Solution of Non-linear Fractional Burger's Type Equations Using The Laplace Transform Decomposition Method

Öz

Our goal in this paper is to use combined Laplace transform (CLT) and Adomian decomposition method
(ADM) (that will be explained in section 3), to study approximate solutions for non-linear time-fractional
Burger's equation, fractional Burger's Kdv equation and the fractional modi?ed Burger's equation for the
Caputo and Conformable derivatives. Comparison between the two solutions and the exact solution is made.
Here we report that the Laplace transform decomposition method (LTDM) proved to be e?cient and be
used to obtain new analytical solutions of nonlinear fractional di?erential equations (FDEs).

Anahtar Kelimeler

• Conformable fractional derivative
• Caputo fractional derivative
• Conformable differential equations
• Burger's equation
• Modified Buerger's equation
• Burger's Kdv equation
• Laplace Transform
• Adomian Decomposition method

Kaynakça

1. [1] T. Abdeljawad, On conformable fractional calculus, Journal of computational and Applied Mathematics. 279 (2015): 57-66.
2. [2] R. Khalil, et al. , A new definition of fractional derivative, Journal of computational and applied mathematics. 264 (2014): 65-70.
3. [3] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and applications of fractional differential equations. Vol. 204. elsevier, 2006.
4. [4] K.S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations. Wiley, 1993.
5. [5] M. Kurulay, The approximate and exact solutions of the space and time-fractional Burger's equations, Ijrras. 3(3) (2010): 257-263.
6. [6] H. Bateman, Some recent researches on the motion of fluids, Monthly Weather Review. 43.4 (1915): 163-170.
7. [7] J. Burgers, A mathematical model illustrating the theory of turbulence, Advances in applied mechanics. Vol. 1. Elsevier, 1948. 171-199.
8. [8] E. Benton, and G.W. Platzman, A Table of solutions of the one-dimensional Burger's equation, Quarterly of Applied Mathematics. 30.2 (1972): 195-212.

9. [9] A. Gorguis, A comparison between Cole-Hopf transformation and the decomposition method for solving Burger's equations, Applied Mathematics and Computation. 173.1 (2006): 126-136.
10. [10] T. Ozis, and A. Ozdes, A direct variational methods applied to Burger's equation, Journal of computational and applied mathematics. 71.2 (1996): 163-175.
11. [11] E.N. Aksan, and A. Ozdes, A numerical solution of Burgers' equation, Applied mathematics and computation. 156.2 (2004): 395-402.
12. [12] S. Kutluay, A. Bahadir, A. Ozdes, Numerical solution of one-dimensional Burger's equation: explicit and exact-explicit finite difference methods, Journal of Computational and Applied Mathematics. 103.2 (1999): 251-261.
13. [13] E. Varoglu, W.D. Liam Finn, Space-time finite elements incorporating characteristics for the burger's equation, Interna- tional Journal for Numerical Methods in Engineering. 16.1 (1980): 171-184.
14. [14] J. Caldwell, P. Wanless, A.E. Cook, A finite element approach to Burger's equation. Applied Mathematical Modelling. 5.3 (1981): 189-193.
15. [15] D.J. Evans, A.R. Abdullah, The group explicit method for the solution of Burger's equation, Computing. 32.3 (1984): 239-253.
16. [16] R.C. Mittal, P. Singhal, Numerical solution of Burger's equation, Communications in numerical methods in engineering. 9.5 (1993): 397-406.
17. [17] M. Safari, D.D Ganji, M. Moslemi, Application of He's variational iteration method and Adomian's decomposition method to the fractional KdV-Burgers-Kuramoto equation, Computers and Mathematics with Applications. 58.11-12 (2009): 2091-2097.
18. [18] F.S. Silva, D.M. Moreira, M.A. Moret, Conformable Laplace transform of fractional differential equations, Axioms. 7.3 (2018): 55.
19. [19] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, 1998.
20. [20] I. Kadri, M. Horani, R. Khalil, Tensor product technique and fractional differential equations, J. Semigroup Theory Appl. 2020 (2020): Article-ID.
21. [21] N.A. Khan, O.A. Razzaq, M. Ayaz, Some properties and applications of conformable fractional Laplace transform (CFLT), J. Fract. Calc. Appl. 9.1 (2018): 72-81.
22. [22] H. Eltayeb, I. Bachar, M. Gad-Allah, Solution of singular one-dimensional Boussinesq equation by using double conformable Laplace decomposition method, Advances in Di?erence Equations. 2019.1 (2019): 1-19.
23. [23] G. Adomian, A review of the decomposition method and some recent results for nonlinear equations, Mathematical and Computer Modelling. 13.7 (1990): 17-43.
24. [24] A. Naghipour, J. Manafian, Application of the Laplace Adomian decomposition and implicit methods for solving Burger's equation, TWMS Journal of Pure and Applied Mathematics. 6.1 (2015): 68-77.
25. [25] D. Kaya, An explicit solution of coupled viscous Burger's equation by the decomposition method, International Journal of Mathematics and Mathematical Sciences. 27.11 (2001): 675-680.
26. [26] K.M. Saad, E.H. Al-Sharif, Analytical study for time and time-space fractional Burger's equation, Advances in Difference Equations. 2017.1 (2017): 1-15.
27. [27] Y. Cenesiz, D. Baleanu, A. Kurt, O. Tasbozan, New exact solutions of Burger's type equations with conformable derivative, Waves in Random and complex Media. 27.1 (2017): 103-116.
28. [28] A. Sonmezoglu, Exact solutions for some fractional di?erential equations, Advances in Mathematical Physics. 2015 (2015).
29. [29] A. Korkmaz, Explicit exact solutions to some one-dimensional conformable time fractional equations, Waves in Random and Complex Media. 29-1 (2019): 124-137.