Objectives and competences
Objectives:
To grasp the basic ideas of systems of ordinary differential equations, Euler and Bessel differential equations, and partial differential equations, in particular wave and heat equations. Apply integral transforms to solve differentail equations.
Competences:
To grasp the basic ideas of mathematical modelling of problems related to differential equations and capability of transferring and applying acquired knowledge for problem solving of engineering problems with mathematical content.
Content (Syllabus outline)
1.Fourier series: Euler's formulas for Fourier's coefficients. Odd and even periodic extension of the function. Transformation of the Dirac delta ''function'' (distribution) and the Heaviside function. Convolution.
2. Ordinary differential equations (ODE's):
- system of ODE's - solution with eigenvectors and root vectors; the theory of stability and the problem of linearization.
-Laplace transformation - transformation of elementary functions, Dirac delta ''function'' and Heaviside function. Basic formulas for transformation of derivatives. Convolution.
- Euler DE.
- special functions - Gamma function, Bessel function, solution of Bessel differential equation.
3. Partial differential equations (PDE's):
- classification
- heat equation
- wave equation
- Laplace equation
- Laplace transformation for solving PDE's
- Fourier transformation for solving PDE's
Learning and teaching methods
Lectures, tutorial, coursework assignments.
Intended learning outcomes - knowledge and understanding
Knowledge and understanding:
At the end of the course student is going to:
1. Analyse the stability of the origin in linear systems of ODE's and find the linearizationof a non-linear system od ODE's near a hyperbolic singular point.
2. Periodically extend any function with finite points of discontinuity on interval [a,b] and analyse its sense of Fourier series.
3. Solve Euler DE.
4. Recognize generelized Bessel differential equation and find its solution.
5. Solve wave and heat equation using different boundary and initial conditions.
6. Distinguish between general and particular solution to PDE and solve it by separation method.
7. Classify 2nd order linear PDEs
8. Apply Laplace's and Fourier transform to solve ODEs and PDEs.
Intended learning outcomes - transferable/key skills and other attributes
Transferable/key competences and other abilities:
• Communication skills: unequivocal and precise expression.
• Computational skills: application of formulas and methods for computing (partial) differential equations.
• Problem solving: translation of problems from technique into mathematical form suitable for the use of analytical methods.
Readings
Osnovna/Basic:
- E. Kreyszig, Advanced Engineering Mathematics, J. Wiley and Sons, 2011.
- M. Mencinger, Uvod v parcialne diferencialne enačbe, Fakulteta za gradbeništvo UM, Maribor, 2011.
Dodatna/Additional:
- G. Tomšič, T. Slivnik, Matematika IV. Fakulteta za elektrotehniko, Založba FE in FRI, Ljubljana, 2004.
- F. John, Partial differential equations. Springer-Verlag, New York, 1991.
Additional information on implementation and assessment Type (examination, oral, coursework, project):
• Written exam (solving computational problems)
• Completed homeworks (to intend the written exam)
Oral exam - theoretica part of exam (if necessarily, in the case of increase of the grade).
The student must finish each part of the exam (written examination and homework) with at least 50%. In the case that a student is offered the opportunity to take a written exam in the form of two midterm tests, each must be graded with at least 30% and both with an average of at least 50% to pass the written part of the exam.
