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MATHEMATICAL METHODS

MATHEMATICAL METHODS For First Year B.Tech, Students of JNTUH, JNTUA, JNTUK and Other Universities

By

Dr. S. SIVAIAH M.Sc., M.Phil., Ph.D., MBA., PGDCP and SA.,(M.Tech) Principal and Professor of Mathematics, Malla Reddy PG College, Maisammaguda, Secunderabad Andhra Pradesh

UNIVERSITY SCIENCE PRESS (An Imprint of Laxmi Publications Pvt. Ltd.) BANGALORE · CHENNAI JALANDHAR

· KOLKATA

· COCHIN

· GUWAHATI

· HYDERABAD

· LUCKNOW

· MUMBAI

· PATNA

RANCHI

· NEW DELHI

Copyright © 2012 by Laxmi Publications Pvt. Ltd. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher.

Published by :

UNIVERSITY SCIENCE PRESS (An Imprint of Laxmi Publications Pvt. Ltd.)

113, Golden House, Daryaganj, New Delhi-110002 Phone : 011-43 53 25 00 Fax : 011-43 53 25 28 www.laxmipublications.com [email protected]

First Edition : 2007; Second Edition : 2012 OFFICES Bangalore Cochin Hyderabad Kolkata Mumbai Ranchi

080-26 75 69 30 0484-237 70 04, 405 13 03 040-24 65 23 33 033-22 27 43 84 022-24 91 54 15, 24 92 78 69 0651-221 47 64

UMM-9410-400-MATHEMATICAL METHODS-SIV Typesetted at : Shubham Composer, New Delhi.

Chennai Guwahati Jalandhar Lucknow Patna

044-24 34 47 26 0361-251 36 69, 251 38 81 0181-222 12 72 0522-220 99 16 0612-230 00 97

CONTENTS Syllabus

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Unit : I Chapter 1. Matrices and Linear System of Equations

1–98

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Introduction Types of Matrices Algebra of Matrices Inverse of a Matrix Rank of a Matrix Elementary Transformations of a Matrix Methods to Find the Rank of a Matrix To Find Two Non-singular Matrices P and Q such that PAQ is Normal Form 1.8 Inverse of a Matrix by Elementary Row Transformations (Gauss-Jordan Method) 1.9 Solution of a System of Linear Equations 1.10 Solutions of Linear System of Equations by Direct Methods 1.11 Consistency of System of Non-homogeneous Linear Equations 1.12 Consistency of a System of Homogeneous Linear Equations Unit : II Chapter 2. Eigen Values, Eigen Vectors and Diagonalization of a Matrix 2.0 2.1 2.2 2.3 2.4 2.5 2.6

Introduction Linear Transformation Eigen Values and Eigen Vectors Properties of Eigen Values and Eigen Vectors Cayley-Hamilton Theorem Diagonalization of a Matrix by Similarity Transformation Diagonalization by Orthogonal Transformation

Unit : III Chapter 3. Real, Complex Matrices and Quadratic Forms 3.0 3.1 3.2 3.3 3.4

Introduction Real Matrices : (Symmetric, Skew-symmetric and Orthogonal) Properties of Real Matrices Complex Matrices Eigen Values, Eigen Vectors and Properties of Complex Matrices

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1 2 5 12 22 23 24 45 50 56 57 69 83 99–168 99 99 100 101 127 140 158 169–213 169 169 170 176 178

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3.5 3.6 3.7 3.8

Quadratic Forms Canonical Form of a Quadratic Form Nature of Quadratic Form Sylvester’s Law of Inertia

Unit : IV Chapter 4. Solution of Algebraic and Transcendental Equations 4.0 4.1 4.2 4.3 4.4

Introduction The Bisection Method or The Bolzano Method Regula-Falsi Method or Method of False Position The Iteration Method or Successive Approximation Method Newton-Raphson Method

Chapter 5. Interpolation 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16

Introduction Finite Differences Forward Differences Differences of a Polynomial Backward Differences Central Differences Symbolic Relations and Separation of Symbols Finite Difference Approximation to Derivatives Factorial Notation or Factorial Power Functions Newton's Forward Difference Interpolation Formula Newton's Backward Interpolation Formula Central Difference Interpolation Formula Gauss's Forward Interpolation Formula Gauss's Backward Interpolation Formula Stirling's Formula Interpolation with Unevenly Spaced Points Lagrange's Interpolation Formula

