JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

Pre-requisites

Mathematical Knowledge at pre-university level

Course Objectives

To learn

• Concept, properties of Laplace transforms
• Solving ordinary differential equations using Laplace transforms techniques.
• Various methods to the find roots of an equation.
• Concept of finite differences and to estimate the value for the given data using interpolation.
• Evaluation of integrals using numerical techniques
• Solving ordinary differential equations using numerical techniques.
• Differentiation and integration of complex valued functions.
• Evaluation of integrals using Cauchy’s integral formula and Cauchy’s residue theorem.
• Expansion of complex functions using Taylor’s and Laurent’s series. Course outcomes: After learning the contents of this paper the student must be able to
• Use the Laplace transforms techniques for solving ODE’s
• Find the root of a given equation.
• Estimate the value for the given data using interpolation
• Find the numerical solutions for a given ODE’s
• Analyze the complex function with reference to their analyticity, integration using Cauchy’s integral and residue theorems.
• Taylor’s and Laurent’s series expansions of complex Function

UNIT - I

Laplace Transforms 10 L Laplace Transforms; Laplace Transform of standard functions; first shifting theorem; Laplace transforms of functions when they are multiplied and divided by‘t’. Laplace transforms of derivatives and integrals of function; Evaluation of integrals by Laplace transforms; Laplace transforms of Special functions; Laplace transform of periodic functions. Inverse Laplace transform by different methods, convolution theorem (without Proof), solving ODEs by Laplace Transform method.

UNIT - II

Numerical Methods – I 10 L Solution of polynomial and transcendental equations – Bisection method, Iteration Method, Newton- Raphson method and Regula-Falsi method. Finite differences- forward differences- backward differences-central differences-symbolic relations and separation of symbols; Interpolation using Newton’s forward and backward difference formulae. Central difference interpolation: Gauss’s forward and backward formulae; Lagrange’s method of interpolation

UNIT - III

Numerical Methods – II 08 L Numerical integration: Trapezoidal rule and Simpson’s 1/3rd and 3/8 rules. Ordinary differential equations: Taylor’s series; Picard’s method; Euler and modified Euler’s methods; Runge-Kutta method of fourth order.

R18 B.TECH ECE

UNIT - IV

Complex Variables (Differentiation) 10 L Limit, Continuity and Differentiation of Complex functions. Cauchy-Riemann equations (without proof), Milne- Thomson methods, analytic functions, harmonic functions, finding harmonic conjugate; elementary analytic functions (exponential, trigonometric, logarithm) and their properties.

UNIT - V

Complex Variables (Integration) 10 L Line integrals, Cauchy’s theorem, Cauchy’s Integral formula, Liouville’s theorem, Maximum-Modulus theorem (All theorems without proof); zeros of analytic functions, singularities, Taylor’s series, Laurent’s series; Residues, Cauchy Residue theorem (without proof)

TEXT BOOKS:

• 1. B.S. Grewal, Higher Engineering Mathematics, Khanna Publishers, 36th Edition, 2010.
• 2. S.S. Sastry, Introductory methods of numerical analysis, PHI, 4th Edition, 2005.
• 3. J. W. Brown and R. V. Churchill, Complex Variables and Applications, 7th Ed., Mc-Graw Hill, 2004.

REFERENCE BOOKS:

• 1. M. K. Jain, SRK Iyengar, R.K. Jain, Numerical methods for Scientific and Engineering Computations , New Age International publishers.
• 2. Erwin kreyszig, Advanced Engineering Mathematics, 9th Edition, John Wiley & Sons,2006. R18 B.TECH ECE