A (3 + 1)-dimensional boundary value problem for equations of Keldysh type (the second kind) is studied. Such problems for equations of Tricomi type (the first kind) or for the wave equation were formulated by M.H. as multidimensional analogues of Darboux or Cauchy-Goursat plane problems. Now, it is well known that Protter problems are not correctly set, and they have singular generalized solutions, even for smooth right-hand sides. In this paper an analogue of the Protter problem for equations of Keldysh type is given. An appropriate generalized solution with possible singularity is defined. Results for uniqueness and existence of such a generalized solution are obtained. Some a priori estimates are stated.
We study a boundary value problem for (3 + 1)-D weakly hyperbolic equations of Keldysh type (problem PK). The Keldysh-type equations are known in some specific applications in plasma physics, optics, and analysis on projective spaces. Problem PK is not well-posed since it has infinite-dimensional cokernel. Actually, this problem is analogous to a similar one proposed by M. Protter in 1952, but for Tricomi-type equations which, in part, are closely connected with transonic fluid dynamics. We consider a properly defined, in a special function space, generalized solution to problem PK for which existence and uniqueness theorems hold. It is known that it may have a strong power-type singularity at one boundary point even for very smooth right-hand sides of the equation. In the present paper we study the asymptotic behavior of the generalized solutions of problem PK at the singular point. There are given orthogonality conditions on the right-hand side of the equation, which are necessary and sufficient for the existence of a generalized solution with fixed order of singularity.
Data from the World Health Organization indicate that Bulgaria has the second-highest COVID-19 mortality rate in the world and the lowest vaccination rate in the European Union. In this context, to find the crucial epidemiological parameters that characterize the ongoing pandemic in Bulgaria, we introduce an extended SEIRS model with time-dependent coefficients. In addition to this, vaccination and vital dynamics are included in the model. We construct an appropriate Cauchy problem for a system of nonlinear ordinary differential equations and prove that its unique solution possesses some biologically reasonable features. Furthermore, we propose a numerical scheme and give an algorithm for the parameters identification in the obtained discrete problem. We show that the found values are close to the parameters values in the original differential problem. Based on the presented analysis, we develop a strategy for short-term prediction of the spread of the pandemic among the host population. The proposed model, as well as the methods and algorithms for parameters identification and forecasting, could be applied to COVID-19 data in every single country in the world.
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