Ukr. Mat. Zh. - 2014. - 66, № 7. - pp. 983–1002
In the paper, we study the boundary-value problems for parameter-dependent anisotropic differential-operator equations with variable coefficients. Several conditions for the uniform separability and Fredholmness in Banach-valued L p -spaces are given. Sharp uniform estimates for the resolvent are established. It follows from these estimates that the indicated operator is uniformly positive. Moreover, it is also the generator of an analytic semigroup. As an application, the maximal regularity properties of the parameter-dependent abstract parabolic problem and infinite systems of parabolic equations are established in mixed L p -spaces.
The block by block method with Romberg quadrature for solving nonlinear Volterra integral equations on the large intervals
Ukr. Mat. Zh. - 2012. - 64, № 7. - pp. 919-931
We investigate the numerical solution of nonlinear Volterra integral equations by block by block method, which is useful specially for solving integral equations on large-size intervals. A convergence theorem is proved that shows that the method has at least sixth order of convergence. Finally, the performance of the method is illustrated by some numerical examples.