Newton's method based on bifurcation for solving multiple solutions of nonlinear elliptic equations

被引:3
作者
Zhu HaiLong [1 ,3 ]
Li ZhaoXiang [2 ]
Yang ZhongHua [2 ]
机构
[1] Anhui Univ Finance & Econ, Dept Math, Bengbu 233030, Peoples R China
[2] Shanghai Normal Univ, Dept Math, Shanghai 200234, Peoples R China
[3] Anhui Univ Finance & Econ, Inst Appl Math, Bengbu 233030, Peoples R China
关键词
Newton's method; symmetry-breaking bifurcation; pseudo-arclength continuation; concave and convex nonlinearities; multiple solutions; SEARCH-EXTENSION METHOD; POSITIVE SOLUTIONS; CONVEX NONLINEARITIES; CONCAVE;
D O I
10.1002/mma.2747
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
On the basis of bifurcation theory, we use Newton's method to compute and visualize the multiple solutions to a series of typical semilinear elliptic boundary value problems with a homogeneous Dirichlet boundary condition in R2. We present three algorithms on the basis of the bifurcation method to solving these multiple solutions. We will compute and visualize the profiles of such multiple solutions, thereby exhibiting the geometrical effects of the domains on the multiplicity. The domains include the square, disk, symmetric or nonsymmetric annuli and dumbbell. The nonlinear partial differential equations include the Lane-Emden equation, concave-convex nonlinearities, Henon equation, and generalized Lane-Emden system. Copyright (c) 2013 John Wiley & Sons, Ltd.
引用
收藏
页码:2208 / 2223
页数:16
相关论文
共 50 条
[21]   Bifurcation method for solving multiple positive solutions to Henon equation on the unit cube [J].
Li, Zhao-xiang ;
Zhu, Hai-long ;
Yang, Zhong-hua .
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION, 2011, 16 (09) :3673-3683
[22]   On nodal solutions for nonlinear elliptic equations with a nonhomogeneous differential operator [J].
He, Tieshan ;
Yan, Hongming ;
Sun, Zhaohong ;
Zhang, Meng .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2015, 118 :41-50
[23]   Newton's method for nonlinear stochastic wave equations [J].
Leszczynski, Henryk ;
Wrzosek, Monika .
FORUM MATHEMATICUM, 2020, 32 (03) :595-605
[24]   POSITIVE SOLUTIONS AND BIFURCATION PHENOMENA FOR NONLINEAR ELLIPTIC EQUATIONS OF LOGISTIC TYPE: THE SUPERDIFFUSIVE CASE [J].
Filippakis, Michael E. ;
O'Regan, Donal ;
Papageorgiou, Nikolaos S. .
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, 2010, 9 (06) :1507-1527
[25]   Abundance of entire solutions to nonlinear elliptic equations by the variational method [J].
Lerman, L. M. ;
Naryshkin, P. E. ;
Nazarov, A., I .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2020, 190
[26]   A Hybrid Approach for Solving Systems of Nonlinear Equations Using Harris Hawks Optimization and Newton's Method [J].
Sihwail, Rami ;
Solaiman, Obadah Said ;
Omar, Khairuddin ;
Ariffin, Khairul Akram Zainol ;
Alswaitti, Mohammed ;
Hashim, Ishak .
IEEE ACCESS, 2021, 9 :95791-95807
[27]   A composite third order Newton-Steffensen method for solving nonlinear equations [J].
Sharma, JR .
APPLIED MATHEMATICS AND COMPUTATION, 2005, 169 (01) :242-246
[28]   A Variant of Newton Method with Eighth-order Convergence for Solving Nonlinear Equations [J].
Fang, Liang .
MECHATRONICS AND INTELLIGENT MATERIALS II, PTS 1-6, 2012, 490-495 :51-55
[29]   A remark on the existence of multiple solutions to a multiparameter nonlinear elliptic system [J].
Afrouzi, G. A. ;
Rasouli, S. H. .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2009, 71 (1-2) :445-455
[30]   Existence of multiple solutions of higher-order nonlinear elliptic equations [J].
V. F. Lubyshev .
Mathematical Notes, 2011, 89 :255-265