ECE 5930 003 / ECE 6930 003  

Fractional order models and fractional differential equations  

in science and engineering 


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)


Course summary 

The course is aimed on introducing the  methods and tools of the fractional-order calculus into engineering education.

Course syllabus  (PDF)

Course outline (slides in PDF )

Overview. 

History, theory, applications. (slides in PDF )

Special Functions of the Fractional Calculus.  

Gamma Function. Mittag-Leffler Function. Wright Function. (slides in PDF ) (Homework #1 )

Fractional Derivatives and Integrals

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 )

Linear Fractional Differential Equations.  

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 )

Fractional Green's Function.  

Definition and Some Properties. One-Term Equation. Two-Term Equation. Three-Term Equation. Four-Term Equation. General Case: n-term Equation.

Other Methods for the Solution of Fractional-order Equations.  

Power Series Method.  (slides in PDF(Homework #4 )

Babenko's Symbolic Calculus Method. Method of Orthogonal Polynomials. The Mellin Transform Method.

Numerical Evaluation of Fractional Derivatives. 

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)

Numerical Solution of Fractional Differential Equations. 

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)

Applications.   

        Fractional order systems and controllers. (slides in PDF)         Some available MATLAB routines and samples.

Supplemental references: 

  1. 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.  

  2. Podlubny I.: Fractional-order systems and PIλDμ–controllers, IEEE Transactions on Automatic Control, vol. 44, no. 1, January 1999, pp. 208-213

  3. 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 .  

  4. Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. Fractional Calculus and Applied Analysis, vol. 5, no. 4, 2002, pp. 367–386

  5. Podlubny, I.: Matrix approach to discrete fractional calculus. Fractional Calculus and Applied Analysis, vol. 3, no. 4, 2000, pp. 359–386

  6. 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

  7. Magin RL, Fractional Calculus in Bioengineering, Critical Reviews in Biomedical Engineering, Part I 32(1): 1-104, 2004, Part II 32(1) : 105-193, 2004, Part III 32(1)  

  8. Mandelbrot BB. The fractal geometry of nature. New York: W. H. Freeman; 1982

  9. Miller KS, Ross B. An introduction to the fractional calculus. New York: John Wiley;  1993 .

  10. Oldham KB, Spanier J. The fractional calculus. New York: Academic Press; 1974

  11. West BJ, Bologna M, Grigolini P. Physics of fractal operators. New York: Springer;  2003

Web site: 

        http://people.tuke.sk/igor.podlubny/

        http://people.tuke.sk/igor.podlubny/USU/

        http://people.tuke.sk/igor.podlubny/fc.html

Homework  

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. 

Grading  

Scores will be weighted as follows:  

Homework & computer assignments         40%

Midterm                                                     20%

            Final exam                                                 40%

Total                                                        100%

Grades will be computed according to the following scale:  

A         > 93%

A-        > 90%

B+       > 87%

B         > 84%

B-        > 80%

C+       > 77%

C         > 74%

C-        > 70%

D+       > 67%

D         > 64%

D-        > 60%

F          < 60%