The mixed boundary value problem for a compressible Stokes system of partial differential equations in a bounded domain is reduced to two different systems of segregated direct Boundary-Domain Integral Equations (BDIEs) expressed in terms of surface and volume parametrix-based potential type operators. Equivalence of the BDIE systems to the mixed BVP and invertibility of the matrix operators associated with the BDIE systems are proved in appropriate Sobolev spaces.2000 Mathematics Subject Classification. Primary: 35J57, 45F15; Secondary: 45P05.

A system of Boundary-Domain Integral Equations is derived from the mixed (Dirichlet-Neumann) boundary value problem for the diffusion equation in inhomogeneous media defined on an unbounded domain. This paper extends the work introduced in [26] to unbounded domains. Mapping properties of parametrix-based potentials on weighted Sobolev spaces are analysed. Equivalence between the original boundary value problem and the system of BDIEs is shown. Uniqueness of solution of the BDIEs is proved using Fredholm Alternative and compactness arguments adapted to weigthed Sobolev spaces.

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The interior Dirichlet boundary value problem for the diffusion equation in nonhomogeneous media is reduced to a system of boundary-domain integral equations (BDIEs) employing the parametrix obtained in Portillo (2019). We further extend the results obtained in Portillo ( 2019) for the mixed problem in a smooth domain with L 2 (Ω) right-hand side to Lipschitz domains and partial differential equation (PDE) right-hand side in the Sobolev space H −1 (Ω), where neither the classical nor the canonical conormal derivatives are well-defined. Equivalence between the system of BDIEs and the original BVP is proved along with their solvability and solution uniqueness in appropriate Sobolev spaces.

The deep neural network (DNN) was applied for estimating a set of unknown parameters of a dynamical system whose measured data are given for a set of discrete time points. We developed a new vectorized algorithm that takes the number of unknowns (state variables) and number of parameters into consideration. The algorithm, first, trains the network to determine weights and biases. Next, the algorithm solves the systems of algebraic equations to estimate the parameters of the system. If the right hand side function of the system is smooth and the system have equal numbers of unknowns and parameters, the algorithm solves the algebraic equation at the discrete point where absolute error between the neural network solutions and the measured data is minimum. This improves the accuracy and reduces computational time. Several tests were carried out in linear and non-linear dynamical systems. Last, we showed that the DNN approach is more successful in terms of computational time as the number of hidden layers increases.