Convergence of the polynomial projection method for the solution of ill-posed integrodifferential equations
The Cauchy problem \(x^{(j)}(-1)=0\), \(j\in\{0,...,m-1\}\), for the following integro-differential equation is considered: \((Gx)(t)+(Bx)(t)+(Hx)(t)=y(t)\), \(t\in [-1,1]\), where \((Gx)(t)=x^{(m)}(t)\), \((Bx)(t)=\sum_{k=1}^m g_k(t)x^{(m-k)}(t)\) and \((Hx)(t)=\sum_{j=0}^p\int_{-1}^1h_j(t,s)x^{(j)}(s)ds\), \(m<p\), and the functions \(y, g_k, h_j\) are known. This problem is ill-posed in standard (for differential equations) pairs of the spaces \((X,Y)\). The author chooses for \(X\) and \(Y\) appropriate Sobolev spaces with special weight functions; in this way, the Cauchy problem becomes well-posed in the sense of Hadamard. As a consequence, a method which uses projections onto subspaces of algebraic polynomials is shown to converge and convergence rates are obtained. In particular, the Galerkin method and the collocation method for the Cauchy problem are discussed.
scientific article; zbMATH DE number 5321001