An initial value problem is a differential equation
- with where is an open set,
together with a point in the domain of ƒ
called the initial condition.
A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies
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.
Read more about Initial Value Problem: Existence and Uniqueness of Solutions, Exponential Smoothing, Examples
Famous quotes containing the words initial and/or problem:
“Capital is a result of labor, and is used by labor to assist it in further production. Labor is the active and initial force, and labor is therefore the employer of capital.”
—Henry George (1839–1897)
“Will women find themselves in the same position they have always been? Or do we see liberation as solving the conditions of women in our society?... If we continue to shy away from this problem we will not be able to solve it after independence. But if we can say that our first priority is the emancipation of women, we will become free as members of an oppressed community.”
—Ruth Mompati (b. 1925)