Local Fractional Aboodh Transform and its Applications to Solve Linear Local Fractional Differential Equations

Abstract

In this work we focus on presenting a method for solving local fractional differential equations. This method based on the combination of the Aboodh transform with the local fractional derivative (we can call it local fractional Aboodh transform), where we have provided some important results and properties. We concluded this work by providing illustrative examples, through which we focused on solving some linear local fractional differential equations in order to obtain nondifferential analytical solutions.

Keywords

• local fractional calculus
• local fractional Laplace transform
• Aboodh transform method
• local fractional differential equations

Kaynakça

1. 1] K.S. Aboodh, The new integrale transform "Aboodh transform", Glob. J. Pure. Appl. Math., 9(1), 35-43, 2013.
2. [2] J. Ahmad, J. Tariq, Application of Aboodh Differential Transform Method on Some Higher Order Problems, Journal of Science and Arts, Year 18, No. 1(42), 5-18, 2018.
3. [3] J. Ahmad, S.T. Mohyud-Din, H.M. Srivastava and X-J. Yang, Analytic solutions of the Helmholtz and Laplace equations by using local fractional derivative operators, Waves Wavelets Fractals Adv. Anal., 1: 22-26, 2015.
4. [4] A.A. Alshikh, M.M.A. Mahgoub, Solving System of Ordinary Differential Equations By Aboodh Transform, World Appl. Sci. J., 34(9): 1144-1148, 2016.
5. [5] A. Ardjouni, A. Djoudi, Existence and uniqueness of solutions for nonlinear hybrid implicit Caputo-Hadamard fractional di?erential equations, Results in Nonlinear Analysis 2 (2019) No. 3, 136-142.
6. [6] S. Benzoni, Analyse de Fourier, Universite de Lyon / Lyon 1, France, 2011.
7. [7] T.M. Elzaki, S.M. Ezaki, On the ELzaki Transform and Ordinary Differential Equation with Variable Coeficients, Adv. Theo. Appl. Math., 6(1), 41-46, 2011.
8. [8] M. Hamdi Cheri, D. Ziane, A New Numerical Technique for Solving Systems of Nonlinear Fractional Partial Differential Equations, Int. J. Anal. Appl., Vol. 15, Nu. 2, 188-197, 2017.

