Course Information

Course Name	Turkish	Diferansiyel Denklemlerde İleri Konular
	English	Advanced Topics in ODE
Course Code	MAT 391E	Credit	Lecture (hour/week)	Recitation (hour/week)	Laboratory (hour/week)
Semester	-	3	3	-	-
Course Language		English
Course Coordinator		Semra Ahmetolan
Course Objectives		1. Teach the basics on stability of dynamical systems. 2. To develop a basic understanding of occurrence of two point boundary value problems, their classification and related problems; such as, initial value, boundary value and initial-boundary value problems in the real world. 3. To develop a basic understanding of the theory and methods of solutions for these problems.
Course Description		Nonlinear Differential Equations and Stability: The Phase Plane-Linear Systems, Autonomous Systems and Stability, Locally Linear Systems, Competing Species, Predator-Prey Equations, Liapunov’s Second Method, Periodic Solutions and Limit Cycles, Chaos and Strange Attractors: The Lorenz Equations. Two-point boundary-value problems; definition, examples, existence and uniqueness of solutions. Linear homogeneous boundary-value problems; eigenvalues and eigenvectors. Sturm-Liouville boundary-value problems; Lagrange identity, orthogonality of eigenfunctions, self-adjoint problems. Nonhomogeneous boundary-value problems; non-homogeneous Sturm-Liouville problems, non-homogeneous heat conduction problems. Singular Sturm-Liouville problems; definition, continuous spectrum, vibration of a circular elastic membrane, Series of orthogonal functions; convergence and completeness. Techniques of Green`s function; generalised functions, Green`s function, modified Green`s function.
Course Outcomes		Students completing this course will be able to: I. Draw the phase portraits of linear systems and determine the qualitative behavior of linear systems, II. Do the local stability analysis of nonlinear dynamical systems, III. Get a deep understanding of dynamical systems like predator-prey equations that find applications in many different areas IV. Apply the Liapunov function method in stability analysis, V. Have the basics in the analysis of systems that exhibit periodic behavior in their solutions and understand bifurcation theory, VI. Recognize the occurrence of two point boundary value problems in mathematical physics and applied mathematics and clasify them as regular, singular and periodic problems. VII. Define eigenvalues and eigenfunctions of homogeneous Sturm-Liouville boundary value problems and know properties of them. VIII. Solve nonhomogeneous two point boundary value problems by eigenfunction expansion method. IX. Solve nonhomogeneous heat conduction problems ( initial and boundary value problems ) by eigenfunction expansion method. X. Get an understanding of the singular Sturm-Liouville problems, XI. Solve boundary value problems defined for more general equations with differend boundary geometries.
Pre-requisite(s)		MAT232 MIN DD / MAT232E MIN DD / MAT201 MIN DD /MAT201E MIN DD / MAT 210 MIN DD / MAT 210E MIN DD
Required Facilities
Other
Textbook		Boyce, W. and Di Prima, R.; Elementary Differential Equations and Boundary Value Problems, 10th Ed., Wiley, New York, 2012.
Other References		Strogatz, S..H., Nonlinear Dynamics and Chaos, with Applications to Physics, Biology, Chemistry, and Engineering, 2nd Ed., 2014, Westview Press. Powers, D.L.; Boundary Value Problems, San Diego, Harcourt Brace Janovich, 1987. Greenberg, M.D.; Application of Green`s Functions in Science and Engineering, Prentice Hall,1971. Stakgold, I.; Green`s Functions and Boundary Value Problems, John Wiley, New York, 1979.

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