The Lie-group shooting method for solving the Bratu equation

Communications in Nonlinear Science and Numerical Simulation

Volumn 16, Issue 11, 2011, Pages 4238-4249

  Abbasbandy, S ^a   Hashemi, M S ^a,b   Liu, Chein Shan ^c  

^a IMAM KHOMEINI INTERNATIONAL UNIVERSITY   (Iran)

^b UNIVERSITY OF BONAB   (Iran)

^c NATIONAL TAIWAN UNIVERSITY   (Taiwan)

Author keywords

Bratu problem; Group preserving scheme; Lie group shooting method

Indexed keywords

BOUNDARY VALUE PROBLEMS; LIE GROUPS;

BRATU EQUATION; BRATU' PROBLEM; GROUP PRESERVING SCHEME; LIE-GROUP SHOOTING METHOD; MATCHINGS; NON LINEAR; SECOND-ORDER BOUNDARY VALUE PROBLEMS;

MATHEMATICAL TRANSFORMATIONS;

EID: 79957917942     PISSN: 10075704     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cnsns.2011.03.033     Document Type: Article

References (35)

1. Wazwaz A.M. Adomian decomposition method for a reliable treatment of the Bratu-type equations. Appl Math Comput 2005, 166:652-663.
2. Syam M.I., Hamdan A. An efficient method for solving Bratu equations. Appl Math Comput 2006, 176:704-713.
3. Buckmire R. Application of a Mickens finite-difference scheme to the cylindrical Bratu-Gelfand problem. Numer Methods Partial Diff Eq 2004, 20(3):327-337.
4. McGough J.S. Numerical continuation and the Gelfand problem. Appl Math Comput 1998, 89:225-239.
5. Mounim A.S., de Dormale B.M. From the fitting techniques to accurate schemes for the Liouville-Bratu-Gelfand problem. Numer Methods Partial Diff Eq 2006, 22(4):761-775.
6. Li S., Liao S.J. An analytic approach to solve multiple solutions of a strongly nonlinear problem. Appl Math Comput 2005, 169:854-865.
7. Liao S., Tan Y. A general approach to obtain series solutions of nonlinear differential equations. Stud Appl Math 2007, 119:297-354.
8. Boyd J.P. Chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one-dimensional Bratu equation. Appl Math Comput 2003, 143:189-200.
9. Aregbesola Y.A.S. Numerical solution of Bratu problem using the method of weighted residual, Electron. J South Afr Math Sci 2003, 3(1):1-7.
10. Jalilian R. Non-polynomial spline method for solving Bratu's problem. Comput Phys Commun 2010, 181:1868-1872.
11. Caglar H., Caglar N., Özer M. Antonios Valaristos, Antonios N. Anagnostopoulos, B-spline method for solving Bratu's problem. Int J Comput Math 2010, 87(8):1885-1891.
12. Khuri S.A. A new approach to Bratu's problem. Appl Math Comput 2004, 147:131-136.
13. Hassan I.H.A.H., Erturk V.S. Applying differential transformation method to the one-dimensional planar Bratu problem. Int J Contemp Math Sci 2007, 2:1493-1504.
14. Deeba E., Khuri S.A., Xie S. An algorithm for solving boundary value problems. J Comput Phys 2000, 159:125-138.
15. Batiha B. Numerical Solution of Bratu-type equations by the variational iteration method. Hacettepe J Math Stat 2010, 39(1):23-29.
16. Aksoy Y., Pakdemirli M. New perturbation-iteration solutions for Bratu-type equations. Comput Math Appl 2010, 59:2802-2808.
17. Caglar H., Caglar N., Özer M., Valaristos A., Miliou A.N., Anagnostopoulos A.N. Dynamics of the solution of Bratu's equation. Nonlinear Anal 2009, 71:672-678.
18. Frank-Kamenetski D.A. Diffusion and Heat Exchange in Chemical Kinetics 1955, Princeton University Press, Princeton, NJ.
19. Liu C.-S. Cone of non-linear dynamical system and group preserving schemes. Int J Non-Linear Mech 2001, 36:1047-1068.
20. Liu C.-S. Efficient shooting methods for the second order ordinary differential equations. CMES: Comput Model Eng Sci 2006, 15:69-86.
21. Liu C.-S. The Lie-group shooting method for singularly perturbed two-point boundary value problems. CMES: Comput Model Eng Sci 2006, 15:179-196.
22. Liu C.-S. The Lie-group shooting method for nonlinear two-point boundary value problems exhibiting multiple solutions. CMES: Comput Model Eng Sci 2006, 13:149-163.
23. Chang C.W., Liu C.-S., Chang J.R. The Lie-group shooting method for quasi-boundary regularization of backward heat conduction problems. ICCES Online J 2007, 3:69-79.
24. Chang J.R., Liu C.-S., Chang C.W. A new shooting method for quasi-boundary regularization of backward heat conduction problems. Int J Heat Mass Transfer 2007, 50:2325-2332.
25. Liu C.-S. An LGEM to identify time-dependent heat conductivity function by an extra measurement of temperature gradient. CMC: Comput Mater Continua 2008, 7:81-95.
26. Liu C.-S. An LGSM to identify nonhomogeneous heat conductivity functions by an extra measurement of temperature. Int J Heat Mass Transfer 2008, 51:2603-2613.
27. Liu C.-S. Solving an inverse Sturm-Liouville problem by a Lie-group method. Boundary Value Prob 2008, Article ID 749865.
28. Liu C.-S. A two-stage LGSM for three-point BVPs of second-order ODEs. Boundary Value Prob 2008, Article ID 963753.
29. Liu C.-S. A two-stage LGSM to identify time-dependent heat source through an internal measurement of temperature. Int J Heat Mass Transfer 2009, 52:1635-1642.
30. Yeih W., Liu C.-S. A three-point BVP of time-dependent inverse heat source problems solving by a TSLGSM. CMES: Comput Model Eng Sci 2009, 46:107-127.
31. Liu C.-S. A two-stage Lie-group shooting method (TSLGSM) to identify time-dependent thermal diffusivity. Int J Heat Mass Transfer 2010, 53:4876-4884.
32. Chang C.-W., Liu C.-S. A new algorithm for direct and backward problems of heat conduction equation. Int J Heat Mass Transfer 2010, 53:5552-5569.
33. Liu C.-S. One-step GPS for the estimation of temperature-dependent thermal conductivity. Int J Heat Mass Transfer 2006, 49:3084-3093.
34. Liu C.-S. An efficient backward group preserving scheme for the backward in time Burgers equation. CMES: Comput Model Eng Sci 2006, 12:55-65.
35. Lee H.C., Chen C.K., Hung C.I. A modified group-preserving scheme for solving the initial value problems of stiff ordinary differential equations. Appl Math Comput 2002, 133:445-459.