9. [9] M. Hamdi Cheri, D. Ziane, Variational iteration method combined with new transform to solve fractional partial differential equations, Univ. J. Math. Appl., 1 (2), 113-120, 2018.
10. [10] Ji-H. He, Asymptotic Methods for Solitary Solutions and Compactons, Abs. Appl. Anal. Vol. 2012, A. ID 916793, 130 pp, 2012.
11. [11] G. Jumarie, Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions, Appl. Math. Lett., 22, 378-385, 2009.
12. [12] E. Karapinar, D. Kumar, R. Sakthivel, N.H. Luc and N.H. Can, Identifying the space source term problem for time-space- fractional diffusion equation, Advances in Di?erence Equations (2020) 2020:557.
13. [13] Z.H. Khan, W.A. Khan, N-transform properties and applications, Nust J. Eng. Sci., 1, 127?133, 2008.
14. [14] D. Lomen, Application of the Mellin Transforin to Boundary Value Problems, Proc. Iowa Acad. Sci, 69(1), 436-442, 1962.
15. [15] M.M. A. Mahgoub, K.S. Aboodh and A.A. Alshikh, On The Solution of Ordinary Differential Equation with Variable Coefficients using Aboodh Transform, Adv. Theo. Appl. Math., Vol.11, Nu. 4 (2016), 383-389, 2016.
16. [16] M.M.A. Mahgoub, A Coupling Method of Homotopy Perturbation and Aboodh Transform for Solving Nonlinear Fractional Heat - Like Equations, Int. J. Sys. Sci. Appl. Math., 1(4) : 63-68, 2016.
17. [17] S. Maitama, W. Zhao, New Integral Transform: Shehu Transform a Generalization of Sumudu and Laplace Transform for Solving differential equations, Int. J. Anal. Appl., 17(2), 167-190, 2019.
18. [18] K. Mpungu, A.M. Nass, Symmetry Analysis of Time Fractional Convection-reaction-diffusion Equation with a Delay, Results in Nonlinear Analysis 2 (2019) No. 3, 113-124.
19. [19] N.T. Negero, Zero-Order Hankel Transform Method for Partial Differential Equations, Int. J. Mod. Sci. Eng. Tech., 3(10), 24-36, 2016.
20. [20] R.I. Nuruddeen, A.M. Nass, Aboodh Decomposition Method and its Application in Solving Linear and Nonlinear Heat Equations, Eur. J. Adv. Eng. Tech., 3(7): 34-37, 2016.
21. [21] N.D. Phuong, L.V.C. Hoan, E. Karapinar, J. Singh, H.D. Binh and N.H. Can, Fractional order continuity of a time semi-linear fractional diffusion-wave system, Alexandria Engineering Journal (2020) 59, 4959-4968.
22. [22] S. Qureshi, M.S. Chandio, A.A. Shaikh and R.A. Memon, On the Use of Aboodh Transform for Solving Non-integer Order Dynamical Systems, Sindhuniv. Res. Jour. (Sci. Ser.) Vol. 51 (01) 53-58 (2019).
23. [23] A.K.H. Sedeeg, M.M.A. Mahgoub, Aboodh Transform Homotopy Perturbation Method For Solving System Of Nonlinear Partial Differential Equations, Math. Theo. Mod., Vol.6, No.8, 108?113, 2016.
24. [24] M.R. Spiegel, Theory and problems of Laplace transform, New York, USA: Schaum's Outline Series, McGraw-Hill., 1965.
25. [25] H.M. Srivastava, A.K. Golmankhaneh, D. Baleanu, X.J. Yang, Local Fractional Sumudu Transform with Application to IVPs on Cantor Sets, Abst. Appl. Anal., Vol. 2014, A. ID 176395, 1-7, 2014.
26. [26] T.G. Thange, A.R. Gade, On Aboodh transform for fractional differential operator, Mal. J. Mat., Vol.8, No.1, 225-229, 2020.
27. [27] N. Tran, Y. Zhou, D. O'Regan and T. Nguyen, On a terminal value problem for pseudoparabolic equations involving Riemann-Liouville fractional derivatives, Applied Mathematics Letters, Volume 106, 2020, 106373.
28. [28] N.H. Tuan, V.V. Au and R. Xu, Semilinear Caputo time-fractional pseudo-parabolic equations, Communications on Pure and Applied Analysis, 2021, 20 (2) : 583-621.
29. [29] G.K. Watugala, Sumudu transform: a new integral transform to solve differentia lequations and control engineering problems, Int. J. Math. Educ. Sci. Tech., 24(1), 35-43, 1993.
30. [30] X-J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic, Hong Kong, 2011.
31. [31] X-J. Yang, Advanced Local Fractional Calculus and Its Applications, World Sci. Pub., New York, NY, USA, 2012.
32. [32] X.J. Yang, D. Baleanu and H.M. Srivastava, Local Fractional Integral Transforms and Their Applications, Academic Press (2015).
33. [33] X-J. Yang, L. Li, R. Yang, Problems of local fractional de?nite integral of the one-variable non-differentiable function, World Sci-Tech R&D, (in Chinese), 31(4), 722-724, 2009.
34. [34] X-J. Yang, Generalized Sampling Theorem for Fractal Signals, Adv. Dig. Mul., Vol.1, No. 2, 88-92, 2012.
35. [35] Z.U. Zafar, ZZ Transform Method, Int. J. Adv. Eng. Glo. Tech., 4(1), 1605-1611, 2016.
36. [36] D. Ziane, M. Hamdi Cherif, Homotopy Analysis Aboodh Transform Method for Nonlinear System of Partial Differential Equations, Univ. J. Math. Appl., 1(4), 244-253, 2018.
37. [37] C.G. Zhao, A.M. Yang, H. Jafari and A. Haghbin, The Yang-Laplace Transform for Solving the IVPs with Local Fractional Derivative, Abs. Appl. Anal., Vol. 2014, A. ID 386459, 1-5, 2014.
38. [38] D. Ziane, The combined of Homotopy analysis method with new transform for nonlinear partial differential equations, Mal. J. Mat., Vol.6, No.1, 34-40, 2018.