1. Home
2. Faculty of Arts and Sciences

MATH723 INT.TO DELAY DIFF.EQUATIONS

Course Objectives

Differential equations with a deviating argument are equations where the unknown function and its derivatives are evaluated at different values of the time. These equations are used to describe various processes that exhibit aftereffects: they commonly appear in fields such as physics and engineering. One example is a problem involving a force acting on a material point, which depends not only on the point's current velocity and position but also on its values at some earlier moment. The course covers standard results regarding the existence and uniqueness of solutions, and introduces the method of steps. Fixed-point theorems are utilized, demonstrating that the process of successive approximations is effective.

Additionally, we address essential concepts such as the continuous dependence of solutions on initial conditions and the continuation of solutions. We then shift our focus to linear equations, primarily examining stability. In practical applications, linear delay equations often arise from linearizing a nonlinear equation around an equilibrium point. Therefore, the emphasis is placed on linear stability analysis and the characteristic equation, as its roots are critical in determining stability. Finally, methods for the study of periodic systems and the existence of periodic solutions are introduced. 

Course Content

Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.

Course Learning Outcomes

By the end of the course, the student will learn the following components of the theory: 

• Statement of the basic initial value problem.  Existence and uniqueness  theorems
• Classification: delay, advanced, and neutral types of differential equations with deviating arguments. The method of steps. Integrable types of equations with a deviating argument
• Differential equations with piecewise constant argument
• Some properties of linear equations. Equations with constant coefficients and constant deviations. Some types of linear equations with variable coefficients and variable deviations
• Basic concepts of the stability theory
• Lyapunov's second method
• Stability in the first approximation
• Some properties of periodic solutions and existence theorems
• Periodic solutions of autonomous, linear, and homogeneous equations
• Periodic solutions of linear equations with variable coefficients and deviations
• Periodic solutions of quasilinear equations

Program Outcomes Matrix

0: No Contribution 1: Little Contribution 2: Partial Contribution 3: Full Contribution