Initial condition

An initial condition for an ordinary differential equation states which functional value the sought solution and, if applicable, its derivative (s) should have at a certain point.

Practically every differential equation allows an infinite number of solutions. An initial condition makes a choice among all of