Programmes → I.M.Sc. in Mathematical Sciences; M.Sc. Mathematics/Applied Mathematics

Functional Differential Equations

Course Code: MA 471; MM 596; AM 596 | Credits: 4 | Prerequisite: Differential Equations, linear-algrbra

Unit I: (Linear differential difference equations) Differential and difference equations, Retarded differential difference equations, Exponential estimates, Characteristic equation, Fundamental solution, Variation-of-constants formula, Neutral differential difference equations.
Unit II: (Retarded functional differential equations basic theory) Definition, Existence, uniqueness, and continuous dependence, Continuation of solutions, Differentiability of solutions, Backward continuation, Caratheodory conditions.
Unit III: (Properties of the solution map) Finite-or infinite-dimensional problem; Equivalence classes of solutions, Exponential decrease for linear systems, Unique backward extensions, Range in R. Compactness and representation.
Unit IV: (Autonomous and periodic processes) Processes, Invariance, Discrete systems maximal compact invariant sets, Fixed points of discrete dissipative processes, Stability and maximal invariant sets in processes, Periodic trajectories of a periodic processes, Convergent systems.
Unit V: (Stability theory) Definitions, The method of Lyapunov functional, Lyapunov functional for autonomous systems, Razumikhin-type theorems.
Unit VI: (General linear systems) Global existence and exponential estimates, Variation-of-constants formula, The formal adjoint equation, The true adjoint Boundary-value problems, Stability and boundedness; Neutral differential equations and applications.

Understand the difference and differential equations and neutral difference equations.
Understand the neutral functional differential equations and its basic theory, existence and uniqueness of solution of these problems and apply some standard techniques to find the solution of a given retarded functional differential equation.
Understand the properties of the solution map and Autonomous and periodic process.
Understand the stability theory of functional differential equations and use this concept to classify the solutions by using the Lyapunov function.
Understand the linear systems and neutral differential equations and its applications.

Hale, J. K., Theory of Functional Differential Equations, Springer-Verlag, 1977.
Kolmanovskii, V. B. and Nosov, V. R., Stability of Functional Differential Equations, Academic Press, 1986.