M. Abell, J. Braselton's Differential Equations with Mathematica PDF

By M. Abell, J. Braselton

ISBN-10: 0120415399

ISBN-13: 9780120415397

Differential Equations with Mathematica thoroughly covers developing, numerical computing, and approximating ideas to boring and partial differential equations. The ebook serves as a hands-on advent to the subject material via a variety of examples that designate how one can resolve vital purposes utilizing Mathematica.

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Additional info for Differential Equations with Mathematica

Example text

Xj As we mentioned before, we claim that Φ (z)[w, w] > c > 0. Let us compute: Φ (z)[w, w] = w1 2 + w2 2 3z12 w12 + 3z22 w22 − 2 w1 w2 dx. − By neglecting the terms which tend to zero, we obtain, m Φ (z)[w, w] ∼ w1 2 + w2 2 −3 w12 Vi2 dx − 3 V 2 w22 dx. 1 to prove that: |∇w2 |2 + w22 − 3V 2 w22 ≥ c w2 2 −C . 10), we argue as in the onedimensional case. Claim: There exists R ∈ (ξ 1/4 , ξ 1/2 ) such that m [|∇wi |2 + wi2 ]dx < 2ξ −1/2 , i = 1, 2 j=1 R<|x−ξpj |

I, Comm. Math. Phys. 235 (2003), 427-466. [8] A. Ambrosetti, A. Malchiodi, S. Secchi, Multiplicity results for some nonlinear Schr¨ odinger equations with potentials, Arch. Rat. Mech. Anal. 159 (2001), 253-271. [9] M. Berti, Ph. Bolle, Variational construction of homoclinics and chaos in presence of a saddle-saddle equilibrium. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), 331–377 (1999). 42 [10] S. Cingolani, M. Nolasco, Multi-peak periodic semiclassical states for a class of nonlinear Schr¨ odinger equations, Proc.

159 (2001), 253-271. [9] M. Berti, Ph. Bolle, Variational construction of homoclinics and chaos in presence of a saddle-saddle equilibrium. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), 331–377 (1999). 42 [10] S. Cingolani, M. Nolasco, Multi-peak periodic semiclassical states for a class of nonlinear Schr¨ odinger equations, Proc. Royal Soc. A, 128 (1998), 1249-1260. M. Coxeter, Regular polytopes, Methuen & Co. , London, 1948. [12] A. Floer, A. Weinstein, Nonspreading wave packets for the cubic Schr¨ odinger equation with a bounded potential, J.