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Laplace transform-homotopy perturbation method as a powerful tool to solve nonlinear problems with boundary conditions defined on finite intervals

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Abstract

This article proposes Laplace transform-homotopy perturbation method (LT-HPM) to solve nonlinear differential equations with Dirichlet, mixed, and Neumann boundary conditions. After comparing figures between approximate and exact solutions, we will see that the proposed solutions are of high accuracy and, therefore, that LT-HPM is extremely efficient.

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References

  • Adomian G (1988) A review of decomposition method in applied mathematics. Math Anal Appl 135:501–544

    Article  MATH  MathSciNet  Google Scholar 

  • Aminikhah H (2011) Analytical approximation to the solution of nonlinear Blasius viscous flow equation by LTNHPM. International Scholarly Research Network ISRN Mathematical Analysis, vol 2012, Article ID 957473. doi:10.5402/2012/957473

  • Aminikhah H (2012) The combined Laplace transform and new homotopy perturbation method for stiff systems of ODE’s. Appl Math Model 36:3638–3644

    Article  MATH  MathSciNet  Google Scholar 

  • Aminikhan H, Hemmatnezhad M (2012) A novel effective approach for solving nonlinear heat transfer equations. Heat Transf Asian Res 41(6):459–466

    Article  Google Scholar 

  • Assas LMB (2007) Approximate solutions for the generalized K-dV–Burgers’ equation by He’s variational iteration method. Phys Scr 76:161–164. doi:10.1088/0031-8949/76/2/008

    Article  MATH  Google Scholar 

  • Babolian E, Biazar J (2002) On the order of convergence of Adomian method. Appl Math Comput 130(2):383–387. doi:10.1016/S0096-3003(01)00103-5

    Article  MATH  MathSciNet  Google Scholar 

  • Belendez A, Pascual C, Alvarez ML, Méndez DI, Yebra MS, Hernández A (2009) High order analytical approximate solutions to the nonlinear pendulum by He’s homotopy method. Physica Scripta 79(1):1–24. doi:10.1088/0031-8949/79/01/015009

    Article  Google Scholar 

  • Biazar J, Aminikhan H (2009) Study of convergence of homotopy perturbation method for systems of partial differential equations. Comput Math Appl 58(11–12): 2221–2230

    Google Scholar 

  • Biazar J, Ghazvini H (2009) Convergence of the homotopy perturbation method for partial differential equations. Nonlinear Anal Real World Appl 10(5): 2633–2640

    Google Scholar 

  • Chowdhury SH (2011) A comparison between the modified homotopy perturbation method and Adomian decomposition method for solving nonlinear heat transfer equations. J Appl Sci 11:1416–1420. doi:10.3923/jas.2011.1416.1420

    Article  Google Scholar 

  • El-Shaed M (2005) Application of He’s homotopy perturbation method to Volterra’s integro differential equation. Int J Nonlinear Sci Numer Simul 6:163–168

    MathSciNet  Google Scholar 

  • Evans DJ, Raslan KR (2005) The Tanh function method for solving some important nonlinear partial differential. Int J Comput Math 82:897–905. doi:10.1080/00207160412331336026

    Article  MATH  MathSciNet  Google Scholar 

  • Fereidon A, Rostamiyan Y, Akbarzade M, Ganji DD (2010) Application of He’s homotopy perturbation method to nonlinear shock damper dynamics. Arch Appl Mech 80(6): 641–649. doi:10.1007/s00419-009-0334-x

    Google Scholar 

  • Filobello-Nino U, Vazquez-Leal H, Castaneda-Sheissa R, Yildirim A, Hernandez Martinez L, Pereyra Diaz D, Perez Sesma A, Hoyos Reyes C (2012) An approximate solution of Blasius equation by using HPM method. Asian J Math Stat 2012: 10. doi:10.3923/ajms.2012. ISSN 1994-5418

  • Filobello-Nino U, Vazquez-Leal H, Khan Y, Yildirim A, Jimenez-Fernandez VM, Herrera May AL, Castaneda-Sheissa R, Cervantes-Perez J (2013) Using perturbation methods and Laplace-Padé approximation to solve nonlinear problems. Miskolc Mathematical Notes 14(1):89–101. e-ISSN: 1787–2413

