Mellin transform and conformable fractional operator: applications

Springer Science and Business Media LLC - Tập 76 - Trang 203-215 - 2018
Mousa Ilie1,2, Jafar Biazar2,3, Zainab Ayati4
1Department of Mathematics, Guilan Science and Research Branch, Islamic Azad University, Rasht, Iran
2Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran
3Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
4Department of Engineering Sciences, Faculty of Technology and Engineering, East of Guilan, University of Guilan, Rudsar-Vajargah, Iran

Tóm tắt

A wide range of fractional differential equations in applied sciences can be solved by integral transformations. In the present work, first some new theorems related to the Mellin transform and the conformable fractional operator are established, and then a few conformable fractional equations such as wave and heat equations are solved through the use of results generated.

Tài liệu tham khảo

Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015) Butzer, P.L., Jansche, S.: A direct approach to the Mellin transform. J. Fourier Anal. Appl. 3(4), 325–376 (1997) Butzer PL, Jansche S, Mellin transform theory and the role of its differential and integral operators. In: Rusev P, Dimovski I, Kiryakova V (eds) Proceedings of the 2nd international workshop in transform methods and special functions (Varna 96). Bulg. Acad. Sci., Sofa, (1998) 63–83 Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Tables of integral transforms, vol. 1. McGraw-Hill, New York City (1954) Flajolet, P., Regnier, M., Sedgewick, R.: Some uses of the Mellin integral transform in the analysis of algorithms. In: Apostolico, A., Galil, Z. (eds.) Combinatorial algorithms on words, pp. 241–254. Springer, Berlin (1985) Flajolet, P., Gourdon, X., Dumas, P.: Mellin transforms and asymptotics: harmonic sums. Theor. Comput. Sci. 144(1–2), 3–58 (1995) Ilie, M., Biazar, J., Ayati, Z.: General solution of Bernoulli and Riccati fractional differential equations based on conformable fractional derivative. Int. J. Appl. Math. Res. 6(2), 49–51 (2017) Ilie, M., Biazar, J., Ayati, Z.: Application of the Lie Symmetry Analysis for second-order fractional differential equations. Iran. J. Optim. 9(2), 79–83 (2017) Ilie, M., Biazar, J., Ayati, Z.: Analytical solutions for conformable fractional Bratu-type equations. Int. J. Appl. Math. Res. 7(1), 15–19 (2018) Ilie, M., Biazar, J., Ayati, Z.: The first integral method for solving some conformable fractional differential equations. Opt. Quantum Electron. 50(2), 55 (2018). https://doi.org/10.1007/s11082-017-1307-x Ilie, M., Biazar, J., Ayati, Z.: Resonant solitons to the nonlinear Schrödinger equation with different forms of nonlinearities. Optik 164, 201–209 (2018) Ilie, M., Biazar, J., Ayati, Z.: Lie Symmetry Analysis for the solution of first-order linear and nonlinear fractional differential equations. Int. J. Appl. Math. Res. 7(2), 37–41 (2018) Ilie, M., Biazar, J., Ayati, Z.: General solution of second order fractional differential equations. Int. J. Appl. Math. Res. 7(2), 56–61 (2018) Ilie M, Biazar J, Ayati Z (2018) Optimal homotopy asymptotic method for conformable fractional Volterra integral equations of the second kind, Accepted for 49th annual Iranian mathematics conference, August 23–26, 2018 Ilie, M., Biazar, J., Ayati, Z.: Optimal homotopy asymptotic method for first-order conformable fractional differential equations. J. Fract. Calculus Appl. 10(1), 33–45 (2019) Ilie, M., Biazar, J., Ayati, Z.: Analytical solutions for second-order fractional differential equations via OHAM. J. Fract. Calculus Appl. 10(1), 105–119 (2019) Khalil, R., Horani, M.A., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014) Kiliçman, A.: A note on Mellin transform and distributions. Math. Comput. Appl. 9(1), 65–72 (2004) Kiliçman, A.: Distributions theory and neutrix calculus. University Putra Malaysia Press (Penerbit UPM), Serdang (2006) Kilicman, A., Omran, M.: Note on fractional Mellin transform and applications. SpringerPlus 5, 100 (2016). https://doi.org/10.1186/s40064-016-1711-x Miller, K.S., Ross, B.: An introduction to the fractional calculus and fractional differential equations. Wiley, New York (1993) Myint-U, Tyn, Debnath, Lokenath: Linear partial differential equations for scientists and engineers, 4th edn. Birkhauser, Boston (2007) Oberhettinger, F.: Tables of Mellin transforms. Springer, New York (1974) Omran, M., Kilicman, A.: On fractional order Mellin transform and some of its properties. Tbil. Math. J. 10(1), 315–324 (2017) Podlubny, I.: Fractional differential equations. Academic Press, San Diego (1999)