DEPARTMENT OF MATHEMATICS, MSRIT, BANGALORE – 560 054 (Revised Syllabus: 2015-16) Subject: Engineering Mathematics-III Code: ML MAT301 Credits: (3:1:0)4 Contact hours: 42L+14T [ Medical Electronics] Course Objectives: The students will 1) Learn to solve algebraic, transcendental and ordinary differential equations numerically. 2) Learn to fit a curve, correlation, regression for a statistical data. 3) Learn the concepts of consistency, methods of solution for linear system of equations and eigen value problems. 4) Learn to represent a periodic function in terms of sines and cosines. 5) Understand the concepts of continuous and discrete integral transforms in the form of Fourier and Z-transforms. 6) Learn the concept of series solutions of ODE and special functions. Course Outcomes: Students are expected to do the following: 1) Should be able to solve the problems of algebraic, transcendental and ordinary differential equations using numerical methods 2)Fit a suitable curve by the method of least squares and determine the lines of regression for a set of statistical data. 3) Find the rank of a matrix and testing the consistency and the solution by Gauss Elimination and Gauss Siedel iteration methods . 4) Find the Fourier series expansion of a function in both full range and half range values of the variable and obtaining the various harmonics of the Fourier series expansion for the given numerical data. 5) Find Fourier transforms, Fourier sine and Fourier cosine transforms of functions and solving difference equations using Z-transforms. 6) Obtain the series solution of ordinary differential equations. Unit I Numerical solution of Algebraic and Transcendental equations: Method of false position, Newton - Raphson method. Numerical solution of Ordinary differential equations: Taylor series method, Euler and modified Euler method, fourth order Runge-Kutta method. Statistics: Curve fitting by the method of least squares, Fitting a linear curve, fitting a parabola, fitting a Geometric curve, Correlation and Regression. Unit II Linear Algebra: Elementary transformations on a matrix, Echelon form of a matrix, rank of a matrix, Consistency of system of linear equations, Gauss elimination and Gauss – Siedal method to solve system of linear equations, eigen values and eigen vectors of a matrix, Rayleigh power method to determine the dominant eigen value of a matrix, diagonalization of a matrix, system of ODEs as matrix differential equations Unit III Fourier series: Convergence and divergence of infinite series of positive terms. Periodic function, Dirichlet conditions, Fourier series of periodic functions of period 2 and arbitrary period, Half range series, Fourier series and Half Range Fourier series of Periodic square wave, Half wave rectifier, Full wave rectifier, Saw-tooth wave with graphical representation, Practical harmonic analysis. Unit IV Fourier Transforms: Infinite Fourier transform, Infinite Fourier sine and cosine transforms, properties, Inverse transform, Convolution theorem, Parseval identity (statements only). Fourier transform of rectangular pulse with graphical representation and its output discussion, Continuous Fourier spectra-Example and physical interpretation. Z-Transforms: Definition, standard Z-transforms, Single sided and double sided, Linearity property, Damping rule, Shifting property, Initial and final value theorem, Inverse Ztransform, Application of Z-transform to solve difference equations. Unit V Series Solution of ODEs and Special Functions: Series solution, Frobenius method, Series solution of Bessel differential equation leading to Bessel function of first kind, Series solution of Legendre differential equation leading to Legendre polynomials, Rodrigues's formula. Text Books: 1. Erwin Kreyszig – Advanced Engineering Mathematics – Wiley publication – 10th edition-2015. 2. B.S.Grewal - Higher Engineering Mathematics - Khanna Publishers – 43rd edition-2014 References: 1. Glyn James – Advanced Modern Engineering Mathematics – Pearson Education – 4th edition – 2010. 2. Dennis G. Zill, Michael R. Cullen - Advanced Engineering Mathematics, Jones and Barlett Publishers Inc. – 3rd edition – 2009.