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 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 -- Appendix A. Solutions to selected exercises. 1. Introduction -- 1.1. A note of caution : rounding errors -- 1.2. More on the limitations of digital computers 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 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 4. Interpolation -- 4.1. Lagrangian interpolation formula -- 4.2. The error caused by interpolation -- 4.3. Newton's form of interpolation polynomial 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 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 -- 6.10. Approximate evaluation of integrals using Taylor series expansion -- 6.11. The Fourier transform -- 6.12. Numerical integration using MATLAB 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 8. Applications of Monte Carlo methods -- 8.1. Random walks -- 8.2. The Ising model -- 8.3. Percolation theory -- 8.4. Simulated annealing 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
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.
Students in the physical sciences and engineering.
Mode of access: World Wide Web. System requirements: Adobe Acrobat Reader, EPUB reader, or Kindle reader.
P.K. Thiruvikraman is currently Professor of Physics at Birla Institute of Technology and Science, Pilani, Hyderabad Campus. He has nearly two decades of experience in teaching courses from many areas of Physics. The author has a PhD in Physics from Mangalore University and a Master's degree in Physics from Indian Institute of Technology, Madras. He has also authored the book A course on Digital Image Processing with MATLAB.