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

Learning outcomes

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

1. Distinguish the difference between Ordinary Differential Equations and Fuzzy 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. Design and conduct research projects independently
   6. Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment
   7. Develop mathematical, communicative, problem-solving, brainstorming skills.
   8. Effectively express his/her research ideas and findings both orally and in writing
   9. Demonstrating professional and ethical responsibility.

   Method of assessment

   1. Written exam
   2. Oral exam
2. Obtain and Explain the Fundamental Definitions, Concepts, Theorems , Stability and Applications of Fuzzy 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. Design and conduct research projects independently

   Method of assessment

   1. Written exam
   2. Oral exam
   3. Homework assignment
3. Generalize, Empasize and Apply the concept of Theory of Ordinary Differential Equations to the Theory of Fuzzy Differential Equations

   Contribution to Program Outcomes

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

   Method of assessment

   1. Written exam
   2. Oral exam
   3. Homework assignment

Contents
Week 1:	Fuzzy Sets.
Week 2:	Calculus of Fuzzy Functions; Convergence of Fuzzy Sets, Measurability, Integrability, Differentiability.
Week 3:	Fundamental Theory; Initial Value Problem, Existence, Comparison Theorems, Convergence of Successive Approximations,
Week 4:	Fundamental Theory; Continuous Dependence, Global Existence, Approximate Solutions, Stability Criteria.
Week 5:	Lyapunov-like Functions;
Week 6:	Stability Criteria. Midterm Exam I.
Week 7:	Nonuniform Stability Criteria, Criteria for Boundedness, Fuzzy Differential Systems,
Week 8:	The Method of Vector Lyapunov Functions, Linear Variation of Parameters Formula.
Week 9:	Miscellaneous Topics; Fuzzy Difference Equations,
Week 10:	Impulsive Fuzzy Differential Equations,
Week 11:	Fuzzy Differential Equations with Delay,
Week 12:	Hybrid Fuzzy Differential Equations. Midterm Exam II.
Week 13:	Fixed Points of Fuzzy Mappings, Boundary Value Problem,
Week 14:	Fuzzy Equations of Volterra Type, A New Concept of Stability.
Week 15*:	-
Week 16*:	Final Exam.
Textbooks and materials:	Lakshmikantham V., Mohapatra R. Theory of Fuzzy Differential Equations and Inclusions (Taylor, 2003)(ISBN 0415300738)
Recommended readings:	Aumann, R.J. Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965) 1-12. Bernfeld, S. and Lakshmikantham, V. An Introduction to Nonlinear Boundary Value Problems. Academic Press, New York, 1974. 3 : Buckley, J.J. and Feuring, T.H. Fuzzy differential equations, Fuzzy Sets and Systems, 110 (2000), 43 -54. 4 : Ding, Z. , Ming, M. and Kandel, A. Existence of solutions of Fuzzy differential equations, Inform. Sci., 99 (1997), 205 - 217. 5 : Dubois, D. and Prade, H. Towards fuzzy differential calculus, Part I, Part II, Part III, Fuzzy Sets and Systems, 8 (1982), 1 - 17, 105 - 116, 225 - 234. 6 : Kaleva, O. Fuzzy differential equations. Fuzzy Sets and Systems 24 (1987) 301--317. 7 : Kaleva, O. On the calculus of fuzzy valued mappings, Appl. Math. Lett., 3 (1990), 55 - 59. 8 : Kaleva, O. The Cauchy problem for fuzzy differential equations. Fuzzy Sets and Systems 35 (1990) 389--396. 9 : Lakshmikantham, V. and Leela, S. Differential and Integral Inequalities, Vol. I. Academic Press, New York, 1969. 10 : Lakshmikantham, V. and Leela, S. Fuzzy differential systems and the new concept of stability. Nonlinear Dynamics and Systems Theory, 1 (2) (2001), 111-119 11 : Lakshmikantham, V. and Leela, S. A new concept unifying Lyapunov and orbital stabilities. Communications in Applied Analysis, (2002), 6 (2). 