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In this paper some ill-posed boundary-value problems (BVPs) for three - dimensional partial differential equations are studied. The situation with them is rather surprising and there is no general understanding even more than sixty years after their statement given by Murray Protter. These problems are multidimensional analogues of classical BVPs on the plane and intuitively the initial expectations was that their properties would be similar. Unexpectedly, it turned out that unlike the two-dimensional variants, the Protter-Morawetz problems are not well posed. The generalized solution is uniquely determined, but it may have a strong singularity at an isolated boundary point even for infinitely smooth right-hand side. Also, the adjoined problem has an infinite number of smooth solution in the kernel. In the present paper such ill-posed problems for 3-D Gellerstedt equation with lower order terms under multidimensional Protter condition are studied. In addition to new results, we also make a survey of the known results concerning the Protter-Morawetz problems for both Tricomi-type equations and Keldysh-type equations. [ABSTRACT FROM AUTHOR]

In this paper we introduce a time-dependent SEIR-based model with vaccination. In the suggested model the host population is divided into seven compartments: susceptible, exposed, infectious, recovered, deceased, vaccinated susceptible individuals and individuals with vaccination-acquired immunity. The dynamics of the infection in these groups is modeled by a Cauchy problem for a system of nonlinear ordinary differential equations. Besides the classical SEIR model parameters (infection rate, incubation rate, recovery rate), the new model involves mortality rate and some additional vaccination parameters such as vaccination rate, vaccines effectiveness and takes into account the time taken for antibodies to develop. Using Bulgarian COVID-19 data we solve the so-called "inverse problem" and find the unknown parameters as a function of time. Then based on these results we solve numerically the Cauchy problem in the model and make computer simulation of the COVID-19 dynamics in Bulgaria. This allows us to calculate the number of people with vaccination-acquired immunity in Bulgaria. The proposed model and method for parameters identification can be applied to COVID-19 data in every single country of the world. [ABSTRACT FROM AUTHOR]

The three-dimensional Protter-Morawetz problem for weakly hyperbolic equations of Keldysh type involving lower order terms is studied. Similar problem for Tricomi-type equations was proposed by M. Protter in connection with the Guderley-Morawetz plane problem that models the transonic flow phenomena. The considered Protter-Morawetz problem for Keldysh-type equations is not Fredholm in the frame of classical solvability, because it has infinite-dimensional co-kernel. In the present paper new nontrivial classical solutions to the homogeneous adjoint problem are found. Further, a generalized solution to the formulated problem is considered, in a special function space, for which existence and uniqueness theorems hold. Under some conditions on lower order terms smooth right-hand side functions are found, such that the corresponding generalized solutions have strong power type singularities. It is interesting that these singularities are isolated at only one boundary point, which makes this case different from the traditional case on the propagation of singularity. [ABSTRACT FROM AUTHOR]

Electronic assessment is a modern form of assessment of students’ knowledge and skills that can be integrated into traditional mathematics education when some of the assessment activities are done online. However, this requires the use of adequate security and protection measures such as various forms for student identification and authorship verification. This article discusses the possibilities and limitations of using a web-based student identification system in mathematics education. We report on the experiment that was conducted in the frame of TeSLA (An Adaptive Trust-based e-assessment System for Learning) project to explore existing problems and to offer new solutions in continuous e-assessment especially in differential equation education. We present an innovative model for e-assessment in the course "Differential equations and applications" in the bachelor program "Software engineering" at Sofia University, based on the long experience we have in the education in this discipline. Some results of a survey of the students opinion in relation to the TeSLA identification software used are presented and analyzed. [ABSTRACT FROM AUTHOR]

This paper deals with Protter problems for Keldysh type equations in R4: Originally such type problems are formulated by M. Protter for equations of Tricomi type. Now it is well known that Protter problems for mixed type equations of the first kind are ill-posed and for smooth right-hand side functions they have singular generalized solutions. In the present paper Protter problem for equations of second kind (Keldysh type) is formulated and it is shown that in the frame of classical solvability this problem is not well posed. Further, a notion for a generalized solution in suitable functional space is given. Results for existence and uniqueness of generalized solution of the considered problem are obtained. Some a priori estimates are stated. [ABSTRACT FROM AUTHOR]