Introduction to the Course and Some Applications of Difference Equations in Engineering, Preliminaries in linear algebra and analysis, Analogies between differential and difference equations, Elementary Difference Operations: the Difference and the Shift operators, The Difference and Summation Calculus, Linear difference equations, First order equations, Higher Order Difference Equations, Linear difference equations with constant coefficients, Linear difference equations with variable coefficients, Method of undetermined coefficients and variation of parameters, Generating functions, The z-transform and its applications, Systems of Linear difference equations and applications, The Sturmian theory and Fourier techniques, Asymptotic methods, Limiting behavior of solutions, Nonlinear difference equations and boundary value problems, Stability theory and relevance to dynamical systems, Partial Difference Equations, Differential-difference equations, Discrete Mechanics, Open problems.

Lecture wise Breakup

I. Lecture 1 (1 Lecture):

• Introduction and Some Applications of Difference Equations in Engineering (1 lecture): Overview of the course. Discussion on some applications, to mechanical systems, of finite difference schemes, Some applications of finite difference schemes in understanding dynamics of discrete mechanical systems as well as discretized systems.

II. Lecture 2–4 (3 Lectures): (Preliminaries in linear algebra and analysis)

• Sets, Mappings and some Sequence Spaces with useful Structure, `p spaces on finite and infinite grids (1 lecture)
• Review of linear algebra and vector spaces, Definition of Banach space and Hilbert space, Discretespaces, Some motivating examples from discrete mechanics, particle mechanics etc, involving such spaces (1 lecture)
• Analogies between: differential and difference equations, integration and summation, Interpolation,Extrapolation (1 lecture)

III. Lecture 5–12 (8 Lectures): (The Difference and Summation Calculus)

• The Difference operator, combinatorial identities (1 lecture)
• The Shift operator, Symbolic calculus (2 lectures)
• Relations between Shift and Difference Operators, Stirling numbers, Factorial polynomial, Taylor’sTheorem and Lagrange identities (3 lectures)
• The Summation operator and applications, Bernoulli polynomials, summation by parts, symbolic operators (2 lectures)

IV. Lecture 13–22 (10 Lectures): (Linear difference equations)

• First order equations, the simplest difference equation (2 lectures)
• Higher Order Difference Equations, existence and uniqueness theorem (1 lecture)
• Linear difference equations with constant coefficients, characteristic roots, fundamental set of solutions(1 lectures)
• Method of undetermined coefficients and variation of parameters, application of symbolic calculus (2 lectures)
• Method of Generating functions (1 lecture)
• Method of z-transform and its applications (1 lecture)
• Linear difference equations with variable coefficients, Gamma function, factorial series, inverse factorialseries (2 lectures)

V. Lecture 23–27 (5 Lectures): (Systems of Linear difference equations)

• As a Generalization of high order difference equations (1 lecture)
• Matrix based methods for constant coefficients (2 lecture)
• Matrix based methods for variable coefficients, applications to mechanics (1 lecture) 4. Method of variation of parameters (1 lecture)

VI. Lecture 28–32 (5 Lectures): (The Sturmian theory and Fourier analysis)

• Adjoint problem, Self-adjoint operator (1 lecture)
• discrete Sturm-Liuoville problems, Comparison and Separation Theorems (2 lectures)
• Eigenvalues and Eigenfunctions, separated boundary conditions and periodic boundary conditions,Spectral Theorem (1 lecture)
• applications to mechanics (1 lecture)

VII. Lecture 33–36 (4 Lectures): (Asymptotics)

• Limiting behavior of solutions (2 lecture): Poincar´e Theorem, Levinson Theorem
• Stability theory and relevance to dynamical systems (1 lecture)
• Nonlinear difference equations and boundary value problems (1 lecture)

VIII. Lecture 37–38 (2 Lectures): (Partial Difference Equations)

• Taylor’s theorem, Partial Difference Equations in two dimensions (1 lecture)
• Partial Difference Equations in arbitrary dimensions with Constant coefficients (1 lecture)

IX. Lecture 39–40 (2 Lectures): (Special Topics)

• Difference Equations with Continuous Time, Differential-difference equations (1 lecture)
• Lagrangian and Hamiltonian Formalism for Difference Equations. Discrete Symmetry, Discrete Mechanics, Open problems in difference equations (1 lecture)

References:

1. An Intoduction to Difference Equations, S. N. Elaydi (Springer) (Textbook)
2. Difference Equation, W. G. Kelley and A. C. Peterson (Academic Press)
3. Finite Difference Equations. H. Levy and F. Lessman (MacMillan)
4. A treatise on the Calculus of Finite Differences. G. Boole. (MacMillan)