Analytical Solution of a Fractional Model of Fluid Flow Through Narrowing System in Terms of Mittag-Leffler Function

Abstract

In this work, we discuss a fractional model of a flow equation in a simple pipeline. Pipeline narrowing is a crucial aspect in drinking water distribution processes, sewage system and in oil-well schemes. The solution of the mathematical model is determined with the aid of the Sumudu transform and finite Hankel transform. The results derived in the current study are in compact and graceful forms in terms of the Mittag-Leffler type function, which are convenient for numerical and theoretical evaluation.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] Landau L.D. and Lifshitz E.M. (1959): Fluid Mechanics. – Pergamon Press.

  • [2] Acheson D.J. (1990): Elementary Fluid Dynamics. – Oxford Applied Mathematics and Computing Science Series, Oxford University Press, ISBN 0-19-859679-0.

  • [3] Batchelor G.K. (1967): An Introduction to Fluid Dynamics. – Cambridge University Press, ISBN 0-521-66396-2.

  • [4] Upadhyay V. (2000): Some Phenomena in Two Phase Blood Flow. – Ph. D. Thesis, Central University, Allahabad.

  • [5] Debnath L. (1976): On a micro - continuum model of pulsatile blood flow. – Acta Mechanica, vol.24, pp.165-177.

  • [6] Jones D.S. and Sleeman B.D. (1976): Differential Equation and Mathematical Biology. – 6 G.A. and V.

  • [7] Upadhyay V. and Pandey P.N. (1999): Newtonian Model of two phase blood flow in aorta and arteries proximate to the heart. – Proc. of Third Con. of Int. Acad. Phy. Sci.

  • [8] Upadhyay V. and Pande P.N. (1999): A power law model of two phase blood flow in arteries remote form the heart. – Proc. of Third Con. of Int. Acad. Phy. Sci..

  • [9] Singh P. and Upadhyay K.S. (1985): A new approach for the shock propagation in two - phase system. – Nat. Acad. Sci. Letters No.2.

  • [10] Debnath L. (1973): On transient flows in non-Newtonian liquids. – Tensor N.S., vol.27, No.2, pp.257-264.

  • [11] Caputo M. (1967): Linear Models of Dissipation whose Q is almost Frequency Independent-II. – Geophysical J. of the Royal Astronomical Soc., vol.13, pp.529-539. (Reprinted in : Fract. Calc. Appl. Anal., 11 no. 1 (2008), 3-14.

  • [12] Hilfer R. (ed.) (2000): Applications of Fractional Calculus in Physics. – World Scientific Publishing Company, Singapore-New Jersey-Hong Kong, pp.87-130.

  • [13] Podlubny I. (1999): Fractional Deferential Equations (An Introduction to Fractional Derivatives, Fractional Deferential Equations, to Methods of Their Solution and some of Their Applications). – Academic Press, San Diego.

  • [14] Miller K.S. and Ross B. (1993): An Introduction to the Fractional Calculus and Fractional Differential Equations. – New York: Wiley.

  • [15] Samko S.G., Kilbas A.A. and Marichev O.I. (1993): Fractional Integrals and Derivatives: Theory and Applications. – Switzerland: Gordon and Breach Science Publishing.

  • [16] Ansari A. (2015): On finite fractional Sturm-Liouville transforms. – Integral Transforms and Special Functions, vol.26, No.1, pp.51-64.

  • [17] Kumar D., Singh J. and Baleanu D. (2017): A hybrid computational approach for Klein-Gordon equations on Cantor sets. – Nonlinear Dynamics, vol.87, pp.511-517.

  • [18] Srivastava H.M., Kumar D. and Singh J. (2017): An efficient analytical technique for fractional model of vibration equation. – Applied Mathematical Modelling, vol.45, pp.192-204.

  • [19] Kumar D., Agarwal R.P. and Singh J. (2017): A modified numerical scheme and convergence analysis for fractional model of Lienard’s equation. – Journal of Computational and Applied Mathematics, http://doi.org/10.1016/j.cam.2017.03.0.11

  • [20] Kumar D., Singh J. and Baleanu D. (2017): A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves. – Mathematical Methods in the Applied Sciences, DOI:10.1002/mma.4414.

  • [21] Singh J., Rashidi M.M., Swroop R. and Kumar D. (2016): A fractional model of a dynamical Brusselator reaction-diffusion system arising in triple collision and enzymatic reactions. – Nonlinear Engineering – Modeling and Application, vol.5, No.4, pp.277-285.

  • [22] Podlubny I. (1999): Fractional differential equations. – Academic Press, San Diego, Calif, USA, vol.198, pp.340 pages.

