DOI QR코드

DOI QR Code

SYSTEMATIC APPROXIMATION OF THREE DIMENSIONAL FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS IN FLUID MECHANICS

  • KHAN, FIRDOUS (DEPARTMENT OF MATHEMATICS, DR. BABASAHEB AMBEDKAR MARATHWADA UNIVERSITY) ;
  • GHADLE, KIRTIWANT P. (DEPARTMENT OF MATHEMATICS, DR. BABASAHEB AMBEDKAR MARATHWADA UNIVERSITY)
  • Received : 2019.08.08
  • Accepted : 2019.09.14
  • Published : 2019.09.25

Abstract

In this article, a systematic solution based on the sequence of expansion method is planned to solve the time-fractional diffusion equation, time-fractional telegraphic equation and time-fractional wave equation in three dimensions using a current and valid approximate method, namely the ADM, VIM, and the NIM subject to the estimate initial condition. By using these three methods it is likely to find the exact solutions or a nearby approximate solution of fractional partial differential equations. The exactness, efficiency, and convergence of the method are demonstrated through the three numerical examples.

References

  1. S. Momani, Analytical and approximate solutions of the space-and time fractional telegraph equations, Applied Mathematics and Computation, 170 (2005), 1126-34. https://doi.org/10.1016/j.amc.2005.01.009
  2. I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
  3. K. P. Ghadle and F. Khan, Solution of FPDE in Fluid Mechanics by ADM, VIM and NIM, American Journal of Mathematical and Computer Modelling, 2 (2017), 13-23.
  4. A. A. Hamoud and K. P. Ghadle, On the Numerical Solution of Nonlinear Volterra-Fredholm Integral Equations by Variational Iteration Method, International Journal of Advanced Scientific and Technical Research, 3 (2016), 45-51.
  5. A. A. Hamoud, A. A Dhurgham and K. P. Ghadle, A Study of some Iterative Methods for solving Fuzzy Volterra-Fredholm Integral Equation, Indonesian Journal of Electrical Engineering and Computer Science, 11 (2018), 1228-1235. https://doi.org/10.11591/ijeecs.v11.i3.pp1228-1235
  6. A. A. Hamoud and K. P. Ghadle, Existence and Uniqueness of the solution for Volterra-Fredholm Integro-Differential Equations, Journal of Siberian Fedreral University Mathematics and Physics, 11 (2018), 692-701. https://doi.org/10.17516/1997-1397-2018-11-6-692-701
  7. A. A. Hamoud, K. P. Ghadle, M. Sh. Bani Issa and Giniswamy. Existence and Uniqueness theorems for Fractional Volterra-Fredholm Integro-Differential Equations, International Journal of Applied Mathematics, 31 (2018), 333-348.
  8. A. A. Hamoud and K. P. Ghadle, Homotopy Analysis Method for the first order Fuzzy Volterra-Fredholm Integro-Differential Equations, Indonesian Journal of Electrical Engineering and Computer Science, 11 (2018), 857-867. https://doi.org/10.11591/ijeecs.v11.i3.pp857-867
  9. A. A. Hamoud, and K. P. Ghadle, Modified Laplace Decomposition Method for Fractional Volterra-Fredholm Integro-Differential Equations, Journal of Mathematical Modeling, 6 (2018), 91-104.
  10. A. A. Hamoud and K. P. Ghadle, Modified Adomian Decomposition Method for Solving Fuzzy Volterra-Fredholm Integral Equations, Journal of the Indian Mathematical Society, 85 (2018), 53-69. https://doi.org/10.18311/jims/2018/16260
  11. S. Fomin, V. Chugunov and T. Hashida, Mathematical Modeling of Anomalous Diffusion in Porous Medium, Fractional Differential Calculus, 1 (2011), 1-28.
  12. S. Momani, Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Elsevier, Physics Letters A, 355 (2006), 271-279. https://doi.org/10.1016/j.physleta.2006.02.048
  13. K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993.
  14. K. B. Oldham, and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
  15. M. Caputo, Linear models of dissipation whose Q is almost frequency indepedent-part II, Geophysical Journal International, 13 (1967), 529-539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x
  16. I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus and Applied Analysis, 5 (2002), 367-386.
  17. Z. Odibat, and S. Momani, The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics, Computers and Mathematics with Applications, 58 (2009), 2199-2208. https://doi.org/10.1016/j.camwa.2009.03.009
  18. A. A.Hemeda, Solution of fractional partial differential equations in fluid mechanics by extension of some iterative method, Abstract and Applied Analysis, 2013 (2013), 1-9.