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Fractional Derivatives with Mittag-Leffler Kernel pp 235–251Cite as

Atangana–Baleanu Derivative with Fractional Order Applied to the Gas Dynamics Equations

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Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 194))

Abstract

We apply the new Atangana–Baleanu derivative in Caputo sense to study gas dynamics equations of arbitrary order using modified homotopy analysis transform method (MHATM). Atangana and Baleanu suggested an interesting fractional operator in 2016 which is based on the exponential kernel. An alternative framework of MHATM with Atangana–Baleanu derivative is presented and the modified Gas dynamics equations are solved numerically and analytically using aforesaid the method. Illustrative examples are included to demonstrate the validity and applicability of the presented technique with new Atangana–Baleanu derivative.

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References

  1. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier (North-Holland) Science Publishers, Amsterdam (2006)

    MATH  Google Scholar 

  2. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications. Academic Press, San Diego, Calif, USA (1999)

    MATH  Google Scholar 

  3. Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A.: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007)

    Book  MATH  Google Scholar 

  4. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  5. Yang, X.J., Baleanu, D., Srivastava, H.M.: Local Fractional Integral Transform and their Applications. Academic Press, New York (2015)

    Google Scholar 

  6. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  7. Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernal: theory and applications to heat transfer model. Therm. Sci. (2016)

    Google Scholar 

  8. Atangana, A., Koca, I.: Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order. Chaos, Solitons & Fractals 89, 447–454 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Atangana, A., Gómez-Aguilar, J.F.: A new derivative with normal distribution kernel: theory, methods and applications. Phys. A: Stat. Mech. Appl. 476, 1–14 (2017)

    Article  MathSciNet  Google Scholar 

  10. Gómez-Aguilar, J.F., Atangana, A.: New insight in fractional differentiation: power, exponential decay and Mittag-Leffler laws and applications. Eur. Phys. J. Plus 132(1), 1–13 (2017)

    Article  Google Scholar 

  11. Atangana, A., Gómez-Aguilar, J.F.: Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena. Eur. Phys. J. Plus 133, 1–22 (2018)

    Article  Google Scholar 

  12. Gómez-Aguilar, J.F.: Irving-Mullineux oscillator via fractional derivatives with Mittag-Leffler kernel. Chaos, Solitons & Fractals 95(35), 1–7 (2017)

    MathSciNet  MATH  Google Scholar 

  13. Cuahutenango-Barro, B., Taneco-Hernández, M.A., Gómez-Aguilar, J.F.: On the solutions of fractional-time wave equation with memory effect involving operators with regular kernel. Chaos, Solitons & Fractals 115, 283–299 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Saad, K.M., Gómez-Aguilar, J.F.: Coupled reaction-diffusion waves in a chemical system via fractional derivatives in Liouville-Caputo sense. Rev. Mex. Fís 64(5), 539–547 (2018)

    Article  MathSciNet  Google Scholar 

  15. Yépez-Martínez, H., Gómez-Aguilar, J.F.: Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and Mittag-Leffler kernel. Math. Model. Nat. Phenom. 13(1), 1–13 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Coronel-Escamilla, A., Gómez-Aguilar, J.F., Torres, L., Escobar-Jiménez, R.F., Valtierra-Rodríguez, M.: Synchronization of chaotic systems involving fractional operators of Liouville-Caputo type with variable-order. Phys. A: Stat. Mech. Appl. 487, 1–21 (2017)

    Article  MathSciNet  Google Scholar 

  17. Coronel-Escamilla, A., Gómez-Aguilar, J.F., Torres, L., Escobar-Jiménez, R.F.: A numerical solution for a variable-order reaction-diffusion model by using fractional derivatives with non-local and non-singular kernel. Phys. A: Stat. Mech. Appl. 491, 406–424 (2018)

    Article  MathSciNet  Google Scholar 

  18. Gómez-Aguilar, J.F., Escobar-Jiménez, R.F., López-López, M.G., Alvarado-Martínez, V.M.: Atangana-Baleanu fractional derivative applied to electromagnetic waves in dielectric media. J. Electromagn. Waves Appl. 30(15), 1937–1952 (2016)

