Solution of Specific Fractional Differential Equations by Natural Integral Transform

G. P. Kamble¹ , B. R. Sontakke²

Section:Research Paper, Product Type: Journal-Paper
Vol.6 , Issue.5 , pp.40-45, Oct-2019

CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i5.4045

Online published on Oct 31, 2019

Copyright © G. P. Kamble, B. R. Sontakke . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
In this paper, we apply the natural integral transform to some specific fractional differential equations which are easy to implement and are very effective and simple in performing a solution of the fractional differential equations. The solution was investigated in details with the help of natural integral transform. Here we apply natural integral transform to the fractional diffusion equation under external force, ∂^α/〖∂t〗^α u(x,t)=D ∂^2/〖∂t〗^2 u(x,t)-∂/∂t (f(x),u(x,t)),0<α≤1,D>0 Where u(x,t) represent the probability density function for finding a particle at the point x at the time instant t, the positive constant D depends on the temperature, the friction coefficient, the universal gas constant and finally on the Avogordo number f(x) are the external forces. The natural integral transform gives the solution of this diffusion equation which is in fractional form

Key-Words / Index Term :
Fractional derivative, Natural transform, Diffusion equation, Fractional differential equation, Homotopy perturbation method

References :
[1]. A. Waqar, H. Zafar and Khan; “N-transform-Properties and Applications”, NUST J. Engg. Sci., pp.127-133, 1 (2008).
[2]. Al-Omari, S. K. Q.; “On the applications of Natural transform”, International Journal of Pure and Applied Mathematics, 85 (4), pp.729-744 (2013).
[3]. Ahmed Safwat Abdel-Rady, Saad Zagloul Rida, Anas Ahmed Mohamed Arafa, Hamdy Ragab Abedl-Rahim; “Natural Transform for Solving Fractional Models”, Journal of Applied Mathematics and Physics, 3, pp. 1633-1644 (2015).
[4]. F. B. Belgacem, M., R. Silambarasan; “Theory of Natural transform”, Math. Engg. Sci. Aerospace (MESA), 3(1), PP.99-124, (2012).
[5]. Deshna Loonkar, P. K. Banerji; “Solution of Fractional Differential Equations by Natural Transform”, International Journal of Mathematical Engineering and Science, Vol.2, Issue 12, (Dec. 2013).
[6]. D. D. Ganji, H. Tari, M. B. Jooybari; “Variational Iteration Method and Homotopy Perturbation Method for Non-linear Evolution Equations”, Computers and Mathematics with Applications, 54, pp.1018-1027 (2007).
[7]. J. H. He; “Homotopy Perturbation Technique”, Computer Methods in Applied Mechanics and Engineering, 178, pp. 257-262 (1999).
[8]. J. H. He; “New Interpretation of Homotopy Perturbation Method”, International Journal of Modern Physics B, 20, pp. 2561-2568.(2006)
[9]. J. H. He; “New Perturbation Technique which is also valid for large Parameters”, Journal of Sound and Vibration, 229, pp. 1257-1263.
[10]. Kumar D. Singh, J. Kumar S.; “Analytic and approximate solutions of space-time fractional telegraph equations via Laplace transform”, Walailak J. Sci. Technol, 11(8),pp. 711-728 (2013).
[11]. Kumar D. Singh, J. Kumar S.; “Numerical Computation of non-linear fractional Zakharov-Kuznetsov equation arising in ion-acoustic waves”, J. Egypt Mat. Soc., 22 (3), pp.373-378 (2014).
[12]. Ma. Hc. Yao, DD. Peng, XF; “Exact solution of non-linear fractional partial differential equations by sub-equation method”, Them. Sci., 19 (4), pp.1239-1244 (2015).
[13]. Mohyud-Din, ST. Nawaz, T. Azhar, E. Akbar, MA; “Fractional sub-equation method to space-time fractional Calogero-Degaspenris and potential kadomtsev-Petviashvili equation”, Journal of Taiban University for Science, (2015).
[14]. Maryam Omran, Adem Kilicman; “Natural Transform of Fractional order and Some Properties”, Omran and Kilicman, Cogent Mathematics, 3: 1251874 (2016).
[15]. S. Z. Rida, A. S. Abedl-Rady, A. A. M. Arafa, H. R. Abedl-Rahim; “Natural Transform for Solving Fractional Systems”, International Journal of Pure and Applied Science and Technology, 30(2) (2015), pp. 64-75.
[16]. D. H. Shou; “The Homotopy Perturbation Method for Non-linear Oscillators”, Computers and Mathematics with Applications, 58, pp.2456-2459. (2009).
[17]. V.R. Lakshmi Gorty; “Continuous generalized Hankel-Clifford wavelet transformation”, International Journal of Computer Sciences and Engineering, Vol.1, Issue 4, pp.1-10 (2003).
[18]. K. Lakshmi, T. Meyyapan; ‘Spanning Tree, Properties, Algorithms and Applications”, International Journal of Computer Sciences and Engineering, Vol.5, Issue10, pp.54-58 (2017).
[19]. N. Jimson, K. Hemachandram; “Digital Image watermarking in Discrete Fourier Transform domain”, International Journal of Computer Sciences and Engineering, Vol.6, Issue 5, pp.16-22, (2018).