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Concept
Initial value problem
Summary
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem.
An initial value problem is a differential equation
with where is an open set of ,
together with a point in the domain of
called the initial condition.
A solution to an initial value problem is a function that is a solution to the differential equation and satisfies
In higher dimensions, the differential equation is replaced with a family of equations , and is viewed as the vector , most commonly associated with the position in space. More generally, the unknown function can take values on infinite dimensional spaces, such as Banach spaces or spaces of distributions.
Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. .
The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y.
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".
Official source
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