A New Analytical Approach for Solving Nonlinear Boundary Value Problems in Finite Domains

ABSTRACT

Based on the homotopy analysis method (HAM), we propose an analytical approach for solving the following type of nonlinear boundary value problems in finite domain. In framework of HAM a convenient way to adjust and control the convergence region and rate of convergence of the obtained series solutions, by defining the so-called control parameter h , is provided. This paper aims to propose an efficient way of finding the proper values of h.Such values of parameter can be determined at the any order of approximations of HAM series solutions by solving of a nonlinear polynomial equation. Some examples of nonlinear initial value problems in finite domain are used to illustrate the validity of the proposed approach. Numerical results confirm that obtained series solutions agree very well with the exact solutions.

Based on the homotopy analysis method (HAM), we propose an analytical approach for solving the following type of nonlinear boundary value problems in finite domain. In framework of HAM a convenient way to adjust and control the convergence region and rate of convergence of the obtained series solutions, by defining the so-called control parameter h , is provided. This paper aims to propose an efficient way of finding the proper values of h.Such values of parameter can be determined at the any order of approximations of HAM series solutions by solving of a nonlinear polynomial equation. Some examples of nonlinear initial value problems in finite domain are used to illustrate the validity of the proposed approach. Numerical results confirm that obtained series solutions agree very well with the exact solutions.

Cite this paper

nullJ. Biazar and B. Ghanbari, "A New Analytical Approach for Solving Nonlinear Boundary Value Problems in Finite Domains,"*Applied Mathematics*, Vol. 2 No. 8, 2011, pp. 987-992. doi: 10.4236/am.2011.28136.

nullJ. Biazar and B. Ghanbari, "A New Analytical Approach for Solving Nonlinear Boundary Value Problems in Finite Domains,"

References

[1] S. Liang and D. J. Jeffrey, “An Analytical Approach for Solving Nonlinear Boundary Value Problems in Finite Domains,” Numerical Algorthms, Vol. 56, No. 1, 2010, pp. 93-106.

[2] H. N. Caglar, S. H. Caglar and E. E. Twizell, “The Numerical Solution of Fifth Order Boundary-Value Problems with Sixth Degree B-Spline Functions,” Applied Mathematics Letters, Vol. 12, No. 5, 1999, pp. 25-30. doi:10.1016/S0893-9659(99)00052-X

[3] A. R. Davies, A. Karageoghis and T. N. Phillips, “Spectral Glarkien Methods for the Primary Two-Point Boundary-Value Problems in Modeling Viscelastic Flows,” International Journal for Numerical Methods in Engineering, Vol. 26, No. 3, 1988, pp. 647-662. doi:10.1002/nme.1620260309

[4] D. J. Fyfe, “Linear Dependence Relations Connecting Equal Interval Nth Degree Splines and Their Derivatives,” Journal of the Institute of Mathematics and Its Applications, Vol. 7, No. 3, 1971, pp. 398-406. doi:10.1093/imamat/7.3.398

[5] S. N. Ha, “A Nonlinear Shooting Method for Two-Point Boundary Value Problems,” Computers & Mathematics with Applications, Vol. 42, No. ER10-11, 2001, pp. 1411-1420. doi:10.1016/S0898-1221(01)00250-4

[6] A. G. Deacon and S. Osher. “Finite-Element Method for a Boundary-Value Problem of Mixed Type,” SIAM Journal on Numerical Analysis, Vol. 16, No. 5, 1979, pp. 756-778. doi:10.1137/0716056

[7] M. El-Gamel, S. H. Behiry and H. Hashish, “Numerical Method for the Solution of Special Nonlinear Fourth-Order Boundary Value Problems,” Applied Mathematics and Computation, Vol. 145, No. 2-3, 2003, pp. 717-734. doi:10.1016/S0096-3003(03)00269-8

[8] A. A. Salama and A. A. Mansour, “Fourth-Order Finite- Difference Method for third-Orderboundary-Value Problems,” Numerical Heat Transfer, Part B, Vol. 47, No. 4, 2005, pp. 383-401. doi:10.1080/10407790590907903

[9] A. M. Wazwaz, “The Numerical Solution of Fifth-Order Boundary-Value Problems by Adomian Decomposition”, Journal of Computational and Applied Mathematics, Vol. 136, No. 1-2, 2001, pp. 259-270. doi:10.1016/S0377-0427(00)00618-X

[10] M. Aslam Noor and S. T. Mohyud-Din, “Variational Iteration Technique for Solving Fifth-Order Boundary Value Problems,” 2006 (Preprint).

[11] C. Chun and R. Sakthivel, “Homotopy Perturbation Technique for Solving Two-Point Boundary Value Problems —Comparison with Other Methods,” Computer Physics Communications, Vol. 181, 2010, pp. 1021-1024. doi:10.1016/j.cpc.2010.02.007

[12] S. Liang and D. J. Jeffrey, “An Efficient Analytical Approach for Solving Fourth Order Boundary Value Problems,” Computer Physics Communications, Vol. 180, 2009, pp. 2034-2040. doi:10.1016/j.cpc.2009.06.006

[13] S. J. Liao, “On the Proposed Homotopy Analysis Technique for Nonlinear Problems and Its Applications,” Ph. D. Dissertation, Shanghai Jiao Tong University, Shanghai, 1992.

