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Volume 2, Issue 6, December 2013, Page: 143-148
The Study of Heat Transfer Phenomena Using PM for Approximate Solution with Dirichlet and Mixed Boundary Conditions
U. Filobello-Nino, Electronic Instrumentation and Atmospheric Sciences School, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, Mexico 91000
H. Vazquez-Leal, Electronic Instrumentation and Atmospheric Sciences School, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, Mexico 91000
A. Sarmiento-Reyes, National Institute for Astrophysics, Optics and Electronics, Luis Enrique Erro #1, Sta. María Tonantzintla. Puebla, México, 72840
A. Perez-Sesma, Electronic Instrumentation and Atmospheric Sciences School, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, Mexico 91000
L. Hernandez-Martinez, National Institute for Astrophysics, Optics and Electronics, Luis Enrique Erro #1, Sta. María Tonantzintla. Puebla, México, 72840
A. Herrera-May, Micro and Nanotechnology Research Center, Universidad Veracruzana, Calzada, Ruiz Cortines, Boca del Rio 94292, Veracruz, Mexico
V. M. Jimenez-Fernandez, Electronic Instrumentation and Atmospheric Sciences School, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, Mexico 91000
A. Marin-Hernandez, Department of Artificial Intelligence, Universidad Veracruzana, Sebastián Camacho No. 5, C.P. 91000 Xalapa, Veracruz, Mexico
D. Pereyra-Diaz, Electronic Instrumentation and Atmospheric Sciences School, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, Mexico 91000
A. Diaz-Sanchez, National Institute for Astrophysics, Optics and Electronics, Luis Enrique Erro #1, Sta. María Tonantzintla. Puebla, México, 72840
Received: Oct. 28, 2013;       Published: Nov. 30, 2013
DOI: 10.11648/j.acm.20130206.16      View  3389      Downloads  247
Abstract
In this paper, we present Perturbation Method (PM) to solve nonlinear problems. As case study PM is employed to obtain approximate solutions for differential equations related with heat transfer phenomena. Comparing figures between approximate and exact solutions, show the effectiveness of the method.
Keywords
Dirichlet Boundary conditions, Mixed Boundary Conditions, Nonlinear Differential Equation, Perturbation Method, Approximate Solutions
To cite this article
U. Filobello-Nino, H. Vazquez-Leal, A. Sarmiento-Reyes, A. Perez-Sesma, L. Hernandez-Martinez, A. Herrera-May, V. M. Jimenez-Fernandez, A. Marin-Hernandez, D. Pereyra-Diaz, A. Diaz-Sanchez, The Study of Heat Transfer Phenomena Using PM for Approximate Solution with Dirichlet and Mixed Boundary Conditions, Applied and Computational Mathematics. Vol. 2, No. 6, 2013, pp. 143-148. doi: 10.11648/j.acm.20130206.16
Reference
[1]
Vasile Marinca and Nicolae Herisanu, Nonlinear Dynamical Systems in Engineering, first edition, Springer-Verlag Berlin Heidelberg, 2011.
[2]
Chow, T.L., Classical Mechanics. John Wiley and Sons Inc., USA. 1995.
[3]
He, J.H., A coupling method of a homotopy technique and a perturbation technique for nonlinear problems. Int. J. Non-Linear Mech., 351 (1998) 37-43. DOI: 10.1016/S0020-7462(98)00085-7
[4]
He, J.H.,. Homotopy perturbation technique, Comput. Methods Applied Mech. Eng., 178 (1999) 257-262. DOI: 10.1016/S0045-7825(99)00018-3
[5]
Assas, L.M.B., Approximate solutions for the generalized K-dV- Burgers’ equation by He’s variational iteration method. Phys. Scr., 76 (2007) 161-164. DOI: 10.1088/0031-8949/76/2/008
[6]
He, J.H., Variational approach for nonlinear oscillators. Chaos, Solitons and Fractals, 34 (2007) 1430-1439. DOI: 10.1016/j.chaos.2006.10.026
[7]
Kazemnia, M., S.A. Zahedi, M. Vaezi and N. Tolou, Assessment of modified variational iteration method in BVPs high-order differential equations. Journal of Applied Sciences, 8 (2008) 4192-4197. DOI:10.3923/jas.2008.4192.4197
[8]
Evans, D.J. and K.R. Raslan, The Tanh function method for solving some important nonlinear partial differential. Int. J. Computat. Math., 82 (2005) 897-905. DOI: 10.1080/00207160412331336026
[9]
Xu, F., A generalized soliton solution of the Konopelchenko-Dubrovsky equation using exp-function method. ZeitschriftNaturforschung - Section A Journal of Physical Sciences, 62(12) (2007) 685-688.
[10]
Mahmoudi, J., N. Tolou, I. Khatami, A. Barari and D.D. Ganji, Explicit solution of nonlinear ZK-BBM wave equation using Exp-function method. Journal of Applied Sciences, 8 (2008) 358-363. DOI:10.3923/jas.2008.358.363
[11]
Adomian, G., A review of decomposition method in applied mathematics. Mathematical Analysis and Applications. 135 (1988) 501-544.
