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Exact Solution for the Conformable Burgers’ Equation by the Hopf-Cole Transform

Year 2016, Volume: 13 Issue: 2, - , 01.11.2016

Abstract

In this paper, we use Hopf-Cole transform to solve conformable Burgers’ equation. After applying
Hopf-Cole transform to conformable Burgers’ equation, we achieve conformable heat equation. Subsequently
by using Fourier transform we have the exact solution of conformable Burgers’ equation with fractional order

References

  • [1] H. Bateman, Some Recent Researches on the Motion of Fluids, Monthly Weather Rev. 43, (1915), 163-170.
  • [2] L. Debnath, Partial Differential Equations for Scientists and Engineers, Birkh ¨auser, Boston (1997).
  • [3] J. M. Burger, Mathematical Examples Illustrating the relations Occurring in the Theory of Turbulent Fluid Motion, Trans. Roy. Neth. Acad. Sci. Amsterdam 17, (1939), 1-53.
  • [4] J. M. Burger, A Mathematical Model Illustrating the Theory of Turbulence, in Advances Applied Mechanics 1, Academic Press, New York (1948).
  • [5] J. D. Cole, On a Quasi Linear Parabolic Equation Occurring in Aerodynamics, Quart. Apply. Math. 9, (1951), 225-236.
  • [6] E. Varoglu, W.D.L. Finn, Space-Time Finite Incorporating Characteristics for the Burgers’ Equation, Internat. J. Number. Methods Engrg. 16, (1980), 171-184.
  • [7] D.T. Blackstock, Termaviscous Attenuation of Plane, Peridodic, Finite-amplitude Sound Waves, The Journal of Acoustical Society of America 36, (1964).
  • [8] Z.A. Goldberg, Finite-Amplitude Waves in Magnetohydrodynamics, Soviet Physics Jetp 15, (1962), 179-181.
  • [9] L.A. Pospelov, Propagation on Finite-amplitude Elastic Waves, Soviet Physics Acoust 11, (1966), 302-304.
  • [10] T. Ozis¸, A. Ozdes¸, A Direct Variational Method Applied to Bugers’ Equation, Journal of Computational and Applied Mathematics 71, (1996), 163-175.
  • [11] J. Caldwell, P. Wanless, A.E. Cook, A Finite Element Approach to Burgers’ Equation, App. Math. Model. 5, (1981), 189-193.
  • [12] D. J. Evans, A.R. Abdullah, The Group Explicit Method for the Solution of the Burgers’ Equation, Computing 32, (1984), 239-253.
  • [13] R.C. Mittal, P. Singhal, Numerical Solution of Burger’s Equation, Commun. Numer. Meth. Engng. 9, (1993), 397-406.
  • [14] A. Esen, N.M. Yagmurlu, O. Tasbozan, Approximate Analytical Solution to Time-Fractional Damped Burger and Cahn-Allen Equations, Appl. Math. Inf. Sci. 7, (2013), 1951-1956.
  • [15] E.A.-B. Abdel-Salam, E.A. Yousif, Y.A.S. Arko, E.A.E. Gumma, Solution of Moving Boundary Space-Time Fractional Burger’s Equation, J. App. Math 2014, http://dx.doi.org/10.1155/2014/218092 (2014).
  • [16] A. Esen, O. Tasbozan, Numerical Solution of Time Fractional Burgers’ Equation by Cubic B-spline Finite Elements, Mediterr. J. Math., DOI 10.1007/s00009-015-0555-x (2015).
  • [17] M. Inc, The Approximate and Exact Solutions of the Space- and Time-fractional Burgers Equations with Initial Conditions by Variational Iteration Method, J. Math. Anal. Appl. 345, (2008), 476-484.
  • [18] E. Hopf, The Partial Differential Equation ut +uux =µuxx, Comm. Pure Appl. Math. 3, (1950), 201-230.
  • [19] T. Ozis, E.N. Aksan, A. Ozdes, A Finite Element Approach for Solution of Burgers Equation, Applied Mathematics and Computation 139, (2003), 417-428.
  • [20] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley &Sons, New York, NY,USA (1993).
  • [21] A.Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego (2006).
  • [22] I. Podlubny, Fractional Differential Equations. Academic Press,San Diego, Calif, USA (1999).
  • [23] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A New Definition of Fractional Derivative, Journal of Computational and Applied Mathematics 264, (2014), 65-70.
  • [24] T. Abdeljawad, On Conformable Fractional Calculus, Journal of Computational and Applied Mathematics, 279, (2015), 57-66.
  • [25] T. Abdeljawad, Mohammad Al Horani, Roshdi Khalil, Conformable Fractional Semigroup Operators, Journal of Semigroup Theory and Applications 2015 (2015) Article ID. 7.
  • [26] D. R. Anderson, D. J. Ulness, Newly Defined Conformable Derivatives, Advances in Dynamical Systems and Applications 10, (2015), 109-137.
Year 2016, Volume: 13 Issue: 2, - , 01.11.2016

