In this paper, we study the inverse problem of identifying the dimensionless thermal conductivity coefficient for the Green–Naghdi equation of type III, which describes the propagation of thermal disturbances with a finite velocity and takes into account the inertial effects of heat flux. For the inverse problem, the stability requirement (Hadamard criteria) is violated, as a result of which even minimal data distortions lead to significant errors in parameter identification. As a solution method, we use an approach based on physically informed neural networks (PINN), which combines the capabilities of deep learning with a priori knowledge of the structure of the differential equation. The parameter is included among the trained variables, and the loss function is formed based on the deviation from the differential equation, boundary conditions, initial conditions, and noisy experimental data from a point sensor. The results of computational experiments are presented, demonstrating high accuracy of parameter recovery (error less than 0.03%) and the stability of the method with respect to the presence of additive Gaussian noise in the data. The PINN method has proven itself to be an effective tool for solving ill-posed inverse problems of mathematical physics.

An ontological design approach was used to describe the semantics of some boundary value problems in the LibMeta digital library. To describe problems in the LibMeta library, connections of terms and concepts with classical definitions of the mathematical encyclopedia and other primary sources have been established. Establishing links allows you to form a dictionary and thesaurus of the applied subject area of new boundary value problems and place the results in the semantic environment of the digital library. Examples of this approach are demonstrated using the capabilities of the LibMeta semantic library, which contains a digitized version of the mathematical encyclopedia, encyclopedia of mathematical physics, classifiers, and applied mathematical thesauri and dictionaries. New terms from publications, after being added to the content of the library, were reflected with links in the mathematical encyclopedia. The thesaurus for problems in the elasticity theory domain was created for the first time by integrating subject dictionaries, classifiers, metadata of specialized journal publications and encyclopedic content of the LibMeta library. The purpose of such research is to provide the user with additional services in the search for publications in the applied scientific field.

The article discusses the construction of a visual solution of boundary value problems for a linear differential equation of the second order by the tridiagonal sweep method in VBA in Excel. Creating macros in VBA allows you to visualize solutions to such boundary value problems.