Unit : V Chapter 6. Curve Fitting 6.0 6.1 6.2 6.3 6.4 6.5 6.6

Introduction Fitting of a Straight Line by the Method of Least Squares Fitting of a Non-linear Curve or Fitting of a Parabolic (Second Order) Curve Fitting of a Power Curve of the Form Y = axb Fitting of an Exponential Curve of the Form Y = aebx Fitting of the Curve of the Form Y = abx Curve Fitting by a Sum of Exponentials

189 191 192 192 214–236 214 214 221 226 230 237–306 237 237 237 240 250 253 255 258 270 273 284 289 290 291 292 298 299

307–335 307 307 314 318 321 323 325

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6.7 6.8 6.9

Weighted Least Squares Approximation Linear Weighted Least Square Approximation Non-linear Weighted Least Square Approximation

Chapter 7. Numerical Differentiation and Integration 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9. 7.10 7.11 7.12

Introduction Numerical Differentiation Derivatives Using Forward Difference Formula Derivatives Using Backward Difference Formula Derivatives Using Central Difference Formula Maximum and Minimum Value of a Tabulated Function Numerical Integration Trapezoidal Rule Simpson’s One-Third Rule Simpson’s Three-Eight Rule Boole’s Rule Weddle’s Rule The Cubic Spline Method

Unit : VI Chapter 8. Numerical Solution of Ordinary Differential Equations 8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8

Introduction Taylor’s Series Method Picard’s Method of Successive Approximation Euler’s Method Modified Euler’s Method Runge-Kutta Methods Predictor-Corrector Methods Milne’s Method Adams-Moulton Method

Unit : VII Chapter 9. Fourier Series 9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8

Introduction Periodic Function Even and Odd Functions Fourier Series Determination of Euler’s Coefficients Dirichlet’s Conditions Fourier Series for Discontinuous Functions Change of Interval Half-Range Fourier Series

330 330 331 336–369 336 336 337 338 340 341 349 349 350 352 352 353 354 370–419 370 370 376 381 382 397 406 407 413 420–485 420 420 421 422 423 439 440 451 467

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Chapter 10. Fourier Integral Transforms 10.0 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9

Introduction Integral Transforms Fourier Integral Theorem Fourier Sine and Cosine Integrals Complex Form of Fourier Integral Fourier Transforms Fourier Sine and Cosine Transforms Properties of Fourier Transforms Parseval’s Identity for Fourier Transforms Finite Fourier Transforms

Unit : VIII Chapter 11. Partial Differential Equations 11.0 Introduction 11.1 Formation of Partial Differential Equations 11.2 Solution of Partial Differential Equation 11.3 Solutions by Direct Integration 11.4 First Order Linear Partial Differential Equations 11.5 Lagrange’s Linear Equation 11.6 Method of Grouping 11.7 Method of Multipliers 11.8 Non-Linear Partial Differential Equation of First Order 11.9 Equations Reducible to Standard Forms 11.10 Charpit’s Method 11.11 Method of Separation of Variables Chapter 12. Z-Transforms 12.0 12.1 12.2 12.3 12.4 12.5 12.6 12.7

Introduction Z-Transforms Z-Transforms for Some Standard Functions Properties of Z-Transforms Initial Value Theorem Final Value Theorem Inverse Z-Transforms Solving Difference Equation by Z-Transforms Examination Papers

486–530 486 486 487 488 489 495 495 496 517 523 531–606 531 532 548 549 551 551 553 560 568 585 590 595 607–641 607 607 608 613 615 616 625 632 643–650

PREFACE TO THE SECOND EDITION I have great pleasure to bringout the second edition of the book “Mathematical Methods” for B.Tech., First year students of JNTUH, JNTUA, JNTUK and other Universities. The earlier edition have received positive response from the teachers and the students. This book has been written strictly according to the revised syllabus 2009–2010 of First year B.Tech students of JNTUH, JNTUA, JNTUK. It has been written in a simple, lucid and easy to understand style. Number of solved examples and the problems given in the exercise have been taken from examination papers of JNTU to make students familiar with the type of questions. Hints and answers are given in the exercise to help students in self learning. Each chapter has been planned as an independent unit so that various topics connected with can also be read separately. At the end of each Unit, objective type questions along with answers are provided to equip the students in facing JNTU on-line internal examinations as well as competitive examinations like GATE, IES, etc. This book will also serve the needs of students pursuing B.Tech (CCC) of JNTU and other Universities. Suggestions for further improvement of the book are most welcome and will be gratefully acknowledged. —Author

PREFACE TO THE FIRST EDITION I have great pleasure in presenting this book “Mathematical Methods” for B.Tech., 1st Year students of JNTU and other Universities. It has been written in a simple, lucid and easy to understand style. Number of problems have been worked out as illustrations. Most of the worked out examplesand the problems