    Google Scholar 

  • Filobello-Nino U, Vazquez-Leal H, Khan Y, Yildirim A, Pereyra-Diaz D, Perez-Sesma A, Hernandez-Martinez L, Sanchez-Orea J, Castaneda-Sheissa R, Rabago-Bernal F (2012) HPM applied to solve nonlinear circuits: a study case. Appl Math Sci 6(85–88):4331–4344

    Google Scholar 

  • Ganji DD, Babazadeh H, Noori F, Pirouz MM, Janipour M (2009) 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). Int J Nonlinear Sci 7(4):309–404

  • Ganji DD, Mirgolbabaei H, Miansari Me, Miansari Mo (2008) Application of homotopy perturbation method to solve linear and non-linear systems of ordinary differential equations and differential equation of order three. J Appl Sci 8:1256–1261. doi:10.3923/jas.2008.1256.1261

    Google Scholar 

  • He JH (2000) A coupling method of a homotopy technique and a perturbation technique for nonlinear problems. Int J Non-Linear Mech 35(1): 37–43. doi: 10.1016/S0020-7462(98)00085-7

    Google Scholar 

  • He JH (2006) Some asymptotic methods for strongly nonlinear equations. Int J Modern Phys B 20(10):1141–1199

    Google Scholar 

  • He JH (1999) Homotopy perturbation technique. Comput Methods Appl Mech Eng 178:257–262. doi:10.1016/S0045-7825(99)00018-3

    Article  MATH  Google Scholar 

  • He JH (2006) Homotopy perturbation method for solving boundary value problems. Phys Lett A 350(1–2):87–88

    Article  MATH  MathSciNet  Google Scholar 

  • He JH (2006) Some asymptotic methods for strongly nonlinear equations. Int J Modern Phys B 20(10):1141–1199. doi:10.1142/S0217979206033796

    Article  MATH  Google Scholar 

  • He JH (2007) Variational approach for nonlinear oscillators. Chaos Solitons Fractals 34:1430–1439. doi:10.1016/j.chaos.2006.10.026

    Article  MATH  MathSciNet  Google Scholar 

  • He JH (2008) Recent development of the homotopy perturbation method. Topol Methods Nonlinear Anal 31(2):205–209

    MATH  MathSciNet  Google Scholar 

  • Kazemnia M, Zahedi SA, Vaezi M, Tolou N (2008) Assessment of modified variational iteration method in BVPs high-order differential equations. J Appl Sci 8:4192–4197. doi:10.3923/jas.2008.4192.4197

    Article  Google Scholar 

  • Khan Y, Qingbiao W (2011) Homotopy perturbation transform method for nonlinear equations using He’s polynomials. Comput Math Appl 61(8):1963–1967

    Article  MATH  MathSciNet  Google Scholar 

  • Khan M, Gondal MA, Vanani K (2011) A new study between homotopy analysis method and homotopy perturbation transform method on a semi infinite domain. Math Comput Model 55:1143–1150

    Article  MathSciNet  Google Scholar 

  • Kooch A, Abadyan M (2011) Evaluating the ability of modified Adomian decomposition method to simulate the instability of freestanding carbon nanotube: comparison with conventional decomposition method. J Appl Sci 11:3421–3428. doi:10.3923/jas.2011.3421.3428

    Google Scholar 

  • Kooch A, Abadyan M (2012) Efficiency of modified Adomian decomposition for simulating the instability of nano-electromechanical switches: comparison with the conventional decomposition method. Trends Appl Sci Res 7:57–67. doi:10.3923/tasr.2012.57.67

    Article  Google Scholar 

  • Madani M, Fathizadeh M, Khan Y, Yildirim A (2011) On the coupling of the homotopy perturbation method and Laplace transformation. Math Comput Model 53(9–10):1937–1945

    Article  MATH  MathSciNet  Google Scholar 

  • Mahmoudi J, Tolou N, Khatami I, Barari A, Ganji DD (2008) Explicit solution of nonlinear ZK-BBM wave equation using exp-function method. J Appl Sci 8:358–363. doi:10.3923/jas.2008.358.363

