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On an integral equation of the problem of heat conduction with domain boundary moving by law of t = x 2In the article it is shown that the homogeneous Volterra integral equation of the second kind, to which the homogeneous boundary value problem of heat conduction in the degenerating domain is reduced, has a nonzero solution. The boundary of the domain moves with a variable velocity. It is shown that the norm of the integral operator acting in classes of continuous functions is equal to 1. Mellin transformation is applied to the obtained integral equation. It is proved that for certain values of the spectral parameter the eigenvalues of the integral equation will be simple.

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In this paper, we study the solvability of a second-kind pseudo-Volterra integral equation. By replacing the right-hand side and the unknown function, the integral equation is reduced to an integral equation, the kernel of which is not compressible . Using the Laplace transform, the obtained equation is reduced to an ordinary first-order differential equation (linear). Its solution is found. The solution of the homogeneous integral equation corresponding to the original nonhomogeneous integral equation found in explicit form. Special cases of a homogeneous integral equation and its solutions are written for different values of the parameter k. Classes are indicated in which the integral equation has a solution. Singular integral equations were considered in works [1][2][3]. Their kernels were also incompressible , but kernels had an another form. In this connection, the weight classes of the solution existence differ from the class of the solution existence for the equation considered in this work.

The study describes an innovative methodology for teaching natural and mathematical sciences in the context of distance learning using modern technological solutions and based on the concepts of active social learning that involves constructivist, problem-oriented, project and research approaches. The proposed methodology was tested on 80 students enrolled in two training courses in [BLINDED] University and [BLINDED] University, respectively: Mathematics Teaching in the Content and Language Integrated Learning (CLIL) context and Molecular Biology. The psychological safety of the proposed pedagogical methodology was investigated by assessing the level of psychological well-being of the participants in the educational process using the Scale of psychological well-being questionnaire developed by C. D. Ryff that was adapted to the educational context. The results of the study showed that the proposed pedagogical methodology was safe in relation to the life of students. It generally improved their perception of learning and themselves in the role of its participants, promoted positive self-esteem in group learning contacts, and, by developing learning involvement and interest through the Clil technology, qualitatively contributed to the achievement of educational progress by each student. The developed innovative methodology for teaching natural and mathematical sciences can be used as a pedagogical model for developing effective training courses. The demonstrated assessment of the level of psychological well-being adapted to the educational context can serve as a basis for the development of motivational learning strategies that support students in crisis learning conditions during the pandemic.