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given in the exercises have been taken from examination papers of JNTU to make students familiar with the type of questions. Hints and answers are given for the problems in the exercise to help students in self-learning. Each chapter has been planned as an independent unit so that various topics connected with can also be read separately. At the end of each unit, objective type questions along with answers are provided to equip the students in facing JNTU on-line examinations as well as competitive examinations like GATE, IES, etc. This book will also serve the needs of students pursuing B.Tech. (CCC) of JNTU and other universities, GATE and IES competitive examinations. Suggestions for further improvement of the book are most welcome and will be gratefully acknowledged. —Author

ACKNOWLEDGEMENT I express my deep sense of gratitude to Ch. Malla Reddy, Chairman, Col. G. Ram Reddy, Director (Admn), Prof R. Madan Mohan Director (Academics), Malla Reddy Group of Institutions, Dr. M.R.K. Murthy, Principal, Malla Reddy Engineering College and all Principal’s of Malla Reddy Group of Institutions for their constant inspiration and encouragement in writing this book. I am thankful to Dr. DRV Prasada Rao, Professor of Mathematics, S.K. University, Anantapur, Dr. D. Rama Murthy, Professor of Mathematics, Dr. V.V. Hara Gopal, Professor of Statistics, Osmania University, Hyderabad and my elder brother Dr. S. Eswaraiah Setty, Reader in Mathematics, Smt. G.S. College, Jaggaiah Peta, Krishna (Dt), Andhra Pradesh for their guidance and encouragement. I am thankful to my colleagues in Malla Reddy PG College and well wishers who helped me a lot in completing this book. My mother Smt. S. Guramma, my wife Smt. S. Amara veni and my daughter Miss S. Sowjanya deserve very special thanks for their cooperation in writing this book, without which I am sure that I would not have been completed the work. —Author

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SYLLABUS MATHEMATICAL METHODS 1st Year B. Tech. (E.C.E., E.E.E., E.I.E., C.S.E., I.T., C.S.S.E., E. CONT. E. E.C.M)

UNIT - I Matrices: Elementary row transformations – Rank – Echelon form, normal form – Solution of Linear System of Homogeneous and Non-homogeneous equations – Direct Methods – Gauss Elimination, Gauss Jordan Methods. Eigen values, Eigen vectors – Properties. Cayley - Hamilton Theorem – Inverse and powers of a matrix by Cayley-Hamilton theorem – Diagonolization of matrix. Calculation of powers of matrix.

UNIT - II Real matrices – Symmetric, Skew - Symmetric, orthogonal matrices – Linear Transformation – Orthogonal Transformation. Complex matrices: Hermitian, Skew-Hermitian and Unitary matrices – Eigen values and Eigen vectors and their properties. Quadratic forms – Reduction of quadratic form to canonical form and their nature.

UNIT - III Solution of Algebraic and Transcendental Equations: Introduction – The Bisection Method – The Method of False Position – The Iteration Method – Newton - Raphson Method. Interpolation: Introduction – Finite differences – Forward differences – Backward differences – Newton’s forward and backward difference formulae for interpolation – Lagrange’s Interpolation formula.

UNIT - IV Curve Fitting: Fitting a straight line – Second degree curve – Exponential curve – Power curve by method of least squares. Numerical Differentiation and Integration – Trapezoidal rule – Simpson's 1/3 Rule – Simpson's 3/8 Rule.

UNIT - V Numerical solution of Ordinary Differential equations : Solution by Taylor’s series – Picard’s Method of Successive Approximations – Euler’s Method – Runge-Kutta Methods – Milne’s Predictor-Corrector Method.

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UNIT - VI Fourier Series: Determination of Fourier coefficients – Fourier series of Even and Odd functions – Fourier series in an arbitrary interval – Even and odd periodic continuation – Half-range Fourier sine and cosine expansions. Fourier integral theorem (statement only) – Fourier sine and cosine integrals. Fourier transform – Fourier sine and cosine transforms – Properties – Inverse transforms – Finite Fourier transforms.

UNIT - VII Formation of partial differential equations by elimination of arbitrary constants and arbitrary functions – Method of separation of variables – Solutions of one dimensional wave equation, heat equation and two-dimensional Laplace equation under initial and boundary conditions.

UNIT - VIII z-transform – Inverse z-transform – Properties – Damping rule – Shifting rule – Initial and final value theorems. Convolution theorem – Solution of difference equations by z-transforms.

Dedicated to My Beloved Parents Smt. S. GURAMMA & Late S. HAMPAIAH Who made me What I am today.

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“Be not afraid of growing slowly, but be afraid of standing still” —Confucius

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