• Basic information
• Learning outcomes
• Contents
• Assessment
• Workload

Learning outcomes

Upon successful completion of this course, students will be able to:

1. Explain the basic concepts of Fractional order derivatives

   Contribution to Program Outcomes

   1. Define and manipulate advanced concepts of Mathematics
   2. Develop mathematical, communicative, problem-solving, brainstorming skills.

   Method of assessment

   1. Written exam
2. Obtain and Explain the Fundamental Definitions, Concepts, Theorems , Stability and Applications of Fractional Dynamical Systems

   Contribution to Program Outcomes

   1. Define and manipulate advanced concepts of Mathematics
   2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
   3. Acquire scientific knowledge and work independently
   4. Design and conduct research projects independently
   5. Develop mathematical, communicative, problem-solving, brainstorming skills.

   Method of assessment

   1. Written exam
3. Gain Experience on Fractional order Differential Equations

   Contribution to Program Outcomes

   1. Define and manipulate advanced concepts of Mathematics
   2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
   3. Acquire scientific knowledge and work independently
   4. Work effectively in multi-disciplinary research teams
   5. Develop mathematical, communicative, problem-solving, brainstorming skills.

   Method of assessment

   1. Homework assignment
   2. Seminar/presentation
   3. Term paper
4. Generilaze, Empasize and Apply the concept of Theory of Ordinary Differential Equations to the Fractional order Differential Equations

   Contribution to Program Outcomes

   1. Define and manipulate advanced concepts of Mathematics
   2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
   3. Acquire scientific knowledge and work independently
   4. Develop mathematical, communicative, problem-solving, brainstorming skills.
   5. Effectively express his/her research ideas and findings both orally and in writing

   Method of assessment

   1. Seminar/presentation
   2. Term paper
5. Interpret the Stability results and Applications of Fractional Dynamical Systems

   Contribution to Program Outcomes

   1. Define and manipulate advanced concepts of Mathematics
   2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
   3. Acquire scientific knowledge and work independently
   4. Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment
   5. Develop mathematical, communicative, problem-solving, brainstorming skills.
   6. Demonstrating professional and ethical responsibility.

