Initial value problem

As an initial value problem ( IVP for short ), sometimes called initial value problem (abbreviated AWA) or Cauchy problem is referred to in the analysis an important class of differential equations. In this article, the initial value problem is first explained for ordinary differential equations, and later also for partial differential equations.

Ordinary Differential Equations

First order initial value problem

An initial value problem of first order is an ordinary first order differential equation with an additional condition. For given initial data, namely the initial value and a time the function is a solution of the initial value problem if the differential equation solve and if additionally applies. The system of equations

So called first order initial value problem.

Initial value problem of order k

Are given and a function. The definition range is a subset of this case, wherein an interval referred to, which comprises. Then say

An initial value problem of order. Each initial value problem of order can be described in a first -order initial value problem.

A special initial value problem is the Riemann problem in which the initial data are constant up to a point of discontinuity.

Initial value problems occur, for example in the natural sciences, if natural processes, a mathematical model is sought.

Important records that affect the solvability of initial value problems for ordinary differential equations are the (local) existence theorem of Peano and the existence and uniqueness theorem of Picard - Lindelöf. One tool is the grönwallsche inequality.

Example

The initial value problem

Which to

Corresponds, has infinitely many solutions, namely, in addition to the trivial solution

Also for any solutions

As well as

This initial value problems have unique solutions, additional capacity shall be proven ( to ). This can happen, for example on the set of Picard - Lindelöf whose conditions are not satisfied in this example.

Numerical solution methods

For the numerical solution of initial value problems one-step or multi- step methods are used. The differential equation is approximated by a discretization.

Abstract Cauchy problem

Be a Banach space and a linear or nonlinear operator. The question of whether a given, and a differentiable function with for all exists, the initial value problem, the

Met, are called abstract Cauchy problem. To their solvability one needs the theory of strongly continuous semigroups and the analytic semigroups. There to the different initial conditions and operators of different types of solution concept, in the linear distributional solutions to the nonlinear integral solution. Offering classically differentiable, or almost everywhere differentiable solutions, employs the downstream regularity.

Partial Differential Equations

Generalizing the Cauchy problem on several variables, such as variables, we obtain partial differential equations. In the following stand for a multi- index of length. Note that there are exactly multi- indices. It was passed on a function in variables. In general Cauchy problem one looks for functions that depend on variables and the equation

. meet Note that the arity has just been chosen by so that you can use and all partial derivatives. In addition, the proposal calls for that the unknown functions satisfy the so-called initial and boundary conditions described below. For their formulation is a hypersurface of class Ck with normal field. With the normal derivatives are denoted. Then are given to defined functions, one calls the general Cauchy problem that the features in addition the conditions

. meet The functions are called the Cauchy- data of the problem, any function that satisfies both conditions ( 1) and (2) is called a solution of the Cauchy problem.

By a suitable coordinate transformation one can retreat to the case. Then the last variable plays a special role, because the initial conditions are given there, where this variable is 0. Since this variable is interpreted in many applications as a time, you can call it like in (Latin tempus = time) to the initial conditions then describe the circumstances at the time. The variables are so. Since the considered hyperplane is given by the condition that the normal derivative is simply according to the derivation. If you write for short and so the Cauchy problem now reads

A typical example is the three-dimensional wave equation

Wherein a constant, a predetermined function and the Laplacian operator were.

If a solution is what should imply sufficient differentiability at the same time, all derivatives are defined with already by the Cauchy data, because it is. Only the derivative is ' set, so here only (1 not by ( 2) ' ) represent a condition. Thus, (1 ') a non-trivial condition, and thus the Cauchy problem is not actually ill-posed from the outset, you will demand that the equation (1' can dissolve ) to. The Cauchy problem then has the form

With a suitable function of arity was. In the last formulation given all occurring derivatives have an order, and the -th derivative by actually occurs, because this is just the left-hand side of ( 1 ") and it does not come on the right side of ( 1" front ). Therefore are also called the order of the Cauchy problem. The above example of the three-dimensional wave equation is obviously easy to bring in this form, thus there is a Cauchy problem of order 2.

Are all Cauchy data analysis, the Cauchy - Kovalevskaya ensures unique solutions of the Cauchy problem.