Dynamical Equations on Time Scales

Syllabus Details

• Unit I: (The Time Scale Calculus) Basic definitions, Differentiation, Examples and applications, Integration, Chain rule, Polynomials, Basic results.
• Unit II: (First Order Linear Equations) Hilger’s complex plane, Exponential functions and examples, Initial value problems.
• Unit III: (Second Order Linear Equations) Wronskians, Hyperbolic and Trigonometric functions, Reduction of order, Method of factoring, Non-constant coefficients, Euler-Cauchy Equations, Variation of parameters, Annihilator method, Laplace transform.
• Unit IV: (Self Adjoint Equations) Preliminaries and examples, Riccati equation, Disconjugacy, Boundary value problems and Green’s function, Eigenvalue problems.
• Unit V: (Linear Systems and Higher Order Equations) Regressive matrices, Constant coefficients, Self-adjoint Matrix equations, Asymptotic behavior of solutions, Higher order linear dynamic equations.
• Unit VI: (Dynamic Inequalities) Gronwall’s inequality, Holder’s and Minkowski’s inequalities, Jensen’s inequality, Opial’s inequalities, Lyapunov inequalities, Upper and Lower solutions.

Course Learning Objective

1. Understand the time scale calculus such as continuity, differentiation, integration etc. and basic results related to this.
2. Understand the first and second order dynamic equations on time scales and find its solutions by using different techniques.
3. Apply time scale calculus to study the results related to BVPs, eigenvalue problems, and Green’s functions.
4. Understand the solutions of the linear systems and higher order equations and analyze qualitative behavior of the solutions.
5. Derive the different inequalities by using time scale calculus.

Recommended Books

1. Bohner, Martin and Peterson, Allan., Dynamic Equations On Time Scales: An Introduction with Applications, Birkhäuser, 2001.
2. Bohner, Martin and Peterson, Allan., Advances in Dynamic Equations On Time Scales, Birkhäuser, 2002.