Let H be a Hilbert space with norm • , A : D(A) ⊂ H → H a positive definite, self-adjoint operator with compact inverse on H, and T and given positive numbers. The ill-posed Cauchy problem for elliptic equationswith a 1 being given and α > 0 the regularization parameter. A priori and a posteriori parameter choice rules are suggested which yield order-optimal regularization methods. Numerical results based on the boundary element method are presented and discussed.

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BackgroundThe risks of fish-borne zoonotic trematodes (FZT) to human health constitute an important problem in Vietnam. The infection of humans with these trematodes, such as small liver trematodes (Clonorchis sinensis and Opisthorchis viverrini), intestinal trematodes (Heterophyidae) and others is often thought to be linked to fish culture in areas where the habit of eating raw fish is common. Juvenile fish produced in nurseries are often heavily infected with FZT and since fishes are sold to aquaculture facilities for growth, control of FZT in these fishes should be given priority. Controlling the first intermediate host (i.e., freshwater gastropods), would be an attractive approach, if feasible. The black carp, Mylopharyngodon piceus, is a well-known predator of freshwater snails and is already used successfully for biological control of snails in various parts of the world including Vietnam. Here we report the first trials using it for biological control of intermediate host snails in nursery ponds stocked with 1-week old fry (10–12 mm in length) of Indian carp, Labeo rohita.MethodsSemi-field and field experiments were set up to test the effect of black carp on snail populations. In the semi-field experiment a known quantity of snails was initially introduced into a pond which was subsequently stocked with black carp. In the field trial in nursery ponds, density of snails was estimated prior to a nursing cycle and at the end of the cycle (after 9 weeks).ResultsThe results showed that black carp affect the density of snail populations in both semi-field and field conditions. The standing crop of snails in nursery ponds, however, was too high for 2 specimens to greatly reduce snail density within the relatively short nursing cycle.ConclusionsWe conclude that the black carp can be used in nursery ponds in Northern Vietnam for snail control. Juvenile black carp weighing 100 - 200g should be used because this size primarily prey on intermediate hosts of FZT and other studies have shown that it does not prey on fish fry of other species. It may be necessary to use a high stocking density of black carp or to reduce snail density in the nursery ponds using other measures (e.g. mud removal) prior to stocking fry in order for the black carp to keep the density of intermediate host snails at a very low level.

Let H be a Hilbert space with the norm • and A(t) (0 t T ) be positive self-adjoint unbounded operators from D(A(t)) ⊂ H to H. In the paper, we establish stability estimates of Hölder type and propose a regularization method for the ill-posed backward parabolic equation with time-dependent coefficientsOur stability estimates improve the related results by Krein (1957 Dokl. Akad. Nauk SSSR 114 1162-5), and Agmon and Nirenberg (1963 Commun. Pure Appl. Math. 16 121-239). Our regularization method with a priori and a posteriori parameter choice yields error estimates of Hölder type. This is the only result when a regularization method for backward parabolic equations with time-dependent coefficients provides a convergence rate.* Dedicated to Professor Rudolf Gorenflo on the occasion of his 80th birthday.

The ill-posed parabolic equation backward in timewith the positive self-adjoint unbounded operator A and E > > 0 being given is regularized by the well-posed non-local boundary value problemThe error estimates of Hölder type of the regularized solutions are obtained. These estimates improve the related results by Mel'nikova, Denche and Bessila.