Numerical Solutions of Boundary Value Problems of Non-linear Differential Equations

The book presents in comprehensive detail numerical solutions to boundary value problems of a number of non-linear differential equations. Replacing derivatives by finite difference approximations in these differential equations leads to a system of non-linear algebraic equations which we have solved using Newton?s iterative method. In each case, we have also obtained Euler solutions and ascertained that the iterations converge to Euler solutions. We find that, except for the boundary values, initial values of the 1st iteration need not be anything close to the final convergent values of the numerical solution. Programs in Mathematica 6.0 were written to obtain the numerical solutions.

1. Introduction. 1.1. The non-linear differential equations we solved in this book. 1.2 Approximation to derivatives. 1.3 Statement of the problem. 1.4 Euler solution of differential equation. 1.5 Newton’s method of solving system of non-linear equations 2. Numerical Solution of Boundary Value Problem of Non-linear Differential Equation: Example I. 2.1 The 1st non-linear differential equation in this book: Euler solution. 2.2 The 1st non-linear differential equation in this book: solution by Newton’s iterative method. 3. Numerical solution of boundary value problem of non-linear differential equation:Example II. 3.1 The 2nd non-linear differential equation in this book: Euler solution. 3.2. The 2nd non-linear differential equation in this book: solution by Newton’s iterative method. 4. Numerical solution of boundary value problem of non-linear differential equation: Example III. 4.1 The 3rd non-linear differential equation in this book: Euler solution. 4.2 The 3rd non-linear differential equation in this book: solution by Newton’s iterative method. 5. Numerical solution of boundary value problem of non-linear differential equation: Example IV. 5.1 The 4th non-linear differential equation in this book: Euler solution . 5.2 The 4th non-linear differential equation in this book: solution by Newton’s iterative method. 6. Numerical solution of boundary value problem of non-linear differential equation: Example V. 6.1 The 5th non-linear differential equation in this book: Euler solution . 6.2 The 5th non-linear differential equation in this book: solution by Newton’s iterative method 7. Numerical solution of boundary value problem of non-linear differential equation: Example VI 7.1 The 6th non-linear differential equation in this book: Euler solution . 7.2 The 6th non-linear differential equation in this book: solution by Newton’s iterative method. 8. Numerical solution of boundary value problem of non-linear differential equation: A laborious exercise. 8.1 The 7th non-linear differential equation in this book: Euler solution. 8.2 The 7th non-linear differential equation in this book: solution by Newton’s iterative method. Concluding remarks. References

Sujaul Chowdhury is a Professor in Department of Physics, Shahjalal University of Science and Technology (SUST), Bangladesh. He obtained a B.Sc. (Honours) in Physics in 1994 and M.Sc. in Physics in 1996 from SUST. He obtained a Ph.D. in Physics from The University of Glasgow, UK in 2001. He was a Humboldt Research Fellow for one year at The Max Planck Institute, Stuttgart, Germany.

Syed Badiuzzaman Faruque is a Professor in Department of Physics, SUST. He has a research interest in Quantum Theory, Gravitational Physics, Material Science etc. He has been teaching Physics at university level for about 27 years. He studied Physics in The University of Dhaka, Bangladesh and in The University of Massachusetts Dartmouth, U.S.A. and did PhD in SUST.

Ponkog Kumar Das is an Assistant Professor in Department of Physics, SUST. He obtained a B.Sc. (Honours) and M.Sc. in Physics from SUST. He is a promising future intellectual.