057c9375ea2a55 Contents Engineering Mathematics II 2nd ed

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Contents Preface ........................................................................................................................ (vii)

UNIT - I

| Solution of Algebraic and Transcendental Equations

1.1 Solution of Algebraic and Transcendental Equations ................................................ 2 1.1.1 Introduction ............................................................................................... 2 1.1.2 Bisection Method ...................................................................................... 3 1.1.3 Method of False Position .......................................................................... 5 1.1.4 The Iteration Method ................................................................................ 8 1.1.5 Newton-Raphson Method (One Variable) .............................................. 10 1.1.6 Newton-Raphson Method for Solving Simultaneous Equations ............. 14 Summary ................................................................................................................ 26 Solved University Questions ................................................................................... 27 Objective Type Questions ....................................................................................... 38

UNIT - II | Interpolation 2.1 Interpolation ............................................................................................................ 42 2.1.1 Introduction ............................................................................................. 42 2.1.2 Finite Differences ................................................................................... 42 2.1.2.1 Forward Differences ........................................................ 42 2.1.2.2 Backward Differences ..................................................... 44 2.1.2.3 Central Differences .......................................................... 45 2.1.2.4 Some Other Operators ..................................................... 46 2.1.2.5 Differences of a Polynomial............................................. 46 2.1.2.6 Newton’s Formulae for Interpolation ............................... 47 2.1.2.7 Newton’s Forward Interpolation Formula ........................ 47 2.1.2.8 Newton’s Backward Interpolation Formula ..................... 50 (ix)

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2.1.3 Central Difference Interpolation Formulae ............................................. 54 2.1.3.1 Gauss Forward Interpolation Formula.............................. 55 2.1.3.2 Gauss Backward Interpolation Formula........................... 57 2.1.4 Lagrange’s Interpolation Formula ........................................................... 61 Summary ................................................................................................................ 66 Solved University Questions ................................................................................... 68 Objective Type Questions ....................................................................................... 75

UNIT - III | Numerical Integration and Numerical Solutions of Ordinary Differential Equations 3.1 3.2 3.3 3.4 3.5 3.6 3.7

Numerical Integration ............................................................................................. 80 Numerical Solution of Ordinary Differential Equations .......................................... 90 Taylor’s Series Method ........................................................................................... 91 Picard’s Method of Successive Approximations .................................................... 94 Euler’s Method ....................................................................................................... 98 Modified Euler’s Method ...................................................................................... 100 Runge-Kutta Methods .......................................................................................... 104 3.7.1 Second Order Runge-Kutta Method ..................................................... 104 3.7.2 Third Order Runge-Kutta Method ........................................................ 105 3.7.3 Fourth Order Runge-Kutta Method ...................................................... 105 Summary ............................................................................................................... 111 Solved University Questions ................................................................................. 114 Objective Type Questions ..................................................................................... 132

UNIT - IV | Fourier Series 4.0 Introduction ........................................................................................................... 140 4.1 Periodic Functions ................................................................................................. 140 4.1.1 Even and Odd Functions ....................................................................... 140 4.2 Fourier Series ........................................................................................................ 145 4.2.1 Introduction ........................................................................................... 145 4.2.2 Trigonometric Series ............................................................................. 145 4.2.3 Theorem ................................................................................................ 145 4.2.4 Fourier Series ........................................................................................ 148 4.2.5 Convergence of Fourier Series ............................................................. 148

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4.2.6 Dirichlet's Theorem ............................................................................... 149 4.2.7 Summary ............................................................................................... 149 4.3 Change of Interval [Functions with Arbitrary Periods] ........................................ 174 4.3.1 Euler's Formulae for Functions having a Period '2l' where l is any Positive Number ............................................................ 174 4.4 Fourier Series of Even and Odd Functions ........................................................... 187 4.4.1 Even Functions ...................................................................................... 187 4.4.2 Odd Functions ....................................................................................... 188 4.5 Half Range Series ................................................................................................. 197 4.5.1 Half Range Sine Series ......................................................................... 197 4.5.2 Half Range Cosine Series ..................................................................... 198 Summary .............................................................................................................. 211 Solved University Questions ................................................................................. 214 Objective Type Questions ..................................................................................... 240

UNIT - V | Applications of Partial Differential Equations 5.1 Introduction ........................................................................................................... 248 5.2 Method of Separation of Variables ....................................................................... 248 5.3 One Dimensional Wave Equation ......................................................................... 259 5.4 One Dimensional Heat Equation ........................................................................... 271 5.5 Heat Equation with Non-homogenous Boundary Conditions................................ 285 5.6 Two Dimensional Heat Flow (Laplace Equation in Two Dimensions)................. 293 Summary .............................................................................................................. 302 Solved University Questions ................................................................................. 302 Objective Type Questions ..................................................................................... 311

UNIT - VI | Fourier Transforms 6.1 Integral Transforms ............................................................................................. 316 6.1.1 Definition of Integral Transform ........................................................... 316 6.1.2 Fourier Integral Theorem ...................................................................... 317 6.1.3 Fourier Sine and Cosine Integrals ......................................................... 319 6.1.4 Complex Form of Fourier Integrals ....................................................... 320 6.2 Fourier Transforms ............................................................................................... 328 6.2.1 Definitions ............................................................................................. 328

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6.2.2 Fourier Cosine Transforms ................................................................... 329 6.2.3 Fourier Sine Transforms ....................................................................... 329 6.2.4 Properties of Fourier Transforms .......................................................... 331 6.3 Finite Fourier Transforms...................................................................................... 345 6.4 Convolution Theorem ............................................................................................ 352 6.4.1 Convolution of Two Functions .............................................................. 352 6.4.2 Parseval's Identity ................................................................................. 353 Summary .............................................................................................................. 364 Solved University Questions ................................................................................. 366 Objective Type Questions ..................................................................................... 380 Model Papers of Engineering Mathematics - II ................................................ 383 Gate Previous Questions on Numerical Methods with Answers .................... 391