    Article  Google Scholar 

  19. Saad, K.M., Gómez-Aguilar, J.F.: Analysis of reaction diffusion system via a new fractional derivative with non-singular kernel. Phys. A: Stat. Mech. Appl. 509, 703–716 (2018)

    Article  MathSciNet  Google Scholar 

  20. Coronel-Escamilla, A., Gómez-Aguilar, J.F., Baleanu, D., Córdova-Fraga, T., Escobar-Jiménez, R.F., Olivares-Peregrino, V.H., Qurashi, M.M.A.: Bateman-Feshbach tikochinsky and Caldirola-Kanai oscillators with new fractional differentiation. Entropy 19(2), 1–21 (2017)

    Article  Google Scholar 

  21. Jafari, H., Chun, C., Seifi, S., Saeidy, M.: Analytical solution for nonlinear gas dynamics equation by homotopy analysis method. Appl. Appl. Math. 4(1), 149–154 (2009)

    MathSciNet  MATH  Google Scholar 

  22. Mohamad-Jawad, A.J., Petkovic, M.D., Biswas, A.: Applications of He’s principles to partial differential equations. Appl. Math. Comput. 217, 7039–7047 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Bogomolov, S.V.: Quasi gas dynamics equations. Mat. Modelirovanie 21(12), 145–151 (2009)

    MATH  Google Scholar 

  24. Evans, D.J., Bulut, H.: A new approach to the gas dynamics equation: an application of the decomposition method. Int. J. Comput. Math. 79(7), 817–822 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Steger, J.L., Warming, R.F.: Flux vector splitting of the inviscid gas dynamic equations with application to finite-difference methods. J. Comput. Phys. 40(2), 263–293 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  26. Aziz, A., Anderson, D.: The use of pocket computer in gas dynamics. Comput. Educ. 9(1), 41–56 (1985)

    Article  Google Scholar 

  27. Rasulov, M., Karaguler, T.: Finite difference scheme for solving system equation of gas dynamics in a class of discontinuous function. Appl. Math. Comput. 143(1), 145–164 (2003)

    MathSciNet  MATH  Google Scholar 

  28. Liu, T.P.: Nonlinear Waves in Mechanics and Gas Dynamics. Defense Technical Information Center, Maryland University, College Park Department of Mathematics (1990)

    Google Scholar 

  29. Biazar, J., Eslami, M.: Differential transform method for nonlinear fractional gas dynamics equation. Int. J. Phys. Sci. 6(5), 1–12 (2011)

    Google Scholar 

  30. Das, S., Kumar, R.: Approximate analytical solutions of fractional gas dynamics. Appl. Math. Comput. 217(24), 9905–9915 (2011)

    MathSciNet  MATH  Google Scholar 

  31. Kumar, S., Kocak, H., Yildirim, A.: A fractional model of gas dynamics equations and its analytical approximate solution using Laplace transform. Z. Nat. 67a, 388–396 (2012)

    Google Scholar 

  32. Kumar, S., Rashidi, M.M.: New method for fractional gas dynamics in shock fronts. Comput. Phys. Commun. 185, 1947–1954 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Liao, S,J.: The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. thesis, Shanghai Jiao Tong University (1992)

    Google Scholar 

  34. Liao, S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. CRC Press, Chapman and Hall, Boca Raton (2003)

    Book  Google Scholar 

  35. Liao, S.J.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499–513 (2004)

    MathSciNet  MATH  Google Scholar 

  36. Liao, S.J.: Notes on the homotopy analysis method: some definition and theorems. Commun. Nonlinear Sci. Numer. Simul. 14, 983–997 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. Liao, S.J.: Homotopy analysis method: a new analytical technique for nonlinear problem. Commun. Nonlinear Sci. Numer. Simul. 2, 95–100 (1997)

    Article  Google Scholar 

  38. Wu, G.-C.: Adomian decomposition method for non-smooth initial value problems. Math. Comput. Model. 54, 2104–2108 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wazwaz, A.M.: The combined Laplace transform-Adomian decomposition method for handling nonlinear Volterra integro-differential equations. Appl. Math. Comput. 216(4), 1304–1309 (2010)

