## 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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