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Analytical Solutions of Some Two-Point Non-Linear Elliptic Boundary Value Problems

Several problems arising in science and engineering are modeled by differential equations that involve conditions that are specified at more than one point. The non-linear two-point boundary value problem (TPBVP) (Bratu’s equation, Troesch’s problems) occurs engineering and science, including the modeling of chemical reactions diffusion processes and heat transfer. An analytical expression pertaining to the concentration of substrate is obtained using Homotopy perturbation method for all values of parameters. These approximate analytical results were found to be in good agreement with the simulation results.

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The authors declare no conflicts of interest.

Cite this paper

V. Ananthaswamy and L. Rajendran, "Analytical Solutions of Some Two-Point Non-Linear Elliptic Boundary Value Problems,"

*Applied Mathematics*, Vol. 3 No. 9, 2012, pp. 1044-1058. doi: 10.4236/am.2012.39154.

[1] | Y. Lin, J. A. Enszer and M. Stadtherr, “Enclosing all Solutions of Two Point Boundary Value Problems for ODEs,” Computers and Chemical Engineering, Vol. 32, 2008, pp.1714-1725. doi:10.1016/j.compchemeng.2007.08.013 |

[2] | J. Newman and W. Tiedmann, “Porous-Electrode Theory with Battery Applications,” AIChE Journal, Vol. 21, No. 1, 1975, pp. 25-41. doi:10.1002/aic.690210103 |

[3] | S. H. Mirmoradi, I. Hosseinpour, S. Ghanbarpour and A. Barari, “Application of an Approximate Analytical Method to Nonlinear Troesch’s Problem,” Applied Mathematical Sciences, Vol. 3, No. 32, 2009, pp. 1579-1585. |

[4] | A. Barari, A. R. Ghotbi, F. Farrokhzad and D. D. Ganji, “Variational Iteration Method and Homotopy-Perturbation Method for Solving Different Types of Wave Equations,” Journal of Applied Sciences, Vol. 8, 2008, pp. 120-126. doi:10.3923/jas.2008.120.126 |

[5] | R. Abdoul, A. Ghotbi, A. Barari and D. D. Ganji, “Solving Ratio-Dependent Redator-Prey System with Constant Effort Harvesting Using Homotopy Perturbation Method,” Journal of Mathematical Problems in Engineering, 2008, Article ID: 945420. |

[6] | A. Barari, A. Janalizadeh and D. D. Ganji, “Application of Homotopy Perturbation Method to Zakharov-Kuznetsov Equation,” Journal of Physics, Vol. 96, 2008, pp. 1-8. doi:10.1088/1742-6596/96/1/012082 |

[7] | A. Barari, D. D. Ganji and M. J. Hosseini, “HomotopyPerturbation Method for a Nonlinear Cerebral ReactionDiffusion Equation,” Arab Journal of Mathematics and Mathematical Sciences, Vol. 1, 2007, pp. 1-9. |

[8] | A. Barari, M. Omidvar, A. R. Ghotbi and D. D. Ganji, “Application of Homotopy-Perturbation Method and Variational Iteration Method to Nonlinear Oscillator Differential Equations,” Acta Applicanda Mathematicae, Vol. 104, 2008, pp. 161-171. doi:10.1007/s10440-008-9248-9 |

[9] | A. Barari, M. Omidvar, D. D. Ganji and Abbas Tahmasebi poor, An Approximate Solution for Boundary Value Problems in Structural Engineering and Fluid Mechanics,” Journal of Mathematical Problems in Engineering, 2008, Article ID: 394103. |

[10] | L. N. Zhang and J. H. He, “Homotopy-Perturbation Method for the Solution of the Electrostatic Potential Differential Equation,” Mathematical Problems in Engineering, 2006, Article ID: 83878. doi:10.1155/MPE/2006/83878 |

[11] | J. H. He, “New Interpretation of Homotopy-Perturbation Method,” International Journal of Modern Physics B, Vol. 20, No. 18, 2006, pp. 256-2568. doi:10.1142/S0217979206034819 |

[12] | J.-H. He, “Homotopy-Perturbation Technique,” Computer Methods in Applied Mechanics and Engineering, Vol. 178, No. 3-4, 1999, pp. 257-262. doi:10.1016/S0045-7825(99)00018-3 |

[13] | J.-H. He, “Homotopy-Perturbation Method: A New Nonlinear Analytical Technique,” Applied Mathematics and Computation, Vol. 135, No. 1, 2003, pp. 73-79. doi:10.1016/S0096-3003(01)00312-5 |

[14] | Y. Chen, “Dynamic System Optimization,” Ph.D. Thesis, University of California, Los Angeles, 2006. |

[15] | Y. Chen and V. Manousiouthakis, “Identification of All Solutions of TPBV Problems,” AIChE Annual Meeting, Cincinnati, Paper #11e, Dover, New York, 2005. |

[16] | M. I. Syam and E. M. Allon, “On the Computations of Fold Points for Non Linear Elliptic Eigen Value Problems,” International Journal of Open Problems in Computer Science and Mathematics, Vol. 4, No. 1, 2011. |

[17] | J. P. Abbott, “An Efficient Algorithm for the Determination of Certain Bifurcation Point,” Journal of Computational and Applied Mathematics, Vol. 4, No. 1, 1978, pp. 19-27. doi:10.1016/0771-050X(78)90015-3 |

[18] | H. Amann, “Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Space,” SIAM Review, Vol. 8, 1978, pp. 19-27. |

[19] | H. B. Keller and D. S. Choen, “Some Positone Problems Suggested by Nonlinear Heat Generation,” Journal of Mathematics and Mechanics, Vol. 16, 1967, pp. 1361-1376. |

