The following results of education should be demonstrated by the post-graduate student after studying of the academic discipline:
1. knowledge of the main principles and ideas of the percolation theory;
2. skills in the application of the percolation theory to the analysis of transport phenomena;
3. knowledge of the main principles and ideas of the fractal analysis;
4. skills in the application of the fractal analysis in scientific investigations;
5. knowledge of the main ideas of the deterministic chaos theory;
6. knowledge of the main principles and concepts of the theory of solitons formation, particularly in solids.
7. knowledge of the main types of equations of mathematical physics, particularly, the ones describing the heat transfer and diffusion processes as well the main approaches of their solving.





Studying of the academic discipline foresees formation and development of following competences of the post-graduate students:
general:
- ability of solving of integrated problems during innovative-research and professional activity;
- thorough knowledge and understanding of the philosophical methodology of cognition;
- ability of initialization and undertaking of the original scientific researches, identification of the actual scientific problems, carrying out of searching and critical analysis of the information, producing of the innovative meaningful ideas and application of the non-standard approaches for solving of the complex and non-typical tasks;
- ability to do the scientific talking and discussion with wide scientific community and public.
professional:
- development of the materials with previously defined properties and carrying out of the corresponding theoretical and experimental investigations;
- realization of projected searching of new materials and physical phenomena that can be used for creation of the materials and components of physical and bio-medical electronics;
- calculation and design of the structures and devices for electronic technique based on modern element base with using of new materials and technologies of their obtaining;
- formulation of the tasks and realization of investigations in the field of physics and technology of the materials and devices of physical and bio-medical electronics;.

The results of studying of this discipline detail the following programme results of studying.
1. The possibility to demonstrate the knowledge of modern concepts and theories used for description of transport phenomena.
2. The possibility to integrate and apply the obtained knowledge from different subject spheres for solving of the theoretical-applied tasks in the specific field of investigation.
3. The possibility to choose and apply the methodology and scientific instrumentation for realization of the theoretical and experimental investigations in the field of solid state physics and related areas.
4. The possibility to realize the scientific investigations and fulfill the scientific projects based on the identification of the actual scientific problems, defining of the objective and the tasks, formation and critical analysis of the informational basis.

are absent

In the frames of the discipline the basic theories of percolation, fractal analysis, the deterministic chaos and solitons formation that are used for the transport phenomena analysis, are considered. The solutions of equations of mathematical physics describing diffusion (heat transfer) processes are analyzed. The objective of the discipline is the introduction of the young scientists to the approaches of simulation and modeling of the transport phenomena as well as to the peculiarities of the existing models. Lections: 1. The elements of percolation theory. The principal points. The clusters. The percolation threshold. The task of the nodes. The task of the connections. The covering lattice. The including lattice. The white and the black percolation. Dual lattice. The oriented percolation. 2. The percolation thresholds for volume lattices. Ferromagnetic with long-range interaction and the task of the spheres. The electrical conductivity of doped semiconductors. Mott transition. The structure of the infinite cluster. Shklovskii – de Gennes model. The hopping conductivity; the description of the phenomena in the frames of the percolation theory. 3. The fractal analysis. The concept of the fractal. The coastline paradox. The Hausdorff – Bezikovich dimension. The triadic Koch curve. The similarity and scaling. The dimension of similarity. Examples of the fractals. The fractal dimension of the clusters. Diffusion-limited aggregation. 4. The formation of the fractal structures during the percolation. Self-similarity of the percolation clusters. The finite clusters of the percolation. The cluster radius of gyration. The fractal diffusion front. The relationship between the perimeter and the area. Fractals in solids. Airgels. The formation of fractal structures during the deformation. 5. Phase points and phase trajectories. The phase portrait of the system. Attractors. The strange attractors and deterministic chaos. Lorenz attractor. The fractal dimension of the strange attractors. Solitons. The solitary Russel wave, its main properties. The Korteweg–de Vries equation, its soliton solutions. Frenkel – Kontorova soliton. Sine – Gordon equation, its soliton solutions. Dislocations and anti-dislocations, their soliton properties. Breathers. 6. The basics of mathematical physics. Diffusion equation. The types of differential equations of mathematical physics. Diffusion equation, its solving for the diffusion into medium in the case when the first or third order boundary conditions are realized. 7. Solving of the diffusion equation for the most common cases. Diffusion from the instantaneous source to infinite and semi-infinite media. Diffusion from the continiuous source to semi-infinite media. Using of Laplace transformation for solving of diffusion equation. Diffusion in the case of an extended initial distribution. Diffusion in presence of a driving force. Simultaneous diffusion and chemical reaction. Practical exercises: 1. Computer simulation of the percolation processes in the flat lattices. 2. The numerical determination of percolation thresholds for volume lattices. 3. The software for fractal analysis. 4. The simulation of clusters formed due to diffusion-limited aggregation. Determination of clusters dimensions. 5. Software for construction of strange attractors. Determination of the fractal dimensions of the strange attractors. 6. Numerical simulation of solitions based on the solutions of sine – Gordon equation. 7. Solving of diffusion (heat transfer) equation by means of Maple computing environment.

