1) to be able to build models of stationary and non-stationary processes of thermal conductivity and stress-strain state in homogeneous media; to describe these processes by equations of elliptic and parabolic types, boundary conditions of the first and second kind;
2) to be able to build models of stationary and non-stationary processes of thermal conductivity and stress-strain state in piecewise-homogeneous media; to describe these processes by systems of equations of elliptic and parabolic types, mixed boundary conditions and ideal contact conditions at the interface;
3) to be able to apply the mathematical apparatus of modern numerical methods (boundary and near-boundary elements) for solving the problems of theories of potential, thermal conductivity, and elasticity;
4) to develop algorithms and software for implementation of the studied methods;
5) to compare the numerical results obtained by both methods and to give recommendations on their optimal combination.

Programming Basics (prerequisite)
Software Architecture and Design (prerequisite)
Designing software (prerequisite)
Object-oriented programming (prerequisite)

Topic 1. Introduction. Some background information on methods for solving boundary-value problems for non-homogeneous media. A technique for solving two-dimensional problems of potential theory and elasticity theory in homogeneous media. Elliptic equations. Boundary condition of the first kind. Mixed boundary conditions. Fundamental solutions. Boundary integral equations. Ways to construct boundary and near-boundary elements for a two-dimensional problem. Topic 2. Comparison of mathematical and computational aspects of indirect methods of boundary and near-boundary elements in solving two-dimensional problems of potential theory and elasticity theory. Differences in theoretical positions of both methods. Calculus of integrals in the sense of Riemann and Cauchy. Consideration of internal sources. Unification of software. Topic 3. Methods for solving non-stationary two-dimensional problems of thermal conductivity in homogeneous media. Comparison of mathematical and computational aspects of indirect methods of boundary and near-boundary elements in solving non-stationary two-dimensional problems of heat conduction. Parabolic equation. Mixed boundary conditions. A fundamental solution. Boundary integral equations. Step-by-step time schemes: a single initial condition and a sequence of initial conditions. Topic 4. Generalization of the above methods for solving problems in homogeneous three-dimensional objects. Elliptic and parabolic equations. Mixed boundary conditions. Fundamental solutions. Boundary integral equations. Ways to construct boundary and near-boundary elements for a three-dimensional problem. Topic 5. The technique of solving two-dimensional problems of potential theory and elasticity theory in piecewise-homogeneous media. Systems of elliptic equations. Mixed boundary conditions. Ideal conditions of thermal and mechanical contact. Boundary integral equations. Construction of a system of linear algebraic equations for finding unknown intensities of sources introduced into boundary and near-boundary elements. Topic 6. The technique of solving non-stationary two-dimensional problems of thermal conductivity in piecewise-homogeneous media. The system of parabolic equations. Mixed boundary conditions. Ideal thermal contact conditions. Boundary integral equations. Construction of a system of linear algebraic equations for finding unknown intensities of sources introduced into the boundary and near-boundary elements for two step-by-step time schemes: a single initial condition and a sequence of initial conditions. Theme 7. Generalization of the listed methods for solving problems in piecewise-homogeneous three-dimensional objects. Systems of elliptic and parabolic equations. Mixed boundary conditions. Ideal conditions of thermal and mechanical contact. Fundamental solutions. Boundary integral equations. Ways to construct boundary and near-boundary elements for a three-dimensional problem. Construction of a system of linear algebraic equations for finding unknown intensities of sources introduced into boundary and near-boundary elements.

1. Zhuravchak L. M., Kruk O. S. Consideration of the nonlinear behavior of environmental material and a three-dimensional internal heat sources in mathematical modeling of heat conduction // Mathematical Modeling and Computing, Vol. 2, No. 1, pp. 107–113 (2015).
2. Sapuzhak Ya. S., Zhuravchak L. M. The Technique of numerical solution of 2-D direct current modeling problem in inhomogeneous media // Acta Geophysica Polonica, 1999, Vol.XLVII, No 2. – P.149-163.
3. Zhuravchak, L., Struk, A., Struk, E. Modeling of Nonstationary Process of Reservoir Pressure Change in Piecewise-homogeneous Medium with Nonlinear Behaviour of Regions Materials. International Journal of Advanced Research in Computer Engineering & Technology, 5(5), 1439-1449 (2016).
4. Liubov Zhuravchak. Mathematical Modelling of Non-stationary Processes in the Piecewise-Homogeneous Domains by Near-Boundary Element Method // Springer Nature Switzerland AG 2020 N. Shakhovska and M. O. Medykovskyy (Eds.): CCSIT 2019, AISC 1080, pp. 64–77, 2020.

Methods of diagnosing knowledge: completing individual homework assignments given in practical classes, performing individual research tasks, examination work.
Criteria for evaluating student learning outcomes:
Maximum score in points 100
Current control: 45
Homeworks 20
research task 20
examination control: 60
written component 50
oral component 10