# Numerical Solutions of Boundary Value Problems with Finite Difference Method

- Sujaul Chowdhury, Ponkog Kumar Das, Syed Badiuzzaman Faruque

- September 2018

### Description

Containing an extensive illustration of the use of finite difference method in solving boundary value problems numerically, a wide class of differential equations have been numerically solved in this book. Starting with differential equations of elementary functions like hyperbolic, sine and cosine, special functions such as Hermite, Laguerre and Legendre are solved. Airy function, stationary localised wavepacket, the quantum mechanical problem of the particle in a 1D box and polar equation of motion under gravitational interaction are also explored. Aimed at ensuring readers become adept in using the method, Mathematica 6.0 is used to solve systems of linear equations, and to plot the numerical data, and comparison with known analytic solutions showed nearly perfect agreement in every case.### About Editors

Sujaul Chowdhury is a Professor in Department of Physics at Shahjalal University of Science and Technology (SUST), Bangladesh. He obtained a BSc in physics in 1994 and a MSc in physics in 1996 from SUST. He obtained a PhD in physics from University of Glasgow, UK, in 2001. He was a Humboldt Research Fellow for one year at The Max Planck Institute, Stuttgart, Germany. He is the author of

*Numerical Solutions of Initial Value Problems Using Mathematica*.Ponkog Kumar Das is an Assistant Professor in Department of Physics at SUST. He obtained a BSc and MSc in physics from SUST. He is a co-author of

*Numerical Solutions of Initial Value Problems Using Mathematica*.Syed Badiuzzaman Faruque is a Professor in the Department of Physics at SUST. He is a researcher with interest in quantum theory, gravitational physics and material science. He has been teaching physics at university level for about 26 years. He studied physics at the University of Dhaka, Bangladesh, and the University of Massachusetts Dartmouth, USA, and obtained his PhD from SUST.

### Table of Contents

Chapter I

Numerical Solution of Boundary Value Problem

Using Finite Difference Method

1.1 Statement of the problem

1.2 Approximation to derivatives

1.3 The finite difference method

Chapter II

Differential Equations of Some Elementary Functions:

Boundary Value Problems Numerically Solved

Using Finite Difference Method

2.1 The differential equation for hyperbolic function

2.2 The differential equation for Cosine function

2.3 The differential equation for Sine function

Chapter III

Differential Equations of Special Functions:

Boundary Value Problems Numerically Solved

Using Finite Difference Method

3.1 The Hermite differential equation

3.1.1 Hermite polynomial H3(x)

3.2 The Laguerre differential equation

3.2.1 Laguerre polynomial L3(x)

3.3 The Legendre differential equation

3.3.1 Legendre polynomial P3(x)

Chapter IV

Differential Equation of Airy Function:

Boundary Value Problem Numerically Solved

Using Finite Difference Method

4.1 The differential equation for Airy function

Chapter V

Differential Equation of Stationary Localised Wavepacket:

Boundary Value Problem Numerically Solved

Using Finite Difference Method

5.1 Differential equation for stationary localised wavepacket

Chapter VI

Particle in a Box:

Boundary Value Problem Numerically Solved

Using Finite Difference Method

6.1 The Quantum Mechanical problem of particle in a onedimensional

Box

Chapter VII

Motion under gravitational interaction:

Boundary Value Problem Numerically Solved

Using Finite Difference Method

7.1 Motion under gravitational interaction

Concluding remarks

References

### Bibliographic

Paperback ISBN: 9780750329385

Ebook ISBN: 9781643272795

DOI: 10.1088/978-1-64327-280-1

Publisher: Morgan & Claypool Publishers