  • [23] Kilbas A.A., Srivastava H.M and Trujillo J.J. (2006): Theory and applications of fractional differential equations. – Elsevier, Amsterdam, vol.20), 540 pages.

  • [24] Srivastava H.M., Golmankhaneh A.K., Baleanu D. and Yang X.J. (2014): Local Fractional Sumudu Transform with Application to IVPs on Cantor Sets. – Abst. Appl. Anal., article ID 620529, 7 pages.

  • [25] Shukla A.K. and Prajapati J.C. (2007): On some properties of class of polynomials suggested by mittal. – Universidad Cat’olica del Norte Progyecciones (Antofagasta-Chile), vol.26, No.2, pp.145-156.

  • [26] Shukla A.K. and Prajapati J.C. (2008): On generalized Mittag-Leffler type function and generated integral operator. – Mathematical Sciences Research Journal, vol.12, No.12, pp.283-290.

  • [27] Prajapati J.C. and Nathwani B.V. (2015): Fractional calculus of a unified Mittag-Leffler function. – Ukrainian Mathematical Journal, vol.66, No.8, pp.1267-1280.

  • [28] Watugala G.K. (1993): Sumudu transform, a new integral transform to solve differential equations and control engineering problems. – International Journal of Mathematical Education in Science and Technology, vol.24, pp.35-43.

  • [29] Weerakoon S. (1994): Application of Sumudu transform to partial differential equations. – International Journal of Mathematical Education in Science and Technology, vol.25, pp.277-283.

  • [30] Asiru M.A. (2001): Sumudu transform and the solution of integral equation of convolution type. – International Journal of Mathematical Education in Science and Technology, vol.32, pp.906-910.

  • [31] Belgacem F.B.M. and Karaballi A.A. (2005): Sumudu transform fundamental properties investigations and applications. – International J. Appl. Math. Stoch. Anal., pp.1-23.

  • [32] Chaurasia V.B.L. and Singh J. (2011): Application of Sumudu transform in fractional kinetic equations. – Gen. Math. Notes, vol.2, No.1, pp.86-95.

  • [33] Belgacem F.B.M. and Al-Shemas E.H.N. (2014): Towards a Sumudu based estimation of large scale disasters environmental fitness changes adversely affecting population dispersal and persistence. – AIP Conf. Proc. 1637, (1442); doi: 10.1063/1.4907311.

  • [34] Rainville E.D. (1960): Special Functions. – New York: The Macmillan Company.

  • [35] Debnath L. (1995): Integral Transforms and their Applications. – New York-London-Tokyo: CRC Press.

  • [36] Mittag-Leffler G.M. (1903): Sur la nouvelle function Eα (x). – C.R. Acad.Sci., Paris, (ser. II), vol.137, pp.554-558.

  • [37] Mittag-Leffler G.M. (1905): Sur la representatin analytique d’une branche uniforme d’une function monogene. – Acta Math., vol.29, pp.101-181.

  • [38] Wiman A. (1905): Uber den fundamentalsatz in der Theorie der Functionen Eα (x). Acta. Math., vol.29, pp.191-201.

  • [39] Shukla K. and Prajapat J.C. (2007): On a generalization of Mittag-Leffler function and its properties. – J. Math. Anal. Appl., vol.336, pp.797-811.

  • [40] Prabhakar T.R. (1971): A singular integral equation with a generalized Mittag-Leffler function in the kernel. – Yokohama Math. J., vol.19, pp.7-15.

  • [41] Garra R. and Garrappa R. (2018): The Prabhakar or three parameter Mittag–Leffler function: Theory and application. – Communications in Nonlinear Science and Numerical Simulation, vol.56, pp.314-329.

  • [42] Chaurasia V.B.L. and Singh J. (2010): Application of Sumudu transform in Schodinger equation occurring in quantum mechanics. – Applied Mathematical Sciences, vol.4, No.57, pp.2843-2850.

  • [43] Belgacem F.B.M., Karaballi A.A. and Kalla S.L. (2003): Analytical investigations of the Sumudu transform and applications to integral production equations. – Mathematical Problems in Engineering, vol.3, pp.103-118.

  • [44] Tomovski Z., Hilfer R. and Srivastava H.M. (2010): Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions. – Integral Transforms Spec. Funct., vol.21, No.11, pp.797-814.

  • [45] Patel A.D., Salehbhai I.A., Banerjee J., Katiyar V.K. and Shukla A.K. (2012): An analytical solution of fluid flow through narrowing systems. – Italian Journal of Pure and Applied Mathematics, vol.29, pp.63-70.

OPEN ACCESS

Journal + Issues

Search