleanu, Machado, J.A.T., Luo, A.C.J., eds. Fractional Dynamics and Control, Springer, 145-155.

Yavuz, M., Ozdemir, N., & Okur, Y.Y. (2016). Generalized differential transform method for fractional partial differential equation from finance, Proceedings, International Conference on Fractional Differentiation and its Applications, Novi Sad, Serbia, 778-785.

Kurulay, M., Secer, A., & Akinlar, M.A. (2013). A new approximate analytical solution of Kuramoto-Sivashinsky equation using homotopy analysis method. Applied Mathematics & Information Sciences, 7(1), 267-271.

Turut, V. and Güzel, N. (2013). Multivariate pade approximation for solving nonlinear partial differential equations of fractional order. Abstract and Applied Analysis, 2013.

Javidi, M. and Ahmad, B. (2013). Numerical solution of fractional partial differential equations by numerical Laplace inversion technique. Advances in Difference Equations, 2013(1), 375.

Madani, M., Fathizadeh, M., Khan, Y., & Yildirim, A. (2011). On the coupling of the homotopy perturbation method and Laplace transformation. Mathematical and Computer Modelling, 53(9), 1937-1945.

Khalil, R., Al Horani, M., Yousef, A., & Sababheh, M. (2014). A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264, 65-70.

Anderson, D. and Ulness, D. (2015). Newly defined conformable derivatives. Adv. Dyn. Syst. Appl, 10(2), 109-137.

Atangana, A., Baleanu, D., & Alsaedi, A. (2015). New properties of conformable derivative. Open Mathematics, 13(1), 889-898.

Çenesiz, Y., Baleanu, D., Kurt, A., & Tasbozan, O. (2017). New exact solutions of Burgers’ type equations with conformable derivative. Waves in Random and Complex Media, 27(1), 103-116.

Avcı, D., Eroglu, B.I., & Ozdemir, N. (2016). Conformable heat problem in a cylinder, Proceedings, International Conference on Fractional Differentiation and its Applications, 572-581.

Avcı, D., Eroğlu, B.B.İ., & Özdemir, N. (2017). Conformable fractional wave-like equation on a radial symmetric plate. In: A. Babiarz, Czornik, A., Klamka, J., Niezabitowski, M., eds. Theory and Applications of Non-Integer Order Systems, Springer, 137-146.

Avci, D., Eroglu, B.B.I., & Ozdemir, N. (2017). Conformable heat equation on a radial symmetric plate. Thermal Science, 21(2), 819-826.

Abdeljawad, T. (2015). On conformable fractional calculus. Journal of computational and Applied Mathematics, 279, 57-66.

Acan, O. and Baleanu, D. (2017). A new numerical technique for solving fractional partial differential equations. arXiv preprint arXiv:1704.02575,

Adomian, G. (1988). A review of the decomposition method in applied mathematics. Journal of Mathematical Analysis and Applications, 135(2), 501-544.

Demir, A., Erman, S., Özgür, B., & Korkmaz, E. (2013). Analysis of fractional partial differential equations by Taylor series expansion. Boundary Value Problems, 2013(1), 68.

Momani, S. and Odibat, Z. (2007). Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations. Computers & Mathematics with Applications, 54(7), 910-919.

DOI: http://dx.doi.org/10.11121/ijocta.01.2018.00540

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