12 : Lakshmikantham, V. and Leela, S. Stability theory of fuzzy differential equations via differential inequalities. Math. Inequalities and Appl. 2 (1999) 551--559. 13 : Lakshmikantham, V., Leela, S. and Martynyuk, A.A. Stability Analysis of Nonlinear System. Marcel Dekker, New York, 1989. 14 : Lakshmikantham, V. and Mohapatra, R. Basic properties of solutions of fuzzy differential equations. Nonlinear Studies 8 (2001) 113--124. 15 : Lakshmikantham, V. and Mohapatra, R. N. Theory of Fuzzy Differential Equations and Inclusions . Taylor and Francis Inc. New York, 2003. 16 : Lakshmikantham, V. and Vatsala, A.S., Differential inequalities with time difference and application, Journal of Inequalities and Applications 3, (1999) 233-244. 17 : Li, A., Feng, E. and Li, S., Stability and boundedness criteria for nonlinear differential systems relative to initial time difference and applications. Nonlinear Analysis: Real World Applications 10 (2009) 1073--1080 18 : Lyapunov, A. Sur les fonctions-vecteurs completement additives. Bull. Acad. Sci. URSS, Ser. Math 4 (1940) 465-478. 19 : Massera, J.L. The meaning of stability. Bol. Fac. Ing. Montevideo 8 (1964) 405--429. 20 : Nieto, J.J. The Cauchy problem for fuzzy differential equations. Fuzzy Sets and Systems, (102 (1999), 259 - 262. 21 : Park, J.Y. and Hyo, K.H. Existence and uniqueness theorem for a solution of Fuzzy differential equations, Inter. J. Math.and Math. Sci, 22 (1999), 271-279. 22 : Puri, M. L.D. and Ralescu, A. Differential of Fuzzy functions, J. Math. Anal. Appl, 91 (1983), 552 - 558. 23 : 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. 24 : 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. 25 : Song, S.J. , Guo, L. and Feng, C.H. Global existence of solutions of Fuzzy differential equations, Fuzzy Sets and Systems, 115 (2000), 371 - 376. 26 : Song, S.J. and Wu, C. Existence and Uniqueness of solutions to Cauchy problem of Fuzzy differential equations, Fuzzy Sets and Systems, 110 (2000), 55 - 67. 27 : Yakar, C. Boundedness criteria with initial time difference in terms of two measures, Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 14, supplement 2, (2007) 270--274, . 28 : Yakar, C. Strict stability criteria of perturbed systems with respect to unperturbed systems in terms of initial time difference. Complex Analysis and Potential Theory, World Scientific, Hackensack, NJ, USA (2007) 239--248. 29 : Yakar, C. and Shaw, M. D. A comparison result and Lyapunov stability criteria with initial time difference. Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 12, no. 6, (2005) 731--737. 30 : Yakar, C. and Shaw, M. D. Initial time difference stability in terms of two measures and a variational comparison result. Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 15, no. 3, (2008) 417--425, . 31 : Yakar, C. and Shaw, M. D. Practical stability in terms of two measures with initial time difference. Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, (2009) e781--e785. 32 : Yoshizawa, T. Stability Theory by Lyapunov's second Method, The Mathematical Society of Japan, Tokyo, 1966. 33 : Zadeh, L.A. Fuzzy Sets, Inform. Control., 8 (1965), 338 - 353. 34 : Zhang, Y. Criteria for boundedness of Fuzzy differential equations, Math. Ineq. Appl., 3 (2000), 399 -410.
	* 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:		0
Project:		0
Homework:	2,3,4,5,8,9,10,11,13,14	5
Quiz:	5,11	5
Final exam:	16	50
	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	10
Term project:	0	0
Term project presentation:	0	0
Quiz:	1	2
Own study for mid-term exam:	15	1
Mid-term:	2	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)