### Modeling and Analysis of Differential Equations

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

#### Lecturers

#### 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

##### University graduate

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[FER3-HR] Software Engineering and Information Systems - profile

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#### Learning Outcomes

- classify basic partial differential equations of second order
- select maximum principle for Laplace equation
- apply Fourier method for solving basic PDEs on specific domains
- define Sobolev spaces
- select weak formulation for basic PDEs of second order
- explain the proof of existence for basic PDEs of second order
- 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

#### Grading Method

Continuous Assessment | Exam | |||||
---|---|---|---|---|---|---|

Type | Threshold | Percent of Grade | Threshold | Percent of Grade | ||

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

- Ordinary differential equations, egzistence theorem.
- String equation, boundary value problem, Green's function
- Laplace equation, divergence theorem, boundary value problem
- Fourier sequence, orthogonal systems of functions, Sturm-Liuviell's problem, Laplace on sphere and cylinder.
- Flow of heat, The maximum principle; The fundamental solution
- 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.
- Midterm exam
- 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;
- Second order parabolic problem; Initial boundary condition; Weak formulation; Existence and uniqueness of solution
- Second order hyperbolic problems; Initial and boundary condition; Weak formulation and uniqueness
- Constitutive assumptions, Isotropic transformations, Navier-Stokes equations
- 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*,#### For students

#### 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

#### Grading System

**85**Excellent

**75**Very Good

**60**Good

**50**Sufficient