Discrete Random Process: Expectation, Variance and Co‐variance, Uniform, Gaussian and Exponentially distributed noise, Hillbert space and inner product for discrete signals, Energy of discrete signals, Parseval’s theorem, Wiener Khintchine relation, power spectral density, Sum decomposition theorem, Spectral factorization theorem.

Introduction of Soft-computing tools - Neural Networks, Fuzzy Logic, Genetic Algorithm, and Probabilistic Reasoning; Neural network approaches in engineering analysis, design, and diagnostics problems; Applications of Fuzzy Logic concepts in Engineering Problems; Engineering optimization problem-solving using genetic algorithm; applications of probabilistic reasoning approaches.

Mathematical review and computer arithmetic – numbers and errors; Nonlinear equations; Direct methods for linear systems; Iterative Methods for Linear Systems; Eigenvalues and Eigenvectors – power method, inverse power method, QR method; Approximation Theory – norms, orthogonaliza- tion, polynomial approximation, piecewise polynomial approximation, trigonometric approximation, rational approximation, wavelet bases; Numerical Differentiation; Numerical Integration- Romberg Integration, Gauss Quadrature, Adaptive Quadrature; Numerical Ordinary Differential Equations – single step and multi-st

Vector Space, norm, and angle – linear independence and orthonormal sets – row reduction and echelon forms, matrix operations, including inverses – effect of round-off error, operation counts – block/banded matrices arising from discretization of differential equations – linear dependence and independence – subspaces and bases and dimensions – orthogonal bases and orthogonal projections – Gram-Schmidt process – linear models and least-squares problems – eigenvalues and eigenvectors – diagonalization of a matrix – symmetric matrices – positive definite matrices – similar matrices – linear tr

Introduction: Simulation classification – Objectives, concepts, and types of models – Modeling: 6- DOF models for aerospace vehicle with prescribed control surface inputs – Control Systems: Mechan- ical (structural), Hydraulic systems and their modeling – Block diagram representation of systems – Dynamics of aerospace vehicles – Pilot station inputs – Cues for the Pilot: Visual, biological, and stick force – Virtual Simulation: Fly-by-wire system simulation – Uncertainty Modeling & Simula- tion: Characterization of uncertainty in model parameters and inputs, use of simulation to propaga

Fundamentals of matrix algebra and vector spaces – Solution of simultaneous equations for square, under-determined, and over-determined systems – Concepts of basis vector transformations – Sim- ilarity and adjoint transformations – Eigenvalues and eigenvectors – Jordan form – Characteristic equation – Analytic functions of square matrices and Cayley-Hamilton theorem – Concepts of state, state-space, state-vector – Methods for obtaining the system mathematical model in the state-space form – State-space Form for Aerospace Systems: Aircraft dynamics, missile dynamics, inertial navi- gation sy

Panel methods – Unsteady potential flows – Compressible flow over wings – Axisymmetric flows and slender body theories – Boundary layer analysis – Viscous-inviscid coupling – Flight vehicle aerody- namics – Rotor aerodynamics – Low Reynolds number aerodynamics – Flapping wings – Two- and three-dimensional flow separation.

Introduction to stability – Review of dynamical systems concepts – Instabilities of fluids at rest – Stability of open shear flows: Inviscid and viscous theory, spatio-temporal stability analysis (absolute and convective instabilities) – Parabolized stability equation – Transient growth – Introduction to global instabilities.

Governing equations for viscous fluid flow – Heat conduction and compressibility – Exact solutions – Laminar boundary layer approximations – Similar and nonsimilar boundary layers – Momentum integral methods – Separation of boundary layer – Compressible boundary layer equations – Recov- ery factor – Reynolds analogy – Similar solutions – Stability of boundary layer flows: Transition prediction and bypass transition – Turbulent Flows: Phenomenological theories – Reynolds stress – Turbulent boundary layer – Momentum integral methods – Turbulent free shear layer – Flow separation.

Multidisciplinary Design Optimization (MDO): Need and importance – Coupled systems – Analyser vs. evaluator – Single vs. bi-level optimisation – Nested vs.

General features and applications of high temperature flows – Equilibrium Kinetic Theory: Maxwellian distribution, collision rates and mean free path – Chemical thermodynamics – Mixture of perfect gases, law of mass action – Statistical Mechanics: Enumeration of micro-states, energy distribution, contribution of internal structure – Equilibrium Flow: Ideal dissociating gas, equilibrium shock wave relations, nozzle flows – Vibrational and chemical rate processes – Flows with vibrational and chemical non-equilibrium.

Launch vehicle ascent trajectory design – Reentry trajectory design – Low thrust trajectory design – Satellite constellation design – Rendezvous mission design – Ballistic lunar and interplanetary trajectory design – Basics of optimal control theory – Mission design elements for various missions – Space flight trajectory optimization – Direct and indirect optimization techniques – Restricted 3-body problem – Lagrangian points – Mission design to Lagrangian point.