Factorization of the Green's operator in the Dirichlet problem for (-1)SUPm/SUP(d/d t)SUP2m/SUP

摘要

In this article we propose a method for solving the Dirichletboundary value problem (??1)mu(2m) = f, u(k)( 1) = 0, k = 0; : : : ;m??1,which is based on the factorization of the Green’s operator, G2m =(??1)mJm1ProjmJm : L2(I) ! Hm0 (I) H2m(I); I = [??1; 1]. Here Jmis a Volterra operator of m-fold integration ànd1Projm operator oforthogonal projection in L2(I). The polynomialseP[2m]2m+N = Jm1ProjmP[m]m+Nform the basis of the Sobolev space Hm0 (I) H2m(I), where P[m]m+N(t) =JmPN(t) =(t ?? 1)mm!Cmm+NP(m;??m)N (t), PN are Legendre polynomials andP(m;??m)N non–classical Jacobi polynomials. The study of polynomialsP[m]m+N occupies the most part of this work including the problem ofexpanding P[m]m+N in Legendre polynomials. The formula for calculatingthe connection coefficients is obtained.