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Lecturer(s)
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Kulish Vladimír, doc. Ing. PhD., DSc.
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Course content
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Annotation: The course offers a comprehensive exploration of partial differential equations (PDEs), focusing on their origins, classifications, and methods of solution. It begins with a review of PDEs, emphasizing their formulation through conservation laws and constitutive relationships. The classification of PDEs into hyperbolic, parabolic, and elliptic types is studied through canonical forms and representative equations such as the wave, diffusion, Laplace, and Schrödinger equations. Various exact solution methods, including d´Alembert´s solution, integral transforms, and separation of variables, are introduced, alongside special functions like Bessel functions, relevant to cylindrical and spherical geometries. The course delves into Green´s functions, highlighting their role in solving PDEs using convolution integrals and exploring their theoretical properties. Nonlinear dynamics is introduced through examples of nonlinear ODEs and PDEs, with a focus on stability, chaos, and real-world phenomena such as epidemics and chemotaxis, providing students with a solid foundation in the mathematical tools and concepts essential for modelling physical systems. 1. A short review of PDEs: definition of PDE, origins of PDEs, conservation law, and the role of constitutive relationship. 2. Classification of the equations: six basic classifications of PDEs; canonical forms of second order PDEs. 3. Classification of the equations: the wave, diffusion and Laplace equations seen as representatives of hyperbolic, parabolic and elliptic types; Schrödinger equation. 4. Exact methods of solutions to PDEs: d'Alembert's solution - wave equation in an infinite domain; diffusion kernel; initial value problems in an infinite space. 5. Exact methods of solutions to PDEs: integral transforms; semi-infinite domains; parameter identification. 6. Exact methods of solutions to PDEs: separation of variables. 7. Special functions arising during solving some PDEs: series method of solution for ODEs; introduction to Bessel functions. 8. Special functions arising during solving some PDEs: separation of variables in cylindrical and spherical geometries. 9. Green's function and its application in solving PDEs: solutions of PDEs written in terms of convolution (memory) integrals; divergence theorem; Green's first and second identities; Green's theorem. 10. Green's function and its application in solving PDEs: properties of Green's function; distributions. 11. Introduction to non-linear ODEs and PDEs; some non-linear phenomena (epidemics, chemotaxis, etc.). 12. Concept of stability and chaos in non-linear systems. Outcomes: After completing the course, students will understand the fundamental principles of partial differential equations, their classification into hyperbolic, parabolic, and elliptic types, and key methods for their solution, including d´Alembert´s solution, integral transforms, and separation of variables. They will learn to use special functions such as Bessel functions and apply Green´s functions in solving PDEs through convolution integrals. Students will gain foundational knowledge of nonlinear ordinary and partial differential equations, comprehend concepts of stability and chaos, and be able to model complex physical systems involving nonlinear dynamics. Content of tutorials/seminar: Defending the course project in the form of a seminar.
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Learning activities and teaching methods
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unspecified
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Learning outcomes
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The course aims to provide doctoral students with a comprehensive understanding of the mathematical foundations of physical modelling through partial differential equations. Emphasis is placed on the classification of PDEs, exact analytical solution techniques, Green´s functions, special functions, and the analysis of nonlinear systems. The course develops the theoretical and analytical skills required for advanced research in applied mathematics, physics, engineering, and related disciplines.
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Prerequisites
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Students are expected to possess prior knowledge of advanced calculus, ordinary differential equations, and vector analysis. Familiarity with mathematical methods in physics is desirable.
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Assessment methods and criteria
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unspecified
Independent study of assigned literature; preparation and presentation of a course project on a selected topic; passing the written examination.
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Recommended literature
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F. John ? Partial Differential Equations, Springer, 1982..
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George F. Simmons ? Differential Equations with Applications and Historical Notes, CRC Press, 1991..
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John D. Logan ? Applied Partial Differential Equations, Springer, 2015..
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Lawrence C. Evans ? Partial Differential Equations, AMS, 2010..
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Vladimir Kulish, Partial differential equations. 2nd edition. Pearson/Prentice Hall, (2010), ISBN 9810684398..
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Walter A. Strauss ? Partial Differential Equations: An Introduction, Wiley, 2007..
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