[20] | U. M. Assher, R. M. Matthij and R. D. Russell, “Numerical Solution of Boundary Value Problems for Ordinary Differential Equations,” Society for Industrial and Applied Mathematics, Philadelphia, 1995. |

[21] | G. Moore and A. Spence, “The Calculation of Turning Points of Nonlinear Equations,” SIAM Journal on Numerical Analysis, Vol. 17, No. 4, 1980, pp. 567-576. doi:10.1137/0717048 |

[22] | E. Deeba, S. A. Khuri and S. Xie, “An Algorithm for Solving Boundary Value Problems,” Journal of Computational Physics, Vol. 159, No. 2, 2000, pp. 125-138. doi:10.1006/jcph.2000.6452 |

[23] | J. B. Rosen, “Approximate Solution and Error Bounds for Quasilinear Elliptic Boundary Value Problems,” SIAM Journal on Numerical Analysis, Vol. 7, No. 1, 1970, pp. 80-103. doi:10.1137/0707004 |

[24] | H. Davis, “Introduction to Nonlinear Differential and Integral Equations,” Dover, New York, 1962. |

[25] | G. Bratu, “Sur Certaines Equations Integrals Non Lineares,” Comptes Rendus, Vol. 150, 1910, pp. 896-899. |

[26] | D. A. Frank-Kamenetskii, “Diffusion and Heat Transfer in Chemical Kinetics,” Plenum Press, New York, 1969. |

[27] | E. S. Weibel, “Confident of a Plasma Column by Radiation Pressure,” In R. K. M. Landshoff, Ed., The Plasma in a Magnetic Field, Stanford University Press, Stanford, 1958, pp. 60-76. |

[28] | S. M. Roberts and J. Shipman, “On the Closed Form Solution of Troesch’s Problem,” Journal of Computational Physics, Vol. 21, No. 3, 1976, pp. 291-304. doi:10.1016/0021-9991(76)90026-7 |

[29] | B. A. Troesch, “A Simple Approach to a Sensitive TwoPoint Boundary Value Problem,” Journal of Computational Physics, Vol. 21, No. 3, 1976, pp. 279-290. doi:10.1016/0021-9991(76)90025-5 |

[30] | S. A. Khuri, “A Numerical Algorithm for Solving Troesch’s Problem,” International Journal of Computer Mathematics, Vol. 80, No. 4, 2003, pp. 493-498. doi:10.1080/0020716022000009228 |

[31] | X. Feng, L. Mei and G. He, “An Efficient Algorithm for Solving Troesch’s Problem,” Applied Mathematics and Computation, Vol. 189, No. 1, 2007, pp. 500-507. doi:10.1016/j.amc.2006.11.161 |

[32] | V. Hlavacek, M. Marek and M. Kubicek, “Modelling of Chemical Reactors, X: Multiple Solutions of Enthalpy and Mass Balance for a Catalytic Reaction within a Porous Catalyst Particle,” Chemical Engineering Science, Vol. 23, No. 9, 1968, pp. 1083-1097. doi:10.1016/0009-2509(68)87093-9 |

[33] | J. H. He, “Homotopy Perturbation Technique,” Computer Methods in Applied Mechanics and Engineering, Vol. 178, No. 3-4, 1999, pp. 257-262. doi:10.1016/S0045-7825(99)00018-3 |

[34] | J. H. He, “Homotopy Perturbation Method: A New Nonlinear Analytical Technique,” Applied Mathematics and Computation, Vol. 135, No. 1, 2003, pp. 73-79. doi:10.1016/S0096-3003(01)00312-5 |

[35] | J. H. He, “A Simple Perturbation Approach to Blasius Equation,” Applied Mathematics and Computation, Vol. 140, No. 2-3, 2003, pp. 217-222. doi:10.1016/S0096-3003(02)00189-3 |

[36] | P. D. Ariel, “Alternative Approaches to Construction of Homotopy-Perturbation Algorithms,” Nonlinear Science Letters A, Vol. 1, 2010, pp. 43-52. |

[37] | V. Ananthaswmy, A. Eswari and L. Rajendran, “Analytical Solution of System of Nonlinear Reaction-Diffusion Equations in a Thin Membrane: Homotopy-Perturbation Approach,” Journal of Physical Chmistry, Vol. 5, No. 2, 2010. |

[38] | S. Loghambal and L. Rajendran, “Mathematical Modeling of Diffusion and Kinetics of Amperometric Immobilized Enzyme Electrodes,” Electrochim Acta, Vol. 55, No. 18, 2010, pp. 5230-5238. doi:10.1016/j.electacta.2010.04.050 |

[39] | A. Meena and L. Rajendran, “Mathematical Modeling of Amperometric and Potentiometric Biosensors and System of Non-Linear Equations, Homotopy-Perturbation Approach,” Journal of Electroanalytical Chemistry, Vol. 644, No. 1, 2010, pp. 50-59. doi:10.1016/j.jelechem.2010.03.027 |

[40] | S. Anitha, A. Subbiah, S. Subramaniam and L. Rajendran, “Analytical Solution of Amperometric Enzymatic Reactions Based on Homotopy-Perturbation Method,” Electrochimica Acta, Vol. 56, No. 9, 2011, pp. 3345-3352. doi:10.1016/j.electacta.2011.01.014 |

[41] | V. Ananthaswamy and L. Rajendran, “Analytical Solution of Two-Point Non Linear Boundary Value Problems in a Porous Catalyst Particles,” International Journal of Mathematical Archive, Vol. 3, No. 3, 2012, pp. 810-821. |

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