Computational Methods Using MATLAB®: An introduction for physicists

Description

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 theoryAccompanying MATLAB programsApplications of computational methods to various areas of physicsWorked examples and end of chapter problemsEnhanced with animation and sound files 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