Course Title: Fractional order models and fractional differential equations in science and engineering
Instructor: Igor Podlubny
Office: L 178
Phone: 797-7118 (Office)
E-mail Address: igor.podlubny@tuke.sk
Office Hours: M W F 10:00–11:30. Other hours by appointment.
Lecture Time: Tu Th 12:30–14:50
Lecture Place: ENGR 204
Pre-requisites: standard calculus and basic numerical methods.
Text: Podlubny I. Fractional Differential Equations . San Diego: Academic Press; 1999 (view at publisher );
Magin R. Fractional Calculus in Bioengineering. Begell House Inc., Redding; 2006 (view at publisher )
Final Exam: 12:30–14:00 on Thursday, August 2. (still subject to change)
The course is aimed on introducing the methods and tools of the fractional-order calculus into engineering education.
Gamma Function. Mittag-Leffler Function. Wright Function. (slides in PDF ) (Homework #1 )
Grünwald-Letnikov Fractional Derivatives. Riemann-Liouville Fractional Derivatives. Some Other Approaches. Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation. (slides in PDF )
Sequential Fractional Derivatives. Left and Right Fractional Derivatives. Properties of Fractional Derivatives. Laplace Transforms of Fractional Derivatives. Fourier Transforms of Fractional Derivatives. Mellin Transforms of Fractional Derivatives. (slides in PDF ) (Homework #2 )
Fractional Differential Equation of a General Form. Existence and Uniqueness Theorem as a Method of Solution. Dependence of a Solution on Initial Conditions. (slides in PDF )
The Laplace Transform Method. Standard Fractional Differential Equations. Sequential Fractional Differential Equations. (slides in PDF ) (Homework #3 )
Definition and Some Properties. One-Term Equation. Two-Term Equation. Three-Term Equation. Four-Term Equation. General Case: n-term Equation.
Power Series Method. (slides in PDF) (Homework #4 )
Babenko's Symbolic Calculus Method. Method of Orthogonal Polynomials. The Mellin Transform Method.
Approximation of Fractional Derivatives. The "Short-Memory" Principle. Calculation of Heat Load Intensity Change in Blast Furnace Walls. Order of Approximation. Computation of Coefficients. Higher-order Approximations. (slides in PDF)
Initial Conditions: Which Problem to Solve? Numerical Solution. Examples of Numerical Solutions. The "Short-Memory" Principle in Initial Value Problems for Fractional Differential Equations. Matrix approach to discrete fractional calculus. Numerical solution of nonlinear problems. (slides in PDF) (more slides in PDF)
Podlubny, I., Heymans, N.: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheologica Acta. vol. 45, 2006, pp. 765–771.
Podlubny I.: Fractional-order systems and PIλDμ–controllers, IEEE Transactions on Automatic Control, vol. 44, no. 1, January 1999, pp. 208-213 .
Podlubny, I., Petras, I., Vinagre, B.M., O'Leary P., Dorcak L.: Analogue realizations of fractional-order controllers. Nonlinear Dynamics, vol. 29, no. 1–4, 2002, pp. 281–296 .
Podlubny, I.: Matrix approach to discrete fractional calculus. Fractional Calculus and Applied Analysis, vol. 3, no. 4, 2000, pp. 359–386 .
Carpinteri A, Mainardi F, editors. Fractals and fractional calculus in continuum mechanics. CISM Courses and Lectures no. 378. International Center for Mechanical Sciences. New York: Springer-Verlag Wien; 1997 .
Mandelbrot BB. The fractal geometry of nature. New York: W. H. Freeman; 1982 .
Miller KS, Ross B. An introduction to the fractional calculus. New York: John Wiley; 1993 .
Oldham KB, Spanier J. The fractional calculus. New York: Academic Press; 1974 .
West BJ, Bologna M, Grigolini P. Physics of fractal operators. New York: Springer; 2003 .
http://people.tuke.sk/igor.podlubny/
http://people.tuke.sk/igor.podlubny/USU/
http://people.tuke.sk/igor.podlubny/fc.html
It is department pedagogical philosophy that students are responsible for their own learning. The instructor may not cover all of the material in each reading assignment in the lecture period. The student is therefore responsible for asking questions about reading material not covered in the lecture. Questions on exams may come from lectures, computer assignments, reading assignments, or supplementary materials given in class. Homework is due at the beginning of class on the due date. No late homework will be graded.
Scores will be weighted as follows:
Homework & computer assignments 40%
Midterm 20%
Final exam 40%
Total 100%
A > 93%
A- > 90%
B+ > 87%
B > 84%
B- > 80%
C+ > 77%
C > 74%
C- > 70%
D+ > 67%
D > 64%
D- > 60%
F < 60%