Numerical solutions of boundary value problems with finite difference method
Boundary value problems Finite differences
IOP Publishing
2018
EISBN 9781643272801
1. A numerical solution of boundary value problem using the finite difference method.
1.1. Statement of the problem.
1.2. Approximation to derivatives.
1.3. The finite difference method
2. 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
3. Differential equations of special functions : boundary value problems numerically solved using finite difference method.
3.1. The Hermite differential equation.
3.2. The Laguerre differential equation.
3.3. The Legendre differential equation
4. Differential equation of Airy function : boundary value problem numerically solved using finite difference method.
4.1. The differential equation for Airy function
5. Differential equation of stationary localised wavepacket : boundary value problem numerically solved using finite difference method.
5.1. Differential equation for stationary localised wavepacket
6. Particle in a box : boundary value problem numerically solved using finite difference method.
6.1. The quantum mechanical problem of a particle in a one-dimensional box
7. Motion under gravitational interaction : boundary value problem numerically solved using finite difference method.
7.1. Motion under gravitational interaction.
8. Concluding remarks.
The book contains an extensive illustration of use of finite difference method in solving the boundary value problem numerically. A wide class of differential equations has been numerically solved in this book. Starting with differential equations of elementary functions like hyperbolic, sine and cosine, we have solved those of special functions like Hermite, Laguerre and Legendre. Those of Airy function, of stationary localised wavepacket, of the quantum mechanical problem of a particle in a 1D box, and the polar equation of motion under gravitational interaction have also been solved. Mathematica 6.0 has been used to solve the system of linear equations that we encountered and to plot the numerical data. Comparison with known analytic solutions showed nearly perfect agreement in every case. On reading this book, readers will become adept in using the method.
1.1. Statement of the problem.
1.2. Approximation to derivatives.
1.3. The finite difference method
2. 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
3. Differential equations of special functions : boundary value problems numerically solved using finite difference method.
3.1. The Hermite differential equation.
3.2. The Laguerre differential equation.
3.3. The Legendre differential equation
4. Differential equation of Airy function : boundary value problem numerically solved using finite difference method.
4.1. The differential equation for Airy function
5. Differential equation of stationary localised wavepacket : boundary value problem numerically solved using finite difference method.
5.1. Differential equation for stationary localised wavepacket
6. Particle in a box : boundary value problem numerically solved using finite difference method.
6.1. The quantum mechanical problem of a particle in a one-dimensional box
7. Motion under gravitational interaction : boundary value problem numerically solved using finite difference method.
7.1. Motion under gravitational interaction.
8. Concluding remarks.
The book contains an extensive illustration of use of finite difference method in solving the boundary value problem numerically. A wide class of differential equations has been numerically solved in this book. Starting with differential equations of elementary functions like hyperbolic, sine and cosine, we have solved those of special functions like Hermite, Laguerre and Legendre. Those of Airy function, of stationary localised wavepacket, of the quantum mechanical problem of a particle in a 1D box, and the polar equation of motion under gravitational interaction have also been solved. Mathematica 6.0 has been used to solve the system of linear equations that we encountered and to plot the numerical data. Comparison with known analytic solutions showed nearly perfect agreement in every case. On reading this book, readers will become adept in using the method.
