Objectives and competences
Knowledge, understanding, analytical solution of ordinary differential equations and numerical solution of ordinary and partial differential equations. Understanding and solving problems of calculus of variations and examples of the use of differential equations in geometry and physics.
Content (Syllabus outline)
1. Existence theorems: Local and global existence theorems for ODE, solution dependence of parameter, ODE of first order.
2. Linear differential equations: Systems of linear differential equations, Liouvill's formula, linear differential equation of n-th order, LDE with real and constant coefficients, Euler-Cauchy equation.
3. Numerical derivation: Basic methods.
4. Numerical solving of ordinary and partial differential equations.
5. Calculus of variations: Calculus of variations tasks, fundamental theorem of calculus of variations, Euler-Lagrange equation, generalizations, dynamic boundary conditions, isoperimetric problem, Lagrange task.
6. Differential equations in complex: Solutions in regular point surroundings, homogeneous linear equation, proper singular point, Frobenius's method.
7. Trigonometric series and transformations: Fourier series, Fourier transformation, discrete Fourier transform
8. Bessel differential equation: Solutions of Bessel DE, integral representations.
Learning and teaching methods
• Lectures
• Seminar
• Teaching and learning are done through the didactic use of ICT
Intended learning outcomes - knowledge and understanding
Knowledge and Understanding:
• Knowledge and understanding of differential equations and methods of their solution.
• Be able to understand and use the calculus of variations, the Frobenious method and numerical methods for solving ordinary and partial differential equations.
Intended learning outcomes - transferable/key skills and other attributes
Transferable/Key Skills and other attributes:
• Critical Thinking Skills (problem solving): solving more demanding physical tasks and practical problems based on the acquired knowledge, linking contents in the field of analysis and algebra.
• Communication skills: public performance at seminar presentation, manner of expression at exams.
• Problem solving: solving more complex differential equations.
Readings
E. Zakrajšek, Analiza III, DMFA Slovenije, Ljubljana, 1998.
F. Križanič, Navadne diferencialne enačbe in variacijski račun, DZS, Ljubljana 1974.
W. Kaplan, Advanced Calculusi, Fourth Edition. Addisson-Wesley Publishing Company, Redwood City, California, 1991.
D. Kincaid, W. Cheney: Numerical Analysis, Brooks/Cole, Pacific Grove, 1996.
W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling: Numerical Recipes in C, Cambridge University Press, New York, 2002.
Prerequisits
Knowledge of differentials and integrals.
Additional information on implementation and assessment The exam may be replaced by written midterm examination in the weight of 25%.
The oral exam may be replaced by written midterm examination in the weight of 50%.
Each of the mentioned commitments must be assessed with a passing grade.
Passing grade of the written exam is required for taking the oral exam.
