### Modeling and Analysis of Differential Equations

Data is displayed for academic year: 2023./2024.

#### Course Description

Ordinary differential equations: basic existence proof, linear systems, exponential function of a matrix, variation of constants formula; equilibrium of a string, boundary value problem, Green's function; Laplace equation, divergence theorem, contact field, equilibrium of a membrane, potential electrostatic field; boundary problem for Laplace equation, Green's formula, harmonic functions; Fourier sequences, orthogonal systems of functions, Sturm Liouviell's problem, Laplace equation on a sphere and cylinder, derivation of heat equation, maximum principle, Fourier method, wave equation derivation on the example of string, Dalambert's, Kirchoff's and Poisson's formula, Fourier method; Elements of functional analysis, Banach and Hilbert spaces, bounded linear operators, compact operators, Fredholm operators, weak convergence, symmetric operators; Elements of measure theory: Lebesgue integral, basic theorems of convergence, Sobolev's spaces and inequalities; Elliptic differential equations of second order, weak solutions, Lax-Milgram theorem, regularity, eigenvalue problems and eigenfunctions; Parabolic differential equations of second order: existence, uniqueness and regularity; Hyperbolic differential equations of second order: existence, uniquenes and regularity; Basic equations of mechanics of continuum: derivation of the equations of elasticity, Navier-Stokes and Maxwell

#### Study Programmes

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#### Learning Outcomes

1. classify basic partial differential equations of second order
2. select maximum principle for Laplace equation
3. apply Fourier method for solving basic PDEs on specific domains
4. define Sobolev spaces
5. select weak formulation for basic PDEs of second order
6. explain the proof of existence for basic PDEs of second order
7. explain the derivations of the equations of elasticity, Navier-Stokes and Maxwell

#### Forms of Teaching

Lectures

Students will 4 hours per week listen to lectures

Exercises

Examples will be shown to students that will help in understanding the lectures.

Independent assignments

Certain topic will be given to student during semester

Continuous Assessment Exam
Seminar/Project 5 % 10 % 45 % 90 %
Mid Term Exam: Written 0 % 45 % 0 %
Final Exam: Written 0 % 45 %
Exam: Written 45 % 90 %

#### Week by Week Schedule

1. Ordinary differential equations, egzistence theorem.
2. String equation, boundary value problem, Green's function
3. Laplace equation, divergence theorem, boundary value problem
4. Fourier sequence, orthogonal systems of functions, Sturm-Liuviell's problem, Laplace on sphere and cylinder.
5. Flow of heat, The maximum principle; The fundamental solution
6. Wave equation derivation on the example of string, Dalambert's, Kirchoff's and Poisson's formula, Fourier method;
7. Elements of functional analysis, Banach and Hilbert spaces, bounded linear operators.
8. Midterm exam
9. Compact operators, Fredholm operators, weak convergence, symmetric operators;
10. Elements of measure theory: Lebesgue integral, basic theorems of convergence, Sobolev's spaces and inequalities;
11. Elliptic differential equations of second order, weak solutions, Lax-Milgram theorem, regularity, eigenvalue problems and eigenfunctions;
12. Second order parabolic problem; Initial boundary condition; Weak formulation; Existence and uniqueness of solution
13. Second order hyperbolic problems; Initial and boundary condition; Weak formulation and uniqueness
14. Constitutive assumptions, Isotropic transformations, Navier-Stokes equations
15. Final exam

#### Literature

(.), Ibrahim Aganović, Krešimir Veselić: Linearne diferencijalne jednadžbe,
(.), Ibrahim aganović: Uvod u rubne zadaće mehanike kontinuuma,
(.), L.C. Evans: Partial Differential Equations: Second Edition,

#### General

ID 222607
Winter semester
5 ECTS
L1 English Level
L1 e-Learning
45 Lectures
0 Seminar
15 Exercises
0 Laboratory exercises
0 Project laboratory