This book provides an introduction to the computational methods commonly employed by physicists and engineers. The book discusses the details of the numerical algorithms involved and also provides MATLAB code for their implementation. Applications of numerical methods to various physical systems including nonlinear systems and fractals are also discussed. Each chapter has a number of solved examples and end of chapter exercises. Solutions to most of the exercises have also been included. The book is suitable for undergraduates in physics or engineering. The methods discussed and some of the examples will also be useful for other scientists and engineers who wish to learn the basics of computational/ numerical methods for solving problems.
Key Features:
- Comprehensive coverage of basic theory
- Accompanying MATLAB programs
- Applications of computational methods to various areas of physics
- Worked examples and end of chapter problems
- Enhanced with animation and sound files
Author(s): P. K. Thiruvikraman
Publisher: IOP Publishing
Year: 2022
Language: English
Pages: 257
City: Bristol
PRELIMS.pdf
Preface
Acknowledgements
Author biography
P K Thiruvikraman
CH001.pdf
Chapter 1 Introduction
1.1 A note of caution: rounding errors
1.2 More on the limitations of digital computers
Exercises:
CH002.pdf
Chapter 2 Introduction to programming with MATLAB
2.1 Computer programming
2.2 Good programming practices
2.3 Introduction to MATLAB
2.4 HELP on MATLAB
2.5 Variables
2.6 Mathematical operations
2.7 Loops and control statements
2.8 Built-in MATLAB functions
2.9 Some more useful MATLAB commands and programming practices
2.10 Functions
2.11 Using MATLAB for visualisation
2.12 Producing sound using MATLAB
Programming exercises
CH003.pdf
Chapter 3 Finding the roots and zeros of a function
3.1 The roots of a polynomial
3.2 Graphical method
3.3 Solution of equations by fixed-point iteration
3.4 Bisection
3.5 Descartes’ rule of signs
3.6 The Newton–Raphson method
3.7 The false position method
3.8 The secant method
3.9 Applications of root finding in physics
3.10 The finite potential well
3.11 The Kronig–Penney model
Exercises
CH004.pdf
Chapter 4 Interpolation
4.1 Lagrangian interpolation formula
4.2 The error caused by interpolation
4.3 Newton’s form of interpolation polynomial
Exercises
CH005.pdf
Chapter 5 Numerical linear algebra
5.1 Solving a system of equations: Gaussian elimination
5.2 Evaluating the determinant of a matrix
5.3 LU decomposition
5.4 Determination of eigenvalues and eigenvectors: the power method
5.5 Convergence of the power method
5.6 Deflation: determination of the remaining eigenvalues
5.7 Curve fitting: the least-squares technique
5.8 Curve fitting: the generalised least-squares technique
Exercises
CH006.pdf
Chapter 6 Numerical integration and differentiation
6.1 Numerical differentiation
6.2 The Richardson extrapolation
6.3 Numerical integration: the area under the curve
6.4 Simpson’s rules
6.5 Comparison of quadrature methods
6.6 Romberg integration
6.7 Gaussian quadrature
6.8 Gaussian quadrature for arbitrary limits
6.9 Improper integrals
6.9.1 Limit comparison test
6.9.2 Direct comparison test
6.10 Approximate evaluation of integrals using Taylor series expansion
6.11 The Fourier transform
6.12 Numerical integration using MATLAB
Exercises
CH007.pdf
Chapter 7 Monte Carlo integration
7.1 Error in multidimensional integration
7.2 Monte Carlo integration
7.3 Error estimate for Monte Carlo integration
7.4 Importance sampling Monte Carlo
7.5 The Box–Muller method
7.6 The Metropolis algorithm
7.7 Random number generators
7.8 The linear congruential method
7.9 Generalised feedback shift register
Exercises
CH008.pdf
Chapter 8 Applications of Monte Carlo methods
8.1 Random walks
8.2 The Ising model
8.3 Percolation theory
8.4 Simulated annealing
Exercises
CH009.pdf
Chapter 9 Ordinary differential equations
9.1 Differential equations in physics
9.2 The simple Euler method
9.3 The modified and improved Euler methods
9.4 Runge–Kutta methods
9.5 The Taylor series method
9.6 The shooting method
9.7 Applications to physical systems
Exercises
CH010.pdf
Chapter 10 Partial differential equations
10.1 Partial differential equations in physics
10.2 Finite difference method for solving ordinary differential equations
10.3 Finite difference method for solving PDEs
10.4 A finite difference method for PDEs involving both spatial and temporal derivatives
Exercises
CH011.pdf
Chapter 11 Nonlinear dynamics, chaos, and fractals
11.1 History of chaos
11.2 The logistic map
11.3 The Lyapunov exponent
11.4 Differential equations: fixed points
11.5 Fractals
Exercises
APP1.pdf
Chapter
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 9
Chapter 10
Chapter 11
