International Scientific Journal

Thermal Science - Online First

online first only

The study of heat transfer phenomena by using modified homotopy perturbation method coupled by Laplace transform

In this paper, we present Modified Homotopy Perturbation Method Coupled by Laplace Transform (MHPMLT) to solve nonlinear problems. As case study MHPMLT is employed in order to obtain an approximate solution for the nonlinear differential equation that describes the steady state of a heat one-dimensional flow. The comparison between approximate and exact solutions shows the practical potentiality of the method.
PAPER REVISED: 2018-07-15
PAPER ACCEPTED: 2018-07-20
  1. Aminikhan, H., Hemmatnezhad, M., A novel Effective Approach for Solving Nonlinear Heat Transfer Equations, Heat Transfer- Asian Research, 41 (2012), 6, pp. 459-466.
  2. Rashidi, M.M., et al., A study of non-newtonian flow and heat transfer over a non-isothermal wedge using the homotopy analysis method, Chemical EngineeringCommunications, 199 (2012), pp. 231-256.
  3. Rashidi, M., et al., Analytic approximate solutions for steady flow over a rotating disk in porous medium with heat transfer by homotopy analysis method, Computers & Fluids, 54 (2012), pp. 1-9.
  4. Spiegel, M.R., Teoría y Problemas de Transformadas de Laplace, primera edición. Serie de compendios Schaum, Mc-Graw Hill, México, 1988.
  5. Mishra, H.K., and Nagar, A.K., He-Laplace method for linear and nonlinear partial differential equations. Journal of Applied Mathematics 2012 (2012), pp. 1-16.
  6. Liu, Z. J., et al., Hybridization of homotopy perturbation method and Laplace transformation for the partial differential equations. Thermal Science, 21 (2017),4, pp. 1843-1846.
  7. He, J.H., Asymptotic methods for solitary solutions and compactons, Abstract and applied analysis. 2012, (2012). DOI: 10.1155/2012/916793
  8. Filobello-Nino U., et al., Laplace transform-homotopy perturbation method as a powerful tool to solve nonlinear problems with boundary conditions defined on finite intervals, Computational and Applied Mathematics, 34 (2015), 1, pp. 1-16. DOI= 10.1007/s40314-013-0073-z.
  9. Marinca, V. and Herisanu, N., Nonlinear Dynamical Systems in Engineering, first edition. (Springer-Verlag, Berlin Heidelberg 2011).
  10. Aminikhah H., Analytical Approximation to the Solution of Nonlinear Blasius Viscous Flow Equation by LTNHPM, International Scholarly Research Network ISRN Mathematical Analysis, 2012 (2012), pp. 1-10 Article ID 957473, doi: 10.5402/2012/957473.
  11. Khan, M., et al., A new study between homotopy analysis method and homotopy perturbation transform method on a semi infinite domain, Mathematical and Computer Modelling, 55 (2011), pp. 1143- 1150.
  12. Assas, L.M.B., Approximate solutions for the generalized K-dV- Burgers' equation by He's variational iteration method, Phys. Scr., 76 (2007), pp. 161-164. DOI: 10.1088/0031-8949/76/2/008
  13. Evans, D.J. and Raslan, K.R. , The Tanh function method for solving some important nonlinear partial differential, Int. J. Computat. Math., 82 (2005), pp. 897-905, DOI:10.1080/00207160412331336026
  14. Xu, F., A generalized soliton solution of the Konopelchenko-Dubrovsky equation using exp-function method, Zeitschrift Naturforschung - Section A Journal of Physical Sciences, 62 (2007), 12, 685-688.
  15. Adomian, G., A review of decomposition method in applied mathematics, Mathematical Analysis and Applications. 135 (1998), pp. 501-544.
  16. Zhang, L.-N. and Xu, L. , 2007. Determination of the limit cycle by He's parameter expansion for oscillators in a potential, Zeitschrift für Naturforschung - Section A Journal of Physical Sciences, 62 (2007), 7-8, pp. 396-398.
  17. He, J.H., Homotopy perturbation technique. Comput. Methods Applied Mech. Eng., 178 (1999), pp. 257-262. DOI: 10.1016/S0045-7825(99)00018-3
  18. Aminikhah, H., The combined Laplace transform and new homotopy perturbation method for stiff systems of ODE s, Applied Mathematical Modelling, 36 (2012), pp. 3638-3644.
  19. He, J.H., A coupling method of a homotopy technique and a perturbation technique for nonlinear problems. Int. J. Non-Linear Mech., 351 (1998) , pp. 37-43. DOI: 10.1016/S0020-7462(98)00085-7
  20. El-Dib, Y.O., and Moatimid, G.M., On the coupling of the homotopy perturbation and Frobenius method for exact solutions of singular nonlinear differential equations, Nonlinear Science Letters A, 9 (2018),3, pp. 219-230
  21. El-Dib, Y.O. Multiple scales homotopy perturbation method for nonlinear oscillators, Nonlinear Sci. Lett. A, 8 (2017),4, pp. 352-364.
  22. Adamu, M. Y., and Ogenyi, P., Parameterized homotopy perturbation method, Nonlinear Sci. Lett. A 8(2017),2, pp. 240-243.
  23. Vazquez-Leal, H., et al., Nonlinearities distribution homotopy perturbation method to find solution for Troesch problem, Nonlinear Science Letters A, 9 (2018),3, pp. 279-291.
  24. Filobello-Niño U., et al., Perturbation method and Laplace-Padé approximation to solve nonlinear problems, Miskolc Mathematical Notes, 14 (2013), 1, pp. 89-101.
  25. Resnick, R., Física Parte 1, CIA. Editorial Continental. S.A de C.V, México 1980.
  26. Murray R. Spiegel, Teoría y Problemas de Análisis de Fourier. Serie de compendios Schaum, McGraw Hill, México, 1978.
  27. Fernández-Rojas, F., et al., Conductividad térmica en metales, semiconductores, dieléctricos y materiales amorfos. Revista de la facultad de ingeniería U.C.V., 23 (2008) , 3, pp. 5-15.
  28. Holmes, M.H., Introduction to Perturbation Methods, Springer-Verlag, New York, USA, 1995.