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Future-sequential regularization methods for ill-posed Volterra equations. Applications to the inverse heat conduction problem. (English) Zbl 0851.65094

To solve the equation of the first kind \[ \int^t_0 k(t- s) u(s) ds= f(t),\quad 0\leq t\leq 1, \] the author introduces a regularized equation of the second kind \[ \alpha(\Delta_r) u(t)+ \int^t_0 \widetilde k(t- s; \Delta_r) u(s) ds= F(t; \Delta_r). \] Here \(\widetilde k\) and \(F\) are functions using “future” values of \(k\) and \(f\), respectively, e.g. \[ \widetilde k(t)= \int^{\Delta_r}_0 k(t+ \rho) d\eta_{\Delta_r}(\rho), \] where \(\eta_{\Delta_r}\) is a suitable measure on the Borel sets of \(\mathbb{R}\) and \(\Delta_r\) is the length of the “future” interval. Regularizing properties of the method are discussed.
Reviewer: G.Vainikko (Espoo)

MSC:

65R20 Numerical methods for integral equations
65R30 Numerical methods for ill-posed problems for integral equations
35K05 Heat equation
35R30 Inverse problems for PDEs
80A23 Inverse problems in thermodynamics and heat transfer
45D05 Volterra integral equations
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