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Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations. (English) Zbl 0951.41006

The paper develops the spectral method for numerical solving the differential equations with singular solutions. The singularities may be not symmetric, and the main idea is to fit singular solutions by Jacobi polynomials, to compare numerical solutions with some unusual orthogonal projections of exact solutions, and to measure the errors in weighted Hilbert spaces \(L^2_\chi((-1,1))\) of functions integrated with square over \(x\in(-1,1)\) with the weight \(\chi(x)=(1-x)^\alpha(1+x)^\beta\).
The author gives several weighted inverse inequalities and considers the Jacobi approximations in the above weighted Hilbert spaces. Such approximations are just the Fourier expansion of the unknown function over Jacobi polynomials. As the applications, in the final part of the paper singular solutions for the second order linear ODE and for evolution semilinear second order PDE are constructed, and the convergence rate of the Jacobi approximations to the exact solutions is estimated.

MSC:

41A10 Approximation by polynomials
46C15 Characterizations of Hilbert spaces
65L20 Stability and convergence of numerical methods for ordinary differential equations
41A25 Rate of convergence, degree of approximation
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
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