Numerical Methods Classical and Advanced Topics

This book presents a pedagogical treatment of a wide range of numerical methods to suit the needs of undergraduate and postgraduate students, and teachers and researchers in physics, mathematics, and engineering. For each method, the derivation of the formula/algorithm, error analysis, case studies, applications in science and engineering and the special features are covered. A detailed presentation of solving time-dependent Schrödinger equation and nonlinear wave equations, along with the Monte Carlo techniques (to mention a few) will aid in students? understanding of several physical phenomena including tunnelling, elastic collision of nonlinear waves, electronic distribution in atoms, and diffusion of neutrons through simulation study.

The book covers advanced topics such as symplectic integrators and random number generators for desired distributions and Monte Carlo techniques, which are usually overlooked in other numerical methods textbooks. Interesting updates on classical topics include: curve fitting to a sigmoid and Gaussian functions and product of certain two functions, solving of differential equations in the presence of noise, and solving the time-independent Schrödinger equation.

Solutions are presented in the forms of tables and graphs to provide visual aid and encourage a deeper comprehension of the topic. The step-by-step computations presented for most of the problems can be verifiable using a scientific calculator and is therefore appropriate for classroom teaching. The readers of the book will benefit from acquiring an acquittance, knowledge, experience and realization of significance of the numerical methods covered, their applicability to physical and engineering problems and the advantages of applying numerical methods over theoretical methods for specific problems.

1.Preliminaries.

2.Solutions of Polynomial and Reciprocal Equations.

3.Solution of General Nonlinear Equations.

4.Solution of Linear Systems AX = B

5.Curve Fitting.

6.Interpolation and Extrapolation.

7.Eigenvalues and Eigenvectors.

8.Numerical Differentiation.

9.Numerical Minimization of Functions.

10.Numerical Integration.

11.Ordinary Differential Equations- Initial-Value Problem.

12.Sympletic Integrators for Hamiltonian Systems.

13.Ordinary Differential Equations- Boundary-Value Problem.

14.Linear Partial Differential Equations.

15.Nonlinear Partial Differential Equations.

16.Fractional Order Ordinary Differential Equations.

17.Fractional Order Partial Differential Equations.

18.Fourier Analysis and Power Spectrum.

19.Random Numbers.

20.Monte Carlo Technique.

21.Answers to Some Selected Problems.

Index

Shanmuganathan Rajasekar was born in Thoothukudi, Tamil Nadu, India, in 1962. He was awarded Ph.D. from Bharathidasan University in 1992 under the supervision of Prof. M. Lakshmanan. In 1993, he joined as a Lecturer at the Department of Physics, Manonmaniam Sundaranar University, Tirunelveli. In 2005, he joined as a Professor at the School of Physics, Bharathidasan University, Tiruchirapalli. His recent research focuses on nonlinear dynamics with a special emphasis on nonlinear resonances. He has authored or co-authored more than 120 research papers in nonlinear dynamics.