Newton's method based on bifurcation for solving multiple solutions of nonlinear elliptic equations |
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Authors: | Zhu HaiLong Li ZhaoXiang Yang ZhongHua |
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Affiliation: | 1. Department of Mathematics, Anhui University of Finance and Economics, , Bengbu 233030, China;2. Institute of Applied Mathematics, Anhui University of Finance and Economics, , Bengbu 233030, China;3. Department of Mathematics, Shanghai Normal University, , Shanghai 200234, China |
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Abstract: | 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 . 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 © 2013 John Wiley & Sons, Ltd. |
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Keywords: | Newton's method symmetry‐breaking bifurcation pseudo‐arclength continuation concave and convex nonlinearities multiple solutions |
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