A big advantage of numerical mathematics is that a numerical solution can be obtained for problems, where an analytical solution does not exist. An additional advantage is, that a numerical method only uses evaluation of standard functions and the operations: addition, subtraction, multiplication and division. Because these are just the operations a computer can perform, numerical mathematics and computers form a perfect combination.
An analytical method gives the solution as a mathematical formula, which is an advantage. From this we can gain insight in the behavior and the properties of the solution, and with a numerical solution (that gives the function as a table) this is not the case. On the other hand some form of visualization may be used to gain insight in the behavior of the solution. To draw a graph of a function with a numerical method is usually a more useful tool than to evaluate the analytical solution at a great number of points.
In this book we discuss several numerical methods for solving ordinary differential equations. We emphasize those aspects that play an important role in practical problems. In this introductory text we confine ourselves to ordinary differential equations with the exception of the last chapter in which we discuss the heat equation, a parabolic partial differential equation. The techniques discussed in
the introductory chapters, for e.g. interpolation, numerical quadrature and the solution of nonlinear equations, may also be used outside the context of differential equations. They have been included to make the book self contained as far as the numerical aspects are concerned.
Contents: Preface | 1. Introduction | 2. Interpolation | 3. Numerical differentiation | 4. Nonlinear equations | 5. Numerical quadrature | 6. Numerical time integration of initial value problems | 7. The finite difference method for boundary value problems | 8. The instationary equation | Literature | Index