Abstract

References

  • [1] H. Bateman, Some Recent Researches on the Motion of Fluids, Monthly Weather Rev. 43, (1915), 163-170.
  • [2] L. Debnath, Partial Differential Equations for Scientists and Engineers, Birkh ¨auser, Boston (1997).
  • [3] J. M. Burger, Mathematical Examples Illustrating the relations Occurring in the Theory of Turbulent Fluid Motion, Trans. Roy. Neth. Acad. Sci. Amsterdam 17, (1939), 1-53.
  • [4] J. M. Burger, A Mathematical Model Illustrating the Theory of Turbulence, in Advances Applied Mechanics 1, Academic Press, New York (1948).
  • [5] J. D. Cole, On a Quasi Linear Parabolic Equation Occurring in Aerodynamics, Quart. Apply. Math. 9, (1951), 225-236.
  • [6] E. Varoglu, W.D.L. Finn, Space-Time Finite Incorporating Characteristics for the Burgers’ Equation, Internat. J. Number. Methods Engrg. 16, (1980), 171-184.
  • [7] D.T. Blackstock, Termaviscous Attenuation of Plane, Peridodic, Finite-amplitude Sound Waves, The Journal of Acoustical Society of America 36, (1964).
  • [8] Z.A. Goldberg, Finite-Amplitude Waves in Magnetohydrodynamics, Soviet Physics Jetp 15, (1962), 179-181.
  • [9] L.A. Pospelov, Propagation on Finite-amplitude Elastic Waves, Soviet Physics Acoust 11, (1966), 302-304.
  • [10] T. Ozis¸, A. Ozdes¸, A Direct Variational Method Applied to Bugers’ Equation, Journal of Computational and Applied Mathematics 71, (1996), 163-175.
  • [11] J. Caldwell, P. Wanless, A.E. Cook, A Finite Element Approach to Burgers’ Equation, App. Math. Model. 5, (1981), 189-193.
  • [12] D. J. Evans, A.R. Abdullah, The Group Explicit Method for the Solution of the Burgers’ Equation, Computing 32, (1984), 239-253.
  • [13] R.C. Mittal, P. Singhal, Numerical Solution of Burger’s Equation, Commun. Numer. Meth. Engng. 9, (1993), 397-406.
  • [14] A. Esen, N.M. Yagmurlu, O. Tasbozan, Approximate Analytical Solution to Time-Fractional Damped Burger and Cahn-Allen Equations, Appl. Math. Inf. Sci. 7, (2013), 1951-1956.
  • [15] E.A.-B. Abdel-Salam, E.A. Yousif, Y.A.S. Arko, E.A.E. Gumma, Solution of Moving Boundary Space-Time Fractional Burger’s Equation, J. App. Math 2014, http://dx.doi.org/10.1155/2014/218092 (2014).
  • [16] A. Esen, O. Tasbozan, Numerical Solution of Time Fractional Burgers’ Equation by Cubic B-spline Finite Elements, Mediterr. J. Math., DOI 10.1007/s00009-015-0555-x (2015).
  • [17] M. Inc, The Approximate and Exact Solutions of the Space- and Time-fractional Burgers Equations with Initial Conditions by Variational Iteration Method, J. Math. Anal. Appl. 345, (2008), 476-484.
  • [18] E. Hopf, The Partial Differential Equation ut +uux =µuxx, Comm. Pure Appl. Math. 3, (1950), 201-230.
  • [19] T. Ozis, E.N. Aksan, A. Ozdes, A Finite Element Approach for Solution of Burgers Equation, Applied Mathematics and Computation 139, (2003), 417-428.
  • [20] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley &Sons, New York, NY,USA (1993).
  • [21] A.Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego (2006).
  • [22] I. Podlubny, Fractional Differential Equations. Academic Press,San Diego, Calif, USA (1999).
  • [23] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A New Definition of Fractional Derivative, Journal of Computational and Applied Mathematics 264, (2014), 65-70.
  • [24] T. Abdeljawad, On Conformable Fractional Calculus, Journal of Computational and Applied Mathematics, 279, (2015), 57-66.
  • [25] T. Abdeljawad, Mohammad Al Horani, Roshdi Khalil, Conformable Fractional Semigroup Operators, Journal of Semigroup Theory and Applications 2015 (2015) Article ID. 7.
  • [26] D. R. Anderson, D. J. Ulness, Newly Defined Conformable Derivatives, Advances in Dynamical Systems and Applications 10, (2015), 109-137.
There are 26 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

