Computational Methods in Engineering

Dedication

Preface

Acknowledgements

Chapter 1. Preliminaries

1.1 Introduction

1.2 Mathematical and computational modeling

1.3 A brief introduction to MATLAB

Suggested reading

Module I: System of Equations and Eigenvalues

Module I: System of Equations and Eigenvalues

Chapter 2. Solution of Linear Equations

2.1 Analytical methods of solving a set of linear equations

2.2 Preliminaries

2.3 Gauss elimination method

2.4 Gauss Jordan method of determining the inverse matrix

2.5 LU decomposition or LU factorization

2.6 Tridiagonal matrix algorithm

2.7 QR Factorization

2.8 Iterative methods of solution

CG Method

2.A MATLAB routines related to Chapter 2

2.B Suggested reading

Chapter 3. Computation of Eigenvalues

3.1 Examples of eigenvalues

3.2 Preliminaries on eigenvalues

3.3 Analytical evaluation of eigenvalues and eigenvectors in simple cases

3.4 Power method

3.5 Rayleigh Quotient Iteration

3.6 Eigenvalue eigenvector pair by QR iteration

3.7 Modification of QR iteration for faster convergence

3.A MATLAB routines related to Chapter 3

3.B Suggested reading

Chapter 4. Solution of Algebraic Equations

4.1 Univariate nonlinear equation

4.2 Multivariate non linear equations

4.3 Root finding and optimization

4.4 Multidimensional unconstrained optimization

4.5 Examples from engineering applications

4.A MATLAB routines related to Chapter 4

4.B Suggested reading

Exercise I

I.1 Solution of linear equations

I.2 Evaluation of eigenvalues

I.3 Solution of algebraic equations

Module II: Interpolation, differentiation and integration

Module II: Interpolation, Differentiation and Integration

Chapter 5. Interpolation

5.1 Polynomial interpolation

5.2 Lagrange interpolating (or interpolation) polynomial

5.3 Newton Polynomials

5.4 Error estimates of polynomial approximations

5.5 Polynomial approximation using Chebyshev nodes

5.6 Piecewise polynomial interpolation

5.7 Hermite interpolation

5.8 Spline interpolation and the cubic spline

5.A MATLAB routines related to Chapter 5

references

Chapter 6. Interpolation in Two and Three Dimensions

6.1 Interpolation over a rectangle

6.2 Interpolation over a triangle

6.3 Interpolation in three dimensions

Chapter 7. Regression or Curve Fitting

7.1 Introduction

7.2 Method of least squares for linear regression

7.3 Multi-linear regression

7.4 Polynomial regression

7.5 Non-linear regression

7.A MATLAB routines related to Chapter 7

7. B Suggested reading

Chapter 8. Numerical Differentiation

8.1 Introduction

8.2 Finite difference formulae using Taylor’s series

8.3 Differentiation of Lagrange and Newton polynomials

8.4 Numerical partial differentiation

8.A MATLAB routines related to Chapter 8

Chapter 9. Numerical Integration

9.1 Introduction

9.2 Trapezoidal rule

9.3 Simpson’s rule

9.4 Integration of functions

9.5 Quadrature using Chebyshev nodes

9.6 Gauss quadrature

9.7 Singular integrals

9.8 Integrals with infinite range

9.9 Adaptive quadrature

9.10 Multiple integrals

9.A MATLAB routines related to Chapter 9

9.B Suggested reading

Exercise II

II.1 Interpolation

II.2 Interpolation in two dimensions

II.3 Regression

II.3 Regression

II.4 Numerical differentiation

II.5 Numerical integration

Module III: Ordinary differential equations

Module III: Ordinary differential equations

Chapter 10. Initial Value Problems

10.1 Introduction

10.2 Euler method

10.3 Modified Euler method or Heun method

10.4 Runge Kutta (RK) methods

10.5 Predictor corrector methods

10.6 Set of first order ODEs

10.7 Higher order ODEs

10.8 Stiff equations and backward difference formulae (BDF) based methods

10.B Suggested reading

Chapter 11. Boundary Value Problems (ODE)

11.1 Introduction

11.2 The ‘shooting method’

11.3 Finite difference method

11.4 Collocation method

11.5 Method of weighted residuals

11.6 Finite element method

11.7 Finite volume method

11.A MATLAB routines related to Chapter 11

11.B Suggested reading

Exercise III

III.1 Initial value problems

III.2 Boundary value problems

Module IV: Partial Differential Equations

Module IV: Partial Differential Equations

Chapter 12. Introduction to PDEs

12.1 Preliminaries

12.2 Second order PDE with constant coefficients

12.3 Numerical solution methods for PDEs

12.4 MATLAB functions related to PDE

12. A Suggested reading

Chapter 13. Laplace and Poisson Equations

13.1 Introduction

13.2 Finite difference solution

13.3 Elliptic equations in other coordinate systems

13.4 Elliptic equation over irregular domain

13.5 FEM and FVM applied to elliptic problems

Chapter 14. Advection and Diffusion Equations

14.1 Introduction

14.2 The advection equation

14.3 Parabolic PDE: Transient diffusion equation

14.4 Advection with diffusion

14.5 Advection equation in multi dimensions

14.6 Diffusion equation in multi dimensions

14.A Suggested reading

Chapter 15. Wave Equation

15.1 Introduction

15.2 General solution of the wave equation

15.3 Numerical solution of the one dimensional wave equation

15.4 Waves in a diaphragm

Exercise IV

IV.1 Types of partial differential equations

IV.2 Laplace and Poisson equations

IV.3 Advection and diffusion equations

IV.4 Wave equation

Chapter 16. Beyond the Book – the Way Ahead

16.1 Researchers developing their own code

16.2 Users of commercial codes

16.3 Where does one look for help?

Index