   Method of assessment

   1. Homework assignment

Contents
Week 1:	Some related functions: Gamma function, Beta function and Mittag-Leffler function. Riemann-Liouville(R-L) fractional integral and derivative of arbitrary order.
Week 2:	Grünwald-Letnikov’s (G-L) definition of fractional derivative. Caputo’s definition of derivative. Mean Value Theorem of fractional derivatives. Dini derivatives and Comparison theorems.
Week 3:	Volterra type fractional integral inequalities. Fractional differential inequalities.
Week 4:	Local existence Theorem and extremal solutions for the IVP. Existence, Uniqueness and Continuous dependence. Approximate solutions and global existence.
Week 5:	Linear fractional order nonhomogeneous differential equations. Finite systems of fractional order differential equations. Existence of Euler solution.
Week 6:	Caputo’s fractional order differential equation. Midterm Exam I.
Week 7:	Theoretical and constructive existence result in a Sector. Generalized monotone iterative technique. Monotone Method for Periodic Boundary Value Problems. Generalized Monotone Iterative Technique for PBVP.
Week 8:	Quasilinearization Method. Generlaized Quasilinearization Method. Stability Criteria. Proximal normal and flow invariance. Relation between fractional and ordinary differential equations.
Week 9:	Basic comparison results of IVP of Caputo’s fractional differential equation.
Week 10:	Stability Criteria and Stability Criteria in terms of two measures.
Week 11:	Boundedness and Lagrange stability in terms of two measures. Several Lyapunov functions. Multi-order fractional differential systems. Stability of multi-order systems via ODEs.
Week 12:	Fractional functional differential equations. Midterm Exam II.
Week 13:	Fractional differential equations involving Causal operators. Fractional Differential Equations in a Banach Space.
Week 14:	Nonlocal boundary value problems. Positive solutions for BVP. BVP for fractional differential inclusions. Almost automorphic solutions of fractional evolution equation.
Week 15*:	-
Week 16*:	Final Exam.
Textbooks and materials:	Lakshmikantham, V. Leela, S. and Vasundhara Devi, J. Theory of Fractional Dynamical Systems,Cambridge Scientific Publishers 2009.
Recommended readings:	Caputo, M. , Linear models of dissipation whose Q is almost independent, II, Geophy. J. Roy. Astronom. 13 (1967) 529--539. 2 : Brauer, F. and Nohel, J., The Qualitative Theory of Ordinary Differential Equations, W.A. Benjamin, Inc., New York 1969. 3 : Oldham, K. B. and Spanier, J., The Fractional Calculus, Academic Press, New York, 1974. 4 : Kilbas, A. A., Srivastava, H. M., Trujillo, J. J. Theory and Applications of Fractional Differential Equations, Amsterdam, The Netherlands, 2006. 5 : Lakshmikantham, V. and Leela, S., Differential and Integral Inequalities, Vol. 1, Academic Press, New York 1969. 6 : Lakshmikantham, V., Leela, S. and Martynyuk, A.A., Stability Analysis of Nonlinear Systems, Marcel Dekker, New York 1989. 7 : Lakshmikantham, V. Leela, S. and Vasundhara Devi, J. Theory of Fractional Dynamical Systems,Cambridge Scientific Publishers 2009. 8 : Lakshmikantham, V. and Vatsala, A.S., Differential inequalities with time difference and application, Journal of Inequalities and Applications 3, (1999) 233-244. 9 : Lakshmikanthama, V. and Vatsala, A.S. Basic theory of fractional differential equations. Nonlinear Analysis 69 (2008) 2677--2682 10 : Podlubny, I., Fractional Differential Equations, Acad. Press, New York 1999. 11 : Samko, S., Kilbas, A. and Marichev, O. Fractional Integrals and Derivatives:Theory and Applications, Gordon and Breach, 1993, 1006 pages, ISBN 2881248640. 12 : Shaw, M.D. and Yakar, C., Generalized variation of parameters with initial time difference and a comparison result in term Lyapunov-like functions, International Journal of Non-linear Differential Equations-Theory Methods and Applications 5, (1999) 86-108. 13 : Shaw, M.D. and Yakar, C., Stability criteria and slowly growing motions with initial time difference, Problems of Nonlinear Analysis in Engineering Systems 1, (2000) 50-66. 14 : Yakar, C. Boundedness Criteria in Terms of Two Measures with Initial Time Difference. Dynamics of Continuous, Discrete and Impulsive Systems. Series A: Mathematical Analysis. Watam Press. Waterloo. Page: 270-275. DCDIS 14 (S2) 1-305 (2007). 15 : Yakar C., Strict Stability Criteria of Perturbed Systems with respect to Unperturbed Systems in term of Initial Time Difference. Proceedings of the Conference on Complex Analysis and Potential Theory. World Scientific Publishing. Page: 239-248 (2007). 16 : Yakar C. and Shaw, M.D., A Comparison Result and Lyapunov Stability Criteria with Initial Time Difference. Dynamics of Continuous, Discrete and Impulsive Systems. A: Mathematical Analysis. Volume 12, Number 6 (2005) (731-741). 17 : Yakar C. and Shaw, M.D., Initial Time Difference Stability in Terms of Two Measures and Variational Comparison Result. Dynamics of Continuous, Discrete and Impulsive Systems. Series A: Mathematical Analysis 15 (2008) 417-425. 18 : Yakar C. and Shaw, M.D., Practical stability in terms of two measures with initial time difference. Nonlinear Analysis: Theory, Methods & Applications. Vol. 71 (2009) e781-e785. 19 : Yakar C., Fractional Differential Equations in Terms of Comparison Results and Lyapunov Stability with Initial Time Difference. Abstract and Applied Analysis. (Accepted) Vol 3. Volume 2010, Article ID 762857, 16 pages doi:10.1155/2010/762857. (2010)
	* Between 15th and 16th weeks is there a free week for students to prepare for final exam.

Assessment
Method of assessment	Week number	Weight (%)
Mid-terms:	6, 12	40
Other in-term studies:	7,13	5
Project:	8,14	5
Homework:	1,2,3,4,9,10,13,15	5
Quiz:	5,11	5
Final exam:	16	40
	Total weight:	(%)

Workload
Activity	Duration (Hours per week)	Total number of weeks	Total hours in term
Courses (Face-to-face teaching):	3	14
Own studies outside class:	3	14
Practice, Recitation:	0	0
Homework:	6	8
Term project:	10	2
Term project presentation:	1	1
Quiz:	1	2
Own study for mid-term exam:	10	1
Mid-term:	1	2
Personal studies for final exam:	15	1
Final exam:	2	1
		Total workload:
		Total ECTS credits:	*
	* ECTS credit is calculated by dividing total workload by 25. (1 ECTS = 25 work hours)