Initial value problem - Definition and Overview

In mathematics, an initial value problem is a statement of a differential equation together with specified value of the unknown function at a given point in the domain of the solution. Calling the given point t₀ and the specified value y₀, the initial value problem is

<math> y'(t) = f(t, y(t)), \quad y(t_0) = y_0 <math>

The problem is then to determine the function y.

This statement subsumes problems of higher order, by interpreting y as a vector. For derivatives of second or higher order, new variables (elements of the vector y) are introduced.

More generally, the unknown function y can take values on infinite dimensional spaces, such as Banach spaces or spaces of distributions.

Existence and uniqueness of solutions

For a large class of initial value problems, the existence and uniqueness of a solution can be demonstrated.

The Picard-Lindelöf theorem guarantees a unique solution on some interval containing t₀ if f and its partial derivative <math>\partial f/\partial y<math> are continuous on a region containing t₀ and y₀. The proof of this theorem proceeds by reformulating the problem as an equivalent integral equation. The integral can be considered an operator which maps one function into another, such that the solution is a fixed point of the operator. The Banach fixed point theorem is then invoked to show that there exists a unique fixed point, which is the solution of the initial value problem.

An older proof of the Picard-Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem. Such a construction is sometimes called "Picard's method" or "the method of successive approximations". This version is essentially a special case of the Banach fixed point theorem.

References

• W. Morris Hirsch and Stephen Smale. Differential Equations, Dynamical Systems, and Linear Algebra. New York: Academic Press, 1974.