摘要:

In this paper we give a complete classification of all Painlevé‐type hyperbolic partial differential equations (PDE's) over the complex domain of the formuxy=F(x,y,u,ux,uywhereFis rational inu,ux, anduy, and locally analytic inxandy. We find exactly 22 equivalence classes of equations (under coordinate changes and Mobius transformations inu), which we denote HS‐I, HS‐II,…, HS‐XXII. A canonical representative of each class is presented and solved by transforming it either to a well‐known soliton equation (sine‐Gordon, Bullough‐Dodd‐Mikhailov) or to a linear equation by means of a Bäcklund correspondence or simpler change of variables. (The parabolic case, in which 10 more canonical equations are obtained, and semilinear PDE's in three or more variables are treated in the accompanying paper II.) The proof that the list is complete involves investigating four sets of necessary conditions in turn, each of which has essentially new features peculiar to the PDE context, as well as familiar features analogous to the corresponding conditions for ordinary differential equations (ODE's) as discussed in the classical literature by Painleve, Gambier, Ince, Bureau, and others. In a setting sufficiently general to embrace ODE's, hyperbolic and parabolic PDE's, and higher dimensional semilinear PDE's, we classify 76 types of O/PDE's, denoted DE‐1,…, according to their Bureau symbols and resonance data, which satisfy those necessary conditions for the Painlevé property common to each of these four Painlev

ISSN:0022-2526    

DOI:10.1002/sapm19938911

年代:1993

数据来源: WILEY

摘要:

The method of decaying residual solution is applied to obtain an approximateinteriorsolution for the torsion of slender prismatic elastic bodies under different end conditions. The approximate solution is generally accurate up to terms that are exponentially small in the length‐to‐cross‐sectional‐width ratio. For stress end conditions, the result is identical to the classical Saint‐Venant torsion solution. Similar types of simple solutions, not known previously, are obtained for different types of mixed end conditions. For displacement conditions at both ends, the corresponding Saint‐Venant type result requires an accurate solution of a canonical problem for a semi‐infinite prismatic body that is to be obtained once and for all. The solution of the canonical problem is elementary for a circular cross section. The approximate interior solution in that case is identical to the known exact inte

ISSN:0022-2526    

DOI:10.1002/sapm199389163

年代:1993

数据来源: WILEY