Ali Kurt This is me

Yucel Cenesiz This is me

Orkun Tasbozan

Publication Date November 1, 2016
Published in Issue Year 2016 Volume: 13 Issue: 2

Cite

APA Kurt, A., Cenesiz, Y., & Tasbozan, O. (2016). Exact Solution for the Conformable Burgers’ Equation by the Hopf-Cole Transform. Cankaya University Journal of Science and Engineering, 13(2).
AMA Kurt A, Cenesiz Y, Tasbozan O. Exact Solution for the Conformable Burgers’ Equation by the Hopf-Cole Transform. CUJSE. November 2016;13(2).
Chicago Kurt, Ali, Yucel Cenesiz, and Orkun Tasbozan. “Exact Solution for the Conformable Burgers’ Equation by the Hopf-Cole Transform”. Cankaya University Journal of Science and Engineering 13, no. 2 (November 2016).
EndNote Kurt A, Cenesiz Y, Tasbozan O (November 1, 2016) Exact Solution for the Conformable Burgers’ Equation by the Hopf-Cole Transform. Cankaya University Journal of Science and Engineering 13 2
IEEE A. Kurt, Y. Cenesiz, and O. Tasbozan, “Exact Solution for the Conformable Burgers’ Equation by the Hopf-Cole Transform”, CUJSE, vol. 13, no. 2, 2016.
ISNAD Kurt, Ali et al. “Exact Solution for the Conformable Burgers’ Equation by the Hopf-Cole Transform”. Cankaya University Journal of Science and Engineering 13/2 (November 2016).
JAMA Kurt A, Cenesiz Y, Tasbozan O. Exact Solution for the Conformable Burgers’ Equation by the Hopf-Cole Transform. CUJSE. 2016;13.
MLA Kurt, Ali et al. “Exact Solution for the Conformable Burgers’ Equation by the Hopf-Cole Transform”. Cankaya University Journal of Science and Engineering, vol. 13, no. 2, 2016.
Vancouver Kurt A, Cenesiz Y, Tasbozan O. Exact Solution for the Conformable Burgers’ Equation by the Hopf-Cole Transform. CUJSE. 2016;13(2).