THERMAL SCIENCE

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Hami Gundogdu

,

Ömer Faruk Gozukizil

ON THE APPROXIMATE NUMERICAL SOLUTIONS OF FRACTIONAL HEAT EQUATION WITH HEAT SOURCE AND HEAT LOSS

ABSTRACT
In this paper, we are interested in obtaining an approximate numerical solution of the fractional heat equation where the fractional derivative is in Caputo sense. We also consider the heat equation with a heat source and heat loss. The fractional Laplace-Adomian decomposition method is applied to gain the approximate numerical solutions of these equations. We give the graphical representations of the solutions depending on the order of fractional derivatives. Maximum absolute error between the exact solutions and approximate solutions depending on the fractional-order are given. For the last thing, we draw a comparison between our results and found ones in the literature.
KEYWORDS
fractional heat equation, Heat source, heat loss, fractional LADM
PAPER SUBMITTED: 2021-07-13
PAPER REVISED: 2021-10-06
PAPER ACCEPTED: 2021-10-12
PUBLISHED ONLINE: 2021-11-06
DOI REFERENCE: https://doi.org/10.2298/TSCI210713321G
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2022, VOLUME 26, ISSUE Issue 5, PAGES [3773 - 3786]
REFERENCES
  1. Weitzner, H., Zaslavskyn, G.M., Some applications of fractional equations, Commun. Nonlinear Sci. Numer. Simulat., 8 (2003), pp. 273-281.
  2. Kilbas, A., Srivastava, H., Trujillo, J., Theory and Applications of Fractional Differential Equations, Elsevier, 2006.
  3. Baleanu, D., Golmankhaneh, A.K., Baleanu, M.C., Fractional electromagnetic equations using fractional forms, Int. J. Theor. Phys., 48 (2009), pp. 3114-3123.
  4. Khater, M.A., Chu, Y.M., Attia, R.A.M., Inc, M., Lu, D., On the Analytical and Numerical Solutions in the Quantum Magnetoplasmas: The Atangana Conformable Derivative ()-ZK Equation with Power-Law Nonlinearity, Adv. Math. Phys., (2020), doi.org/10.1155/2020/5809289
  5. Calandra, H., Gratton, S., Pinel, X., Vasseur, X., An improved two grid preconditioner for the solution of three dimensional Helmholtz problems in heterogeneous media, Numer. Linear Algebra Appl., 20 (2013), pp. 663-688.
  6. Yang, X.J., Baleanu, D., Srivastava, H.M., Local Fractional Integral Transforms and Their Applications, London and New York Academic Press, 2016.
  7. Nuruddeen, R.I., Aboodh, K.S. Analytical solution for time-fractional diffusion equation by Aboodh decomposition method, Int. J. Math. Appl., 5 (2017), pp. 115-122.
  8. Iyiola, O.S., Zaman, F.D., A note on analytical solutions of nonlinear fractional 2-D heat equation with non-local integral terms, Pramana Springer, 2016.
  9. Aksoy, Y., New perturbation iteration solutions for Bratu-type equations, Comput. Math. Appl., 59 (2010), pp. 2802-2808.
  10. Abu-Gdairi, R., Al-Smadi, M., Gumah, M., An expansion iterative technique for handling fractional differential equations using fractional power series scheme, J. Math. Stat., 11 (2015), pp. 29-38.
  11. Mohyud-Din, S.T., Noor, M.A., Noor, K.I., Hosseini, M.M. Variational iteration method for reformulated partial differential equations, Int. J. Nonlinear Sci. Numer. Simul., 11 (2010), pp. 87-92.
  12. Gündoˇgdu, H., Gözükızıl, O.F., Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex mKdV Equations, Konuralp J. Math., 7 (2019), pp. 7-15.
  13. Nuruddeen, R.I., Zaman, F.D., Zakariya, Y.F., Analyzing the fractional heat diffusion equation solution in comparison with the new fractional derivative by decomposition method, Malaya J. Mat., 7 (2019), pp. 213-222.
  14. Gündoˇgdu, H., Gözükızıl, O.F., Solving Nonlinear Partial Differential Equations by Using Adomian Decomposition Method, Modified Decomposition Method and Laplace Decomposition Method, MANAS J. Eng., 5 (2017), pp. 1-13.
  15. Gündoˇgdu, H., Gözükızıl, O.F.,Obtaining the solution of Benney-Luke Equation by Laplace and adomian decomposition methods, SAUJS, 21 (2017), pp. 1524-1528.
  16. Gündoˇgdu, H., Gözükızıl, O.F., Applications of the decomposition methods to some nonlinear partial differential equations, New Trend Math. Sci., 6 (2018), pp. 57-66.
  17. Fourier, J., The Analytical Theory of Heat, Cambridge University Press, 1878.
  18. Potter, M.C., Goldberg, J.L., Aboufadel, E., Advanced Engineering Mathematics, Oxford University Press, 2005.
  19. F. Black and M. Scholes, "The Pricing of Options and Corporate Liabilities", Journal of Political Economy, Vol. 81, No. 3, pp. 637-654, The University of Chicago Press (1973).
  20. L. Debnath, "Nonlinear Partial Differential Equations for Scientists and Engineers", Birkhäuser; 3rd edition, ISBN-13: 978-0817682644 (2012).
  21. K. Huang, "Statistical Mechanics", Wiley, 2nd Edition, ISBN-13: 978-0471815181 (1987).
  22. Farcas, A., Lesnic, D., The boundary-element method for the determination of a heat source dependent on one variable, J. Engrg. Math., 54 (2006), 375-388.
  23. Farcas, A., Lesnick, D., Identification of an unknown time-dependent heat source term from overspecified Dirichlet boundary data by conjugate gradient method, Comput. Math. Appl., 65 (2013), pp. 42-57.
  24. Hao, D.N., Thanh, P.X., Lesnick, D., Ivanov, M., Determination of a source in the heat equation from integral observations, J. Comput. Appl. Math., 264 (2014), pp. 82-98.
  25. Yang, X.J., Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, USA, 2012.
  26. Boyadjiev, l., Scherer, R., Fractional extensions of the temperature field problem in oil strata, Kuwait J. Sci. Eng., 31 (2004), pp. 15-32.
  27. Farid, G., Latif, N., Anwar, M., Imran, A., Ozair, M., Nawaz, M., On applications of Caputo kfractional derivatives, Adv Differ Equ., 439 (2019), doi.org/10.1186/s13662-019-2369-9
  28. Jumarie, G., Modified Riemann-Liouville Derivative and Fractional Taylor Series of Non-differentiable Functions Further Results, Comp. Math. Appl., 51 (2006), pp. 1137-1376.
  29. He, J.H., A new fractional derivative and its application to explanation of polar bear hairs, J. King Saud Univ. Sci., 2015.
  30. Çetinkaya, S., Demir, A., Time Fractional Diffusion Equation with Periodic Boundary Conditions, Konuralp J. Math., 2 (2020), pp. 337-342.
  31. Alqahtani, O. Analytical solution of non-linear fractional diffusion equation, Adv. Differ. Equ., 327 (2021), doi.org/10.1186/s13662-021-03480-z
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On the approximate numerical solutions of fractional heat equation with heat source and heat loss

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