Course Syllabus

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. Lectures 1-2:

• Revision of continuum mechanics (kinematics, singular surfaces, compatibility)

II. Lectures 3-4:

• Revision of continuum mechanics (balance laws, stress)

III. Lectures 5-6:

• The first law of thermodynamics (energy, work, heat)

IV. Lectures 7-10:

• The second law of thermodynamics (temperature, entropy, Clausius-Duhem inequality)

V. Lectures 11-13:

• Constitutive theory (frame indifference, material symmetry, equilibrium response)

VI. Lectures 14-16:

• Dynamic response (Onsager’s relations, dissipation potential, maximum dissipation principle)

VII. Lectures 17-19:

• Thermodynamic equilibrium and stability

VIII. Lectures 20-23:

• Rigid heat conductors (Fourier’s law, second sound, problems)

IX. Lectures 24-27:

• Thermoelasticity (linearized response, boundary value problems in the decoupled theory)

X. Lectures 28-31:

• Thermoplasticity (simple phenomenological models with work hardening, onedimensional problems, adiabatic shear bands)

XI. Lectures 32-33:

• Thermodynamics of surfaces (Gibbs concept of excess quantities, surface energy, surface tension, surface stress, capillarity)

XII. Lectures 34-35:

• Cahn-Hilliard type of theories for diffusive interfaces

XIII. Lectures 36-40:

• Simple one-dimensional problems related to dynamic propagation of phase boundaries and adiabatic shock waves

References:

1. The Mechanics and Thermodynamics of Continuous Media, M. Sˇilhavy ́, Springer, 1997.
2. An Introduction to Thermomechanics, H. Ziegler, North-Holland, 1983.
3. The Mechanics and Thermodynamics of Continua, M. E. Gurtin, E. Fried, and L. Anand, Cambridge, 2010.
4. Theory of Thermal Stresses, B. A. Boley and J. H. Weiner, Dover, 1997.