A Contractor Iteration Method Using Eigenpairs for Positive Solutions of Nonlinear Elliptic Equation
By means of Contractor Iteration Method, we solve and visualize the Lane-Emden(-Fowler) equation Δu + up = 0, in Ω, u = 0, on ∂Ω. It is shown that the present method converges quadratically as Newton’s method and the computation of Contractor Iteration Method is cheaper than the Newton’s method.
[1] Fowler R. Further studies of Emden-s and similar differential equations,
Quart.J.Math. 2: 259-288 (1931).
[2] Chandrasekhar S. An Introduction to the Study of Stellar Structure,
University of Chicago Press. (1939).
[3] Kajikiya R. Non-radial solutions with group invariance for the sublinear
Emden-Fowler equation, Nonlinear Analysis. 47: 3759-3770 (2001).
[4] Flores I. A resonance phenomenon for ground states of an elliptic
equation of Emden-Fowler type, J.Differential Equations. 198: 1-15
(2004).
[5] Pao C V. Block monotone iterative methods for numerical solutions of
nonlinear elliptic equations, Numer Math. 72: 239-262 (1995).
[6] Deng Y, Chen G, Ni W M, et al. Boundary element monotone iteration
scheme for semilinear elliptic partial differential equations, Math Comput.
65: 943-982 (1996).
[7] Choi Y S, McKenna P J. A mountain pass method for the numerical
solutions of semilinear elliptic problems, Nonlinear Anal. 20: 417-437
(1993).
[8] Ding Z H, Costa D, Chen G. A high-linking algorithm for sign-changing
solutions of semilinear elliptic equations, Nonlinear Anal. 38: 151-172
(1999).
[9] Li Y, Zhou J X. A minimax method for finding multiple critical points
and its applications to semilinear PDEs, SIAM J Sci Comput. 23: 840-865
(2002).
[10] Yao X D, Zhou J X. A minimax method for finding multiple critical
points in Banach spaces and its application to quasi-linear elliptic PDE,
SIAM J Sci Comput. 26: 1796-1809 (2005).
[11] Chen C M, Xie Z Q. Search-extension method for multiple solutions of
nonlinear problem, Comp Math Appl. 47: 327-343 (2004).
[12] Yang Z H, Li Z X, Zhu H L. Bifurcation method for solving multiple
positive solutions to Henon equation, Science in China Series A: Mathematics.
51: 2330-2342 (2008).
[13] Altman M. Contractors and Contractor Directions, Theory and Applications.
Marcel Dekker, New York, 1977.
[14] Takanobu G, Masaaki S. A contractor iteration method of solving
nonlinear equations, Engineering Analysis with Boundary Element. 18:
239-244 (1996).
[15] Scheurle J. Newton iterations without inverting the derivative, Math.
Meth. Appl. Sci. 514-529 (1979).
[16] Zhu H L, Li Z X. A Numerical Algorithm for Positive Solutions of
Concave and Convex Elliptic Equation on R2, International Journal of
Computational and Mathematical Sciences, 4: 173-176 (2010).
[17] Yang Z H. Non-linear Bifurcation: Theory and Computation (in Chinese).
Beijing: Science Press, 2007.
[18] Henon M. Numerical experiments on the stability of spherical stellar
systems. Astronom Astrophys Lib, 24: 229-238 (1973).
[19] Lieb E, Yau H T. The Chandrasekhar theory of stellar collapse as the
limit of quantum mechanics, Commun.Math.Phys. 112: 147-174 (1987).
[20] Philip K and Shi J P. On Lane-Emden type systems, discrete and
continuous dynamical systems. 510-517 (2005).
[1] Fowler R. Further studies of Emden-s and similar differential equations,
Quart.J.Math. 2: 259-288 (1931).
[2] Chandrasekhar S. An Introduction to the Study of Stellar Structure,
University of Chicago Press. (1939).
[3] Kajikiya R. Non-radial solutions with group invariance for the sublinear
Emden-Fowler equation, Nonlinear Analysis. 47: 3759-3770 (2001).
[4] Flores I. A resonance phenomenon for ground states of an elliptic
equation of Emden-Fowler type, J.Differential Equations. 198: 1-15
(2004).
[5] Pao C V. Block monotone iterative methods for numerical solutions of
nonlinear elliptic equations, Numer Math. 72: 239-262 (1995).
[6] Deng Y, Chen G, Ni W M, et al. Boundary element monotone iteration
scheme for semilinear elliptic partial differential equations, Math Comput.
65: 943-982 (1996).
[7] Choi Y S, McKenna P J. A mountain pass method for the numerical
solutions of semilinear elliptic problems, Nonlinear Anal. 20: 417-437
(1993).
[8] Ding Z H, Costa D, Chen G. A high-linking algorithm for sign-changing
solutions of semilinear elliptic equations, Nonlinear Anal. 38: 151-172
(1999).
[9] Li Y, Zhou J X. A minimax method for finding multiple critical points
and its applications to semilinear PDEs, SIAM J Sci Comput. 23: 840-865
(2002).
[10] Yao X D, Zhou J X. A minimax method for finding multiple critical
points in Banach spaces and its application to quasi-linear elliptic PDE,
SIAM J Sci Comput. 26: 1796-1809 (2005).
[11] Chen C M, Xie Z Q. Search-extension method for multiple solutions of
nonlinear problem, Comp Math Appl. 47: 327-343 (2004).
[12] Yang Z H, Li Z X, Zhu H L. Bifurcation method for solving multiple
positive solutions to Henon equation, Science in China Series A: Mathematics.
51: 2330-2342 (2008).
[13] Altman M. Contractors and Contractor Directions, Theory and Applications.
Marcel Dekker, New York, 1977.
[14] Takanobu G, Masaaki S. A contractor iteration method of solving
nonlinear equations, Engineering Analysis with Boundary Element. 18:
239-244 (1996).
[15] Scheurle J. Newton iterations without inverting the derivative, Math.
Meth. Appl. Sci. 514-529 (1979).
[16] Zhu H L, Li Z X. A Numerical Algorithm for Positive Solutions of
Concave and Convex Elliptic Equation on R2, International Journal of
Computational and Mathematical Sciences, 4: 173-176 (2010).
[17] Yang Z H. Non-linear Bifurcation: Theory and Computation (in Chinese).
Beijing: Science Press, 2007.
[18] Henon M. Numerical experiments on the stability of spherical stellar
systems. Astronom Astrophys Lib, 24: 229-238 (1973).
[19] Lieb E, Yau H T. The Chandrasekhar theory of stellar collapse as the
limit of quantum mechanics, Commun.Math.Phys. 112: 147-174 (1987).
[20] Philip K and Shi J P. On Lane-Emden type systems, discrete and
continuous dynamical systems. 510-517 (2005).
@article{"International Journal of Engineering, Mathematical and Physical Sciences:61073", author = "Hailong Zhu and Zhaoxiang Li and Kejun Zhuang", title = "A Contractor Iteration Method Using Eigenpairs for Positive Solutions of Nonlinear Elliptic Equation", abstract = "By means of Contractor Iteration Method, we solve and visualize the Lane-Emden(-Fowler) equation Δu + up = 0, in Ω, u = 0, on ∂Ω. It is shown that the present method converges quadratically as Newton’s method and the computation of Contractor Iteration Method is cheaper than the Newton’s method.
", keywords = "Positive solutions, newton's method, contractor iteration method, Eigenpairs.", volume = "4", number = "7", pages = "991-4", }
