×

An analytic approach to solve multiple solutions of a strongly nonlinear problem. (English) Zbl 1151.35354

Summary: Based on a new kind of analytic method, namely the homotopy analysis method, an analytic approach to solve multiple solutions of strongly nonlinear problems is described by using Gelfand equation as an example. Its validity is verified by comparing the approximation series with the known exact solution. And different from perturbation techniques, this approach is independent upon any small/large perturbation quantities. So, the basic ideas of this approach can be employed to search for multiple solutions of strongly nonlinear problems in science and engineering.

MSC:

35J60 Nonlinear elliptic equations
34B15 Nonlinear boundary value problems for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Gelfand, I. M., Some problems in the theory of quasi-linear equations, Am. Math. Soc. Transl. Ser., 2, 29, 295-381 (1963) · Zbl 0127.04901
[2] Jacobsen, J., The Liouville-Bratu-Gelfand problem for radial operators, J. Diff. Equat., 184, 283-298 (2002) · Zbl 1015.34013
[3] Frank-Kamenetskii, D. A., Diffusion and Heat Transfer in Chemical Kinetics (1955), Princeton University Press: Princeton University Press Princeton,NJ
[4] Korman, P., An accurate computation of the global solution curve for the Gelfand problem through a two point approximation, Appl. Math. Computat., 139, 363-369 (2003) · Zbl 1030.65105
[5] McGough, J. S., Numerical continuation and the gelfand problem, Appl. Math. Computat., 89, 225-239 (1998) · Zbl 0908.65094
[6] Plum, M.; Wieners, C., New solution of the Gelfand problem, J. Math. Anal. Appl., 269, 588-606 (2002) · Zbl 1003.65131
[7] Plum, M., Computer-assisted enclosure methods for elliptic differential equations, Linear Algebra Appl., 324, 147-187 (2001) · Zbl 0973.65100
[8] Balakrishnan, E.; Swift, A.; Wake, G. C., Critical values for some non-class a geometries in thermal ignition theory, Math. Comput. Model., 24, 8, 1-10 (1996) · Zbl 0880.35123
[9] Joseph, D. D.; Lundgren, T. S., Quasilinear dirichlet problem driven by positive source, Arch. Rat. Mech. Anal., 49, 241-269 (1973) · Zbl 0266.34021
[10] Liouville, J., Sur i’équation aux différence partielles \(\frac{d^2 \log \lambda}{d u d v} \pm \lambda 2 a^2 = 0\), J. Math Pures Appl., 18, 71-72 (1853)
[11] Liao, S. J., Beyond Perturbation: Introduction to Homotopy Analysis Method (2003), Chapman & Hall/ CRC Press: Chapman & Hall/ CRC Press Boca Raton
[12] Liao, S. J., On the homotopy analysis method for nonlinear problems, Appl. Math. Computat., 147, 499-513 (2004) · Zbl 1086.35005
[13] A.M. Lyapunov, (1892). General Problem on Stability of Motion, London, Taylor & Francis, 1992 (English translation).; A.M. Lyapunov, (1892). General Problem on Stability of Motion, London, Taylor & Francis, 1992 (English translation).
[14] Karmishin, A. V.; Zhukov, A. T.; Kolosov, V. G., Methods of Dynamics Calculation and Testing for Thin-walled Structures (1990), Mashinostroyenie: Mashinostroyenie Moscow, (in Russian)
[15] Adomian, G., Nonlinear stochastic differential equations, J. Math. Anal. Appl., 55, 441-452 (1976) · Zbl 0351.60053
[16] Liao, S. J., An analytic approximate approach for free oscillations of self-excited systems, Int. J. Non-Linear Mech., 39, 2, 271-280 (2004) · Zbl 1348.34071
[17] Liao, S. J.; Cheung, K. F., Homotopy analysis of nonlinear progressive waves in deep water, J. Eng. Math., 45, 2, 105-116 (2003) · Zbl 1112.76316
[18] Liao, S. J., On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet, J. Fluid Mech., 488, 189-212 (2003) · Zbl 1063.76671
[19] Liao, S. J., An explicit analytic solution to the Thomas-Fermi equation, Appl. Math. Computat., 144, 495-506 (2003) · Zbl 1034.34005
[20] Wang, C., On the explicit analytic solution of Cheng-Chang equation, Int. J. Heat Mass Transfer, 46, 10, 1855-1860 (2003) · Zbl 1029.76050
[21] Ayub, M.; Rasheed, A.; Hayat, T., Exact flow of a third grade fluid past a porous plate using homotopy analysis method, Int. J. Eng. Sci., 41, 2091-2103 (2003) · Zbl 1211.76076
[22] Hayat, T.; Khan, M.; Ayub, M., On the explicit analytic solutions of an Oldroyd 6-constant fluid, Int. J. Eng. Sci., 42, 123-135 (2004) · Zbl 1211.76009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.