[12]
Babolian, E. and J. Biazar, On the order of convergence of Adomian method. Applied Mathematics and Computation, 130(2) (2002) 383-387. DOI: 10.1016/S0096-3003(01)00103-5
[13]
Kooch, A. and M. Abadyan, Efficiency of modified Adomian decomposition for simulating the instability of nano-electromechanical switches: comparison with the conventional decomposition method. Trends in Applied Sciences Research, 7 (2012) 57-67. DOI:10.3923/tasr.2012.57.67
[14]
Kooch, A. and M. Abadyan, Evaluating the ability of modified Adomian decomposition method to simulate the instability of freestanding carbon nanotube: comparison with conventional decomposition method. Journal of Applied Sciences, 11 (2011) 3421-3428. DOI:10.3923/jas.2011.3421.3428
[15]
Vanani, S. K., S. Heidari and M. Avaji, A low-cost numerical algorithm for the solution of nonlinear delay boundary integral equations. Journal of Applied Sciences, 11 (2011) 3504-3509. DOI:10.3923/jas.2011.3504.3509
[16]
Chowdhury, S. H., A comparison between the modified homotopy perturbation method and Adomian decomposition method for solving nonlinear heat transfer equations. Journal of Applied Sciences, 11 (2011) 1416-1420. DOI:10.3923/jas.2011.1416.1420
[17]
Zhang, L.-N. and L. Xu, Determination of the limit cycle by He’s parameter expansion for oscillators in a potential. ZeitschriftfürNaturforschung - Section A Journal of Physical Sciences, 62(7-8) (2007) 396-398.
[18]
He, J.H., Homotopy perturbation method for solving boundary value problems. Physics Letters A, 350(1-2) (2006) 87-88.
[19]
He, J.H., Recent Development of the Homotopy Perturbation Method. Topological Methods in Nonlinear Analysis, 31(2) (2008) 205-209.
[20]
Belendez, A., C. Pascual, M.L. Alvarez, D.I. Méndez, M.S. Yebra and A. Hernández, High order analytical approximate solutions to the nonlinear pendulum by He’s homotopy method. PhysicaScripta, 79(1) (2009) 1-24. DOI: 10.1088/0031-8949/79/01/015009
[21]
He, J.H., A coupling method of a homotopy and a perturbation technique for nonlinear problems. International Journal of Nonlinear Mechanics, 35(1) (2000) 37-43.
[22]
El-Shaed, M., Application of He’s homotopy perturbation method to Volterra’s integro differential equation. International Journal of Nonlinear Sciences and Numerical Simulation, 6 (2005) 163-168.
[23]
He, J.H., Some Asymptotic Methods for Strongly Nonlinear Equations. International Journal of Modern Physics B, 20(10) (2006) 1141-1199. DOI: 10.1142/S0217979206033796
[24]
Ganji, D.D, H. Babazadeh, F Noori, M.M. Pirouz, M Janipour. An Application of Homotopy Perturbation Method for Non linear Blasius Equation to Boundary Layer Flow Over a Flat Plate, ACADEMIC World Academic Union, ISNN 1749-3889(print), 1749-3897 (online). International Journal of Nonlinear Science, 7 (4) (2009) 309-404.
[25]
Ganji, D.D., H. Mirgolbabaei , Me. Miansari and Mo. Miansari , Application of homotopy perturbation method to solve linear and non-linear systems of ordinary differential equations and differential equation of order three. Journal of Applied Sciences, 8 (2008) 1256-1261. DOI:10.3923/jas.2008.1256.1261.
[26]
Fereidon, A., Y. Rostamiyan, M. Akbarzade and D.D. Ganji, Application of He’s homotopy perturbation method to nonlinear shock damper dynamics. Archive of Applied Mechanics, 80(6) (2010) 641-649. DOI: 10.1007/s00419-009-0334-x.
[27]
Sharma, P.R. and G. Methi, Applications of homotopy perturbation method to partial differential equations. Asian Journal of Mathematics & Statistics, 4 (2011) 140-150. DOI:10.3923/ajms.2011.140.150
[28]
Aminikhah Hossein, Analytical Approximation to the Solution of Nonlinear Blasius Viscous Flow Equation by LTNHPM. International Scholarly Research Network ISRN Mathematical Analysis, (2012), 10 pages, Article ID 957473, DOI: 10.5402/2012/957473
[29]
Noorzad, R., A. Tahmasebi Poor and M. Omidvar, Variational iteration method and homotopy-perturbation method for solving Burgers equation in fluid dynamics.Journal of Applied Sciences, 8 (2008) 369-373. DOI:10.3923/jas.2008.369.373
[30]
Patel, T., M.N. Mehta and V.H. Pradhan, The numerical solution of Burger’s equation arising into the irradiation of tumour tissue in biological diffusing system by homotopy analysis method. Asian Journal of Applied Sciences, 5 (2012) 60-66. DOI:10.3923/ajaps.2012.60.66
[31]
Vazquez-Leal H., U. Filobello-Niño, R. Castañeda-Sheissa, L. Hernandez Martinez and A. Sarmiento-Reyes, Modified HPMs inspired by homotopy continuation methods . Mathematical Problems in Engineering, (2012) 20 pages, Article ID 309123, DOI:10.155/2012/309123.
