1. Tensors calculus.
Tensors of the first and the second orders. Operations, properties, differential operations.
2. Theory of field
Scalar and vector fields. Integral theorems: Stokes and Gauss-Ostogradski’s. Differential operators in curvilinear orthogonal coordinates.
Application: Equation of continuity. Heat, mass and momentum transfer.
3. Boundary value problems for ordinary differential equations
Boundary value problem formulation. Orthogonal systems of functions. Fourier series. Sturm-Liouville’s problem. The methods of homogeneous and nonhomogeneous boundary value problem solutions.
Application: Stationary conduction of heat and diffusion in plane sheet, cylinder and sphere.
4. Boundary value problems for partial differential equations
Linear partial differential equations of the second order and their classification. Boundary conditions. Boundary value problem formulation. The methods of boundary value problem solutions.
5. The methods of parabolic equations solution
Method of separation of variables, method of similarity transformation, method of Green’s function, finite-difference method.
Application: Nonstationary heat conduction and diffusion in plane sheet, cylinder and sphere.
Nonstationary heat conduction and diffusion in bodies of finite dimensions.
6. The methods of elliptic equations solution
Method of separation of variables, method of analytic functions, finite-difference method.
Application: Stationary models of two-dimensional diffusion and two-dimensional heat conduction.
7. The methods of hyperbolic equations solution
Method of characteristic, method of separation of variables, finite-difference method.
Tensors of the first and the second orders. Operations, properties, differential operations.
2. Theory of field
Scalar and vector fields. Integral theorems: Stokes and Gauss-Ostogradski’s. Differential operators in curvilinear orthogonal coordinates.
Application: Equation of continuity. Heat, mass and momentum transfer.
3. Boundary value problems for ordinary differential equations
Boundary value problem formulation. Orthogonal systems of functions. Fourier series. Sturm-Liouville’s problem. The methods of homogeneous and nonhomogeneous boundary value problem solutions.
Application: Stationary conduction of heat and diffusion in plane sheet, cylinder and sphere.
4. Boundary value problems for partial differential equations
Linear partial differential equations of the second order and their classification. Boundary conditions. Boundary value problem formulation. The methods of boundary value problem solutions.
5. The methods of parabolic equations solution
Method of separation of variables, method of similarity transformation, method of Green’s function, finite-difference method.
Application: Nonstationary heat conduction and diffusion in plane sheet, cylinder and sphere.
Nonstationary heat conduction and diffusion in bodies of finite dimensions.
6. The methods of elliptic equations solution
Method of separation of variables, method of analytic functions, finite-difference method.
Application: Stationary models of two-dimensional diffusion and two-dimensional heat conduction.
7. The methods of hyperbolic equations solution
Method of characteristic, method of separation of variables, finite-difference method.