Basic
1. Mott N.F., Davis E.A., Electronic Processes in Non-Crystalline Materials, OUP Oxford, 2012.
2. Shkovskii B.I., Efros A.L., Electronic Properties of Doped Semiconductors, Moscow, 1979 (in Russian).
3. Zaiman J.M., Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems. Cambridge University Press, 1979.
4. Efros A.L., Physics and Geometry of Disorder. Moscow, 1982 (in Russian).
5. Feder J. Fractals. Springer, 1988.
6. Mandelbrot B.B., The Fractal Geometry of Nature. New York, W.H. Freeman and Company, 1983.
7. Lamb G.L., jr., Elements of Soliton Theory. New York?Chichester?Brisbane?Toronto, John Wiley & Sons, 1980.
8. Philippov A.T. The Many-faced soliton. Мoscow, Science, 1990 (in Russian).
9. Gleick J., Chaos: Making a New Science. Penguin (Non-Classics), 2008.
10. Tikhonov A.N.., Samarskii A.A., Equations of mathematical physics. Мoscow, Science, 1972 (in Russian).
11. Markovych B.M., Equations of mathematical physics. Lviv, Lviv polytechnic National University, 2010 (in Ukrainian).

Additional
1. Sokolov I.M., Dimensions and other geometric critical indicators in percolation theory. Physics-Uspekhi (Advances in Physical Sciences) , 1986, vol. 150, p. 221 (in Russian).
2. Schroeder M..R. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York, W.H. Freeman and Company, 1992.
3. Fractals in Physics. Proceedings of 6th International Symposium on Fractals in Physics, 1985, Moscow, Mir, 1988 (in Russian).
4. Peitgen H.-O., Richter P.H., The Beauty of Fractals. Springer–Verlag, 1986.
5. Nicolis G., Prigogine I., Exploring Complexity: an Introduction. New York, W.H. Freeman and Company, 1989.
6. Prigogine I., Stengers I., Order out of Chaos. Man’s New Dialog with Nature. London, Heinemann, 1984.
7. Crank J., The mathematics of diffusion. Oxford, Claredon Press, 1955.
8. Manning J.R., Diffusion Kinetics for Atoms in Crystals. Princeton, Toronto, D. Van Nostrand Company, Inc., 1968.
9. Carslaw H.S., Jaeger J.C., Conduction of Heat in Solids. Oxford, Claredon Press, 1986.

The sites dedicated to fractals: http://fraktalz.narod.ru/, http://ns1.npkgoi.ru/r_1251/investigations/fractal_opt/data3/data3.html, http://fractals.nsu.ru/, http://www.pbs.org/wgbh/nova/physics/hunting-hidden-dimension.html, etc.

Ongoing monitoring is carried out for determination of the readiness for the class in the following forms:
• selective oral questioning before the class;
• frontal standardized questioning by the cards, tests during 5-10 minutes;
• the estimation of the activity during the class, proposals, original solutions, clarifications and definitions, additions to the previous answers, etc.
The control questions are divided into:
• test tasks – choose correct answers;
• problematic – creation of the situation of problematic character;
• questions-lines – to reveal the causal links.
The final control is carried out in accordance with the results of ongoing monitoring and the results of exam.