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FSI-SPDAcad. year: 2023/2024
The course deals with the following topics: Ordinary differential equations - a brief survay of material studied within the 3rd semester subject and extending of the subject matter (theorems on existence and uniqueness of the solution, stability of the solution, boundary value problems).Partial differential equations - basic concepts. The first-order equations. The Cauchy problem for the k-th order equation. Transformation, classification and canonical form of the second-order equations. Derivation of selected equations of mathematical physics (heat conduction, wave equation, variational prinsiple), formulation of initial and boundary value problems. The classical methods: method of characteristics, The Fourier series method, integral transform method, the Green function method. Maximum principles. Properties of the solutions to the elliptic, parabolic and hyperbolic equations.
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1 Revision of O.D.E. - 1st order and higher order linear equations.2 Systems of linear O.D.E., existence and uniqueness of the solution.3 Test 1: O.D.E. Elements of P.D.E., 4 The 1st order equations.5 The Cauchy problem, classification of 2nd order equations.6 Mathematical Physics Equations: derivation of the heat equation.7 Derivation of the equation of string vibration and wave equations.8 Derivation of membrane equation via variational principle.9 Method of characteristics for 1D wave equation.10 Fourier series method.11 Integral transform method. Test 2 of P. D. E.12 Green function method and the maximum principles.13 Properties of the solutions, reserve.
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