    Article  Google Scholar 

  • Murray RS (1988) Teoría y Problemas de Transformadas de Laplace, primera edición. Serie de compendios Schaum

  • Noorzad R, Tahmasebi Poor A, Omidvar M (2008) Variational iteration method and homotopy-perturbation method for solving Burgers equation in fluid dynamics. J Appl Sci 8:369–373. doi:10.3923/jas.2008.369.373

    Article  Google Scholar 

  • Patel T, Mehta MN, Pradhan VH (2012) The numerical solution of Burger’s equation arising into the irradiation of tumour tissue in biological diffusing system by homotopy analysis method. Asian J Appl Sci 5:60–66. doi:10.3923/ajaps.2012.60.66

    Article  Google Scholar 

  • Sharma PR, Methi G (2011) Applications of homotopy perturbation method to partial differential equations. Asian J Math Stat 4:140–150. doi:10.3923/ajms.2011.140.150

    Article  Google Scholar 

  • Vanani SK, Heidari S, Avaji M (2011) A low-cost numerical algorithm for the solution of nonlinear delay boundary integral equations. J Appl Sci 11:3504–3509. doi:10.3923/jas.2011.3504.3509

    Article  Google Scholar 

  • Vasile M, Nicolae H (2011) Nonlinear dynamical systems in engineering, 1 st edn. Springer, Berlin

  • Vazquez-Leal H, Castañeda-Sheissa R, Filobello-Nino U, Sarmiento-Reyes A, Sánchez-Orea J (2012) High accurate simple approximation of normal distribution related integrals. Mathematical Problems in Engineering, vol. 2012, Article ID 124029. doi: 10.1155/2012/124029

  • Vazquez-Leal H, Filobello-Niño U, Castañeda-Sheissa R, Hernandez Martinez L, Sarmiento-Reyes A (2012a) Modified HPMs inspired by homotopy continuation methods. Mathematical Problems in Engineering, vol. 2012, Article ID 309123. doi:10.155/2012/309123

  • Xu F (2007) A generalized soliton solution of the Konopelchenko–Dubrovsky equation using exp-function method. Zeitschrift Naturforschung: Sect A J Phys Sci 62(12):685–688

    MATH  Google Scholar 

  • Zhang L-N, Xu L (2007) Determination of the limit cycle by He’s parameter expansion for oscillators in a \(u^3/1+u^2\) potential. Zeitschrift für Naturforschung: Sect A J Phys Sci 62(7–8):396–398

    MATH  Google Scholar 

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Authors and Affiliations

  1. Electronic Instrumentation and Atmospheric Sciences School, University of Veracruz, Circuito Gonzalo Aguirre Beltrán S/N, 91000 , Xalapa, Veracruz, Mexico

    U. Filobello-Nino, H. Vazquez-Leal, A. Perez-Sesma, V. M. Jimenez-Fernandez, D. Pereyra-Diaz, J. M. Mendez-Perez & J. Sanchez-Orea

  2. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    Y. Khan

  3. National Institute for Astrophysics, Optics and Electronics, Luis Enrique Erro #1, Sta. María Tonantzintla, 72840 , Puebla, Mexico

    A. Diaz-Sanchez

  4. Micro and Nanotechnology Research Center, University of Veracruz, Calzada Ruiz Cortines 455, 94292 , Boca del Rio, Veracruz, Mexico

    A. Herrera-May

Authors
  1. U. Filobello-Nino
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  2. H. Vazquez-Leal
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  3. Y. Khan
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  4. A. Perez-Sesma
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  5. A. Diaz-Sanchez
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  6. V. M. Jimenez-Fernandez
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  7. A. Herrera-May
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  8. D. Pereyra-Diaz
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  9. J. M. Mendez-Perez
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  10. J. Sanchez-Orea
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Correspondence to H. Vazquez-Leal.

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Communicated by Cristina Turner.

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Filobello-Nino, U., Vazquez-Leal, H., Khan, Y. et al. Laplace transform-homotopy perturbation method as a powerful tool to solve nonlinear problems with boundary conditions defined on finite intervals. Comp. Appl. Math. 34, 1–16 (2015). https://doi.org/10.1007/s40314-013-0073-z

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  • DOI: https://doi.org/10.1007/s40314-013-0073-z

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