JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

Pre-requisites

Mathematics courses in first two years of study

Objectives

To learn

• Most of the random situations are described as functions of many single random variables. In this unit, the objective is to learn functions of many random variables through joint distributions and ANOVA
• The random processes, The classification of random processes, Markov chain, Classification of states
• Stochastic matrix (transition probability matrix) Limiting probabilities, Applications of Markov chains
• Various methods to find the roots of an equation
• Fitting linear, non- linear and exponential curves for the given data.
• The concept of interpolation and fitting a interpolating polynomial.
• Concept of finite differences and to estimate the value for the given data using interpolation.
• Finding the derivatives and proper integrals of given functions using finite differences.
• Solving ordinary differential equations using numerical techniques. Course outcomes: After learning the contents of this paper the student must be able to
• Formulate and solve problems involving the multiple random variables and the ANOVA for analysing the experimental data.
• The student would able to understand about the random process, Markov process and Markov chains which are essentially models of many time dependent processes such as signals in communications, time series analysis, queuing systems. The student would be able to find the limiting probabilities and the probabilities in nth state.
• Find a better approximate root of a given equation.
• Fit a linear, non-linear and exponential curve for the given data.
• Find the finite difference operators in a given data or value of the dependent variable for a given independent variable
• Evaluate the derivative at a given value and integral of a function.
• Solve the initial value problems.

UNIT - I: Multiple Random variables, Correlation & Regression

Joint probability distributions- Joint probability mass / density function, Marginal probability mass / density functions, Covariance of two random variables- ANOVA for one-way classified data.

UNIT - II: Stochastic processes

Introduction to Stochastic Processes –Classification of Random processes, Methods of description of random processes-Stationary and non-stationary random process-Average values of single random process and two or more random processes. Markov process, Markov chain, classification of states – Examples of Markov Chains, Stochastic Matrix.

UNIT - III: Solution of Algebraic, Transcendental Equations and Curve fitting

Bisection method, Regula-Falsi method, Iteration method, Newton-Raphson method. Curve Fitting: Method of least squares, Fitting a straight line, Second degree parabola and Non-linear bx curves of the form y = ae , y = ab x , y = ax b by the method of least squares.

R18 B.TECH. AERONAUTICAL ENGG.

UNIT - IV: Interpolation and Numerical Differentiation

Finite differences: Forward, Backward and Central differences, - Newton’s interpolation formulae, Central difference interpolation formulae: Gauss’s forward and backward interpolation formulae, Interpolation with unequal intervals: Lagrange’s interpolation formula. Numerical differentiation: Derivatives using Newton’s interpolation formulae.

UNIT - V: Integration and Solution of Ordinary Differential Equations of First Order

Trapezoidal rule Simpson’s 1/3rd and 3/8th rule- Solution of ordinary differential equations by Taylor’s series, Picard’s method of successive approximations, Euler’s method, Runge-Kutta method (second and fourth order)

TEXT BOOKS:

• 1. Probability and Statistics for Engineers and Scientists by Sheldon M Ross, Academic Press
• 2. S S Sastry, Introductory Methods of Numerical Analysis, Fifth Edition, PHI Learning Pvt. Ltd, New Delhi, 2012.
• 3. B.S. Grewal, Higher Engineering Mathematics, Khanna Publishers, 36th Edition, 2010.

REFERENCES:

• 1. T. T. Soong, Fundamentals of Probability and Statistics for Engineers, John Wiley & Sons, Ltd, 2004.
• 2. B V Ramana, Higher Engineering Mathematics, Tata McGraw Hill Education Private Limited, New Delhi, 29th Reprint, 2017 R18 B.TECH. AERONAUTICAL ENGG.