    MathSciNet  MATH  Google Scholar 

  40. Pandey, R.K., Singh, O.P., Baranwal, V.K., Tripathi, M.P.: An analytic solution for the space-time fractional advection-dispersion equation using the optimal homotopy asymptotic method. Comput. Phys. Commun. 183, 2098–2106 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  41. Golbabai, A., Fardi, M., Sayevand, K.: Application of the optimal homotopy asymptotic method for solving a strongly nonlinear oscillatory system. Math. Comput. Model. 58(11–12), 1837–1843 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. Gómez-Aguilar, J.F., Yépez-Martínez, H., Torres-Jiménez, J., Córdova-Fraga, T., Escobar-Jiménez, R.F., Olivares-Peregrino, V.H.: Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel. Adv. Differ. Equ. 2017(1), 1–18 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  43. Morales-Delgado, V.F., Gómez-Aguilar, J.F., Yépez-Martínez, H., Baleanu, D., Escobar-Jiménez, R.F., Olivares-Peregrino, V.H.: Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular. Adv. Differ. Equ. 2016(1), 1–16 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  44. Chen, Y.M., Liu, J.K., Meng, G.: Relationship between the homotopy analysis method and harmonic balance method. Commun. Nonlinear Sci. Numer. Simul. 15, 2017–2025 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  45. Liao, H.: Piece wise constrained optimization harmonic balance method for predicting the limit cycle oscillations of an airfoil with various nonlinear structures. J. Fluid Struct. 55, 324–346 (2015)

    Article  Google Scholar 

  46. Kumar, S., Kumar, A., Odibat, Z.: A nonlinear fractional model to describe the population dynamics of two interacting species. Math. Methods Appl. Sci. 40(11), 4134–4148 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  47. Kumar, S., Kumar, A., Argyros, I.K.: A new analysis for the Keller Segel model of fractional order. Numer. Algorithms 75(1), 213–228 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  48. Li, C., Kumar, A., Kumar, S., Yang, X.J.: On the approximate solution of nonlinear time-fractional KdV equation via modified homotopy analysis Laplace transform method. J. Nonlinear Sci. Appl. 9, 5463–5470 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  49. Odibat, Z., Bataineh, A.S.: An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: construction of homotopy polynomials. Math. Methods Appl. Sci. 38, 991–1000 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first author Sunil Kumar would like to acknowledge the financial support received from the National Board for Higher Mathematics. Department of Atomic Energy, Government of India (Approval No. 2/48(20)/2016/NBHM(R.P.)/R and D II/1014).

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Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology, Jamshedpur, 831014, Jharkhand, India

    Sunil Kumar & B. Sharma

  2. Department of Mathematics, Balarampur College Purulia, Balarampur, Purulia, 723143, West Bengal, India

    Amit Kumar

  3. Department of Statistics, Mathematical Analysis and Optimization, University of Santiago de Compostela, 15782, Santiago de Compostela, Spain

    J. J. Nieto

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  1. Sunil Kumar
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  2. Amit Kumar
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  3. J. J. Nieto
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  4. B. Sharma
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Correspondence to Sunil Kumar .

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Editors and Affiliations

  1. CONACYT-Tecnológico Nacional de México, Centro Nacional de Investigación y Desarrollo Tecnológico, Cuernavaca, Morelos, Mexico

    José Francisco Gómez

  2. CONACYT-Instituto de Ingeniería, Universidad Nacional Autónoma de México, Mexico City, Mexico

    Lizeth Torres

  3. Tecnológico Nacional de México, Centro Nacional de Investigación y Desarrollo Tecnológico, Cuernavaca, Morelos, Mexico

    Ricardo Fabricio Escobar

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Kumar, S., Kumar, A., Nieto, J.J., Sharma, B. (2019). Atangana–Baleanu Derivative with Fractional Order Applied to the Gas Dynamics Equations . In: Gómez, J., Torres, L., Escobar, R. (eds) Fractional Derivatives with Mittag-Leffler Kernel. Studies in Systems, Decision and Control, vol 194. Springer, Cham. https://doi.org/10.1007/978-3-030-11662-0_14

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  • DOI: https://doi.org/10.1007/978-3-030-11662-0_14

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  • Online ISBN: 978-3-030-11662-0

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