Topological Methods in Nonlinear Analysis

Application of homotopy perturbation method to the Bratu-type equations

Xinlong Feng, Yinnian He, and Jixiang Meng

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Abstract

A new algorithm is presented for solving the Bratu-type equations. The numerical scheme based on the homotopy perturbation method is deduced. Two boundary value problems and an initial value problem are given to illustrate effectiveness and convenience of the proposed scheme. Our results agree very well with the numerical solutions showing that the homotopy perturbation method is a promising method.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 31, Number 2 (2008), 243-252.

Dates
First available in Project Euclid: 13 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1463150268

Mathematical Reviews number (MathSciNet)
MR2432082

Zentralblatt MATH identifier
1158.34306

Citation

Feng, Xinlong; He, Yinnian; Meng, Jixiang. Application of homotopy perturbation method to the Bratu-type equations. Topol. Methods Nonlinear Anal. 31 (2008), no. 2, 243--252. https://projecteuclid.org/euclid.tmna/1463150268


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References

  • C. Arslanturk, A decomposition method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity , Internat. Commun. Heat Mass Transfer, 32 (2005), 831–841
  • \ref ––––, Optimum design of space radiators with temperature-dependent thermal conductivity , Appl. Thermal Engineering, 26 (2006), 1149–1157
  • \ref N. Bildik and A. Konuralp, The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations , Internat. J. Nonlinear Sci. Numer. Simul., 7 (2006), 65–70
  • \ref M. H. Chang, A decomposition solution for fins with temperature dependent surface heat flux , Internat. J. Heat Mass Transfer., 48 (2005), 1819–1824
  • \ref M. El-Shahed, Application of He's homotopy perturbation method to Volterra's integro-differential equation , Internat. J. Nonlinear Sci. Numer. Simul., 6 (2005), 163–168
  • \ref J. H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems , Internat. J. Non-Linear Mech., 35 (2000), 37–43 \ref ––––, Application of homotopy perturbation method to nonlinear wave equations , Chaos Solitons Fractals, 26 (2005), 695–700 \ref ––––, Homotopy perturbation method for bifurcation of nonlinear problems , Internat. J. Nonlinear Sci. Numer. Simul., 6 (2007), 207–208 \ref ––––, Homotopy perturbation method for solving boundary value problems , Phys. Lett. A, 350 (2006), 87–88 \ref ––––, Homotopy perturbation method: a new nonlinear analytical technique , Appl. Math. Comput., 135 (2003), 73–79 \ref ––––, Homotopy perturbation technique , Comput. Methods Appl. Mech. Engrg., 178 (1999), 257–262 \ref ––––, Limit cycle and bifurcation of nonlinear problems , Chaos Solitons Fractals, 26 (2005), 827–833 \ref ––––, Modified Lindsted–Poincaré methods for some strongly nonlinear oscillations, Part \romIII: Double series expansion, Internat. J. Nonlinear Sci. Numer. Simul., 2 (2001), 317–320 \ref ––––, Modified Lindstedt–Poincaré methods for some strongly non-linear oscillations, Part \romI: expansion of a constant, Internat. J. Nonlinear Mech., 37 (2002), 309–314 \ref ––––, Modified Lindstedt–Poincaré methods for some strongly non-linear oscillations, Part \romII: a new transformation, Internat. J. Nonlinear Mech., 37 (2002), 315–320 \ref ––––, New interpretation of homotopy perturbation method , Internat. J. Modern Phys. B., 20 (2006), 2561–2568 \ref ––––, Non-perturbative methods for strongly nonlinear problems , Berlin, dissertation. de–Verlag im Internet GmbH (2006) \ref ––––, Periodic solutions and bifurcations of delay-differential equations , Phys. Lett. A, 347 (2005), 228–230 \ref ––––, Some asymptotic methods for strongly nonlinear equations , Internat. J. Modern Phys. B, 20 (2006), 1141–1199 \ref ––––, homotopy perturbation method for nonlinear oscillators with discontinuities , Appl. Math. Comput., 151 (2004), 287–292 \ref ––––, Variational iteration method –- a kind of non-linear analytical technique: Some examples , Internat. J. Nonlinear Mech., 34 (1999), 699–708 \ref ––––, Variational principles for some nonlinear partial differential equations with variable coefficients , Chaos Solitons Fractals, 19 (2004), 847–851 \ref ––––, Variational iteration method –- some recent results and new interpretations , J. Comput. Appl. Math., 207 (2007), 3–17
  • \ref J. H. He and Wu XH, Construction of solitary solution and compacton-like solution by variational iteration method , Chaos Solitons Fractals, 29 (2006), 108–113
  • \ref H. M. Liu, Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt–Poincaré method , Chaos Solitons Fractals, 23 (2005), 577–579
  • \ref S. Momani and S. Abuasad, Application of He's variational iteration method to Helmholtz equation , Chaos Solitons Fractals, 27 (2006), 1119–1123
  • \ref Z. M. Odibat and S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Internat. J. Nonlinear Sci. Numer. Simul., 7 (2006), 27–34
  • \ref A. M. Siddiqui, M. Ahmed and Q. K. Ghori, Couette and Poiseuille flows for non-Newtonian fluids , Internat. J. Nonlinear Sci. Numer. Simul., 7 (2006), 15–26
  • \ref A. M. Siddiqui, R. Mahmood and Q. K. Ghori, Thin film flow of a third grade fluid on a moving belt by He's homotopy perturbation method , Internat. J. Nonlinear Sci. Numer. Simul., 7 (2006), 7–14
  • \ref Y. Wu, Variational approach to higher-order water-wave equations , Chaos Solitons Fractals, 32 (2007), 195–198
  • \ref L. Xu, He's parameter-expanding methods for strongly nonlinear oscillators , J. Comput. Appl. Math., 207 (2007), 148–154 \ref ––––, Variational approach to solitons of nonlinear $K(m,n)$ equations , Chaos Solitons Fractals, to appear \ref ––––, Variational principles for coupled nonlinear Schrödinger equations , Phys. Lett. A, 359 (2006), 627–629 \ref ––––, Variational iteration method-reality, potential, and challenges , J. Comput. Appl. Math., 207 (2007), 1–2