[32]
Vazquez-Leal H., R. Castañeda-Sheissa, U. Filobello-Niño, A. Sarmiento-Reyes, and J. Sánchez-Orea, High accurate simple approximation of normal distribution related integrals. Mathematical Problems in Engineering, (2012) 22 pages, Article ID 124029, DOI: 10.1155/2012/124029.
[33]
Filobello-Niño U., H. Vazquez-Leal, R. Castañeda-Sheissa, A. Yildirim, L. HernandezMartinez, D. Pereyra Díaz, A. Pérez Sesma and C. Hoyos Reyes 2012. An approximate solution of Blasius equation by using HPM method. Asian Journal of Mathematics and Statistics, Vol. 2012, 10 pages, DOI: 10.3923 /ajms.2012, ISSN 1994-5418.
[34]
Biazar, J. and H. Aminikhan, Study of convergence of homotopy perturbation method for systems of partial differential equations. Computers and Mathematics with Applications, 58 (11-12) (2009) 2221-2230.
[35]
Biazar, J. andH. Ghazvini, Convergence of the homotopy perturbation method for partial differential equations. Nonlinear Analysis: Real World Applications, 10 (5) (2009) 2633-2640.
[36]
Filobello-Niño U., H. Vazquez-Leal, D. Pereyra Díaz, A. Pérez Sesma, R. Castañeda-Sheissa, Y. Khan, A. Yildirim, L. Hernandez Martinez, and F. Rabago Bernal, HPM Applied to Solve Nonlinear Circuits: A Study Case. Applied Mathematics Sciences, 6 (87) (2012) 4331-4344.
[37]
Holmes, M.H., Introduction to Perturbation Methods. Springer-Verlag, New York, 1995.
[38]
D.D. Ganji, A.R. Sahouli and M. Famouri, A New modification of He’s homotopy perturbation method for rapid convergence of nonlinear undamped oscillators, Journal of Applied Mathematics and Computing, 30 (2009), 181-192.
[39]
Filobello-Niño U., H. Vazquez-Leal, Y. Khan, A. Yildirim, V.M. Jimenez-Fernandez, A. L Herrera May, R. Castañeda-Sheissa, and J.Cervantes Perez. Perturbation Method and Laplace-Padé Approximation to solve nonlinear problems. Miskolc Mathematical Notes, 14(1) (2013) 89-101, ISSN: 1787-2405.
[40]
Filobello-Niño U., H. Vazquez-Leal, Y. Khan, A. Perez-Sesma, A. Diaz-Sanchez, A. Herrera-May, D. Pereyra-Diaz, R. Castañeda-Sheissa, V.M. Jimenez-Fernandez, and J. Cervantes-Perez, A handy exact solution for flow due to a stretching boundary with partial slip. Revista Mexicana de Física E, 59 (2013) 51-55, ISSN 1870-3542.
[41]
Filobello-Niño U., H. Vazquez-Leal, K. Boubaker, Y. Khan, A. Perez-Sesma, A.Sarmiento Reyes, V.M. Jimenez-Fernandez, A Diaz-Sanchez, A. Herrera-May, J. Sanchez-Orea and K. Pereyra-Castro, Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand,s Equation. Asian Journal of Mathematics and Statistics, (2013) 7 pages, DOI: 10.3923 /ajms.2013, ISSN 1994-5418, 2013.
[42]
Filobello-Nino U., H. Vazquez-Leal, Y. Khan, A. Perez-Sesma, A. Diaz-Sanchez, V.M. Jimenez-Fernandez, A. Herrera-May, D. Pereyra-Diaz, J.M. Mendez-Perez and J. Sanchez-Orea, Laplace transform-homotopy perturbation method as a powerful tool to solve nonlinear problems with boundary conditions defined on finite intervals, Computational and Applied Mathematics, (2013) 1-16 ISSN: 0101-8205, DOI= 10.1007/s40314-013-0073-z.
[43]
Rajabi A., Ganji, D.D, H.Taherian, Application of Homotopy Perturbation Method in nonlinear heat conduction and convection equations. Physic Letters A, Elsevier 360 (2007) 570-573. DOI: 10.1016/ j.physleta.2006.08.079.
[44]
D.D Ganji A Rajabi., Assessment of Homotopy Perturbation and Perturbation methods in heat radiation equations. International communication in heat and mass transfer, Elsevier 360 (2006) 391-400. DOI: 10.1016/ j.icheatmasstransfer. 2005.11.001.
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