[14] S. J. Liao, “General Boundary Element Method for Non-linear Heat Transfer Problems Governed by Hyperbolic Heat Conduction Equation,” Computational Mechanics, Vol. 20, No. 5, 1997, pp. 397-406. doi:10.1007/s004660050260

[15] S. J. Liao, “Beyond Perturbation: Introduction to Homotopy Analysis Method,” Chapman & Hall/CRC Press, Boca Raton, 2003. doi:10.1201/9780203491164

[16] S. J. Liao, “Numerically Solving Nonlinear Problems by the Homotopy Analysis Method,” Computational Mechanics, Vol. 20, No. 6, 1997, pp. 530-540. doi:10.1007/s004660050273

[17] S. J. Liao, “Notes on the Homotopy Analysis Method: Some Definitions and Theorems,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 4, 2009, pp. 983-997. doi:10.1016/j.cnsns.2008.04.013

[18] A. Mehmood, S. Munawar and A. Ali, “Comments to: Homotopy Analysis Method for Solving the MHD Flow over a Non-Linear Stretching Sheet,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 6, 2009, pp. 2653-2663. doi:10.1016/j.cnsns.2009.12.039

[1] S. Liang and D. J. Jeffrey, “An Analytical Approach for Solving Nonlinear Boundary Value Problems in Finite Domains,” Numerical Algorthms, Vol. 56, No. 1, 2010, pp. 93-106.

[2] H. N. Caglar, S. H. Caglar and E. E. Twizell, “The Numerical Solution of Fifth Order Boundary-Value Problems with Sixth Degree B-Spline Functions,” Applied Mathematics Letters, Vol. 12, No. 5, 1999, pp. 25-30. doi:10.1016/S0893-9659(99)00052-X

[3] A. R. Davies, A. Karageoghis and T. N. Phillips, “Spectral Glarkien Methods for the Primary Two-Point Boundary-Value Problems in Modeling Viscelastic Flows,” International Journal for Numerical Methods in Engineering, Vol. 26, No. 3, 1988, pp. 647-662. doi:10.1002/nme.1620260309

[4] D. J. Fyfe, “Linear Dependence Relations Connecting Equal Interval Nth Degree Splines and Their Derivatives,” Journal of the Institute of Mathematics and Its Applications, Vol. 7, No. 3, 1971, pp. 398-406. doi:10.1093/imamat/7.3.398

[5] S. N. Ha, “A Nonlinear Shooting Method for Two-Point Boundary Value Problems,” Computers & Mathematics with Applications, Vol. 42, No. ER10-11, 2001, pp. 1411-1420. doi:10.1016/S0898-1221(01)00250-4

[6] A. G. Deacon and S. Osher. “Finite-Element Method for a Boundary-Value Problem of Mixed Type,” SIAM Journal on Numerical Analysis, Vol. 16, No. 5, 1979, pp. 756-778. doi:10.1137/0716056

[7] M. El-Gamel, S. H. Behiry and H. Hashish, “Numerical Method for the Solution of Special Nonlinear Fourth-Order Boundary Value Problems,” Applied Mathematics and Computation, Vol. 145, No. 2-3, 2003, pp. 717-734. doi:10.1016/S0096-3003(03)00269-8

[8] A. A. Salama and A. A. Mansour, “Fourth-Order Finite- Difference Method for third-Orderboundary-Value Problems,” Numerical Heat Transfer, Part B, Vol. 47, No. 4, 2005, pp. 383-401. doi:10.1080/10407790590907903

[9] A. M. Wazwaz, “The Numerical Solution of Fifth-Order Boundary-Value Problems by Adomian Decomposition”, Journal of Computational and Applied Mathematics, Vol. 136, No. 1-2, 2001, pp. 259-270. doi:10.1016/S0377-0427(00)00618-X

[10] M. Aslam Noor and S. T. Mohyud-Din, “Variational Iteration Technique for Solving Fifth-Order Boundary Value Problems,” 2006 (Preprint).

[11] C. Chun and R. Sakthivel, “Homotopy Perturbation Technique for Solving Two-Point Boundary Value Problems —Comparison with Other Methods,” Computer Physics Communications, Vol. 181, 2010, pp. 1021-1024. doi:10.1016/j.cpc.2010.02.007

[12] S. Liang and D. J. Jeffrey, “An Efficient Analytical Approach for Solving Fourth Order Boundary Value Problems,” Computer Physics Communications, Vol. 180, 2009, pp. 2034-2040. doi:10.1016/j.cpc.2009.06.006

[13] S. J. Liao, “On the Proposed Homotopy Analysis Technique for Nonlinear Problems and Its Applications,” Ph. D. Dissertation, Shanghai Jiao Tong University, Shanghai, 1992.

[14] S. J. Liao, “General Boundary Element Method for Non-linear Heat Transfer Problems Governed by Hyperbolic Heat Conduction Equation,” Computational Mechanics, Vol. 20, No. 5, 1997, pp. 397-406. doi:10.1007/s004660050260

[15] S. J. Liao, “Beyond Perturbation: Introduction to Homotopy Analysis Method,” Chapman & Hall/CRC Press, Boca Raton, 2003. doi:10.1201/9780203491164

[16] S. J. Liao, “Numerically Solving Nonlinear Problems by the Homotopy Analysis Method,” Computational Mechanics, Vol. 20, No. 6, 1997, pp. 530-540. doi:10.1007/s004660050273

[17] S. J. Liao, “Notes on the Homotopy Analysis Method: Some Definitions and Theorems,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 4, 2009, pp. 983-997. doi:10.1016/j.cnsns.2008.04.013

[18] A. Mehmood, S. Munawar and A. Ali, “Comments to: Homotopy Analysis Method for Solving the MHD Flow over a Non-Linear Stretching Sheet,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 6, 2009, pp. 2653-2663. doi:10.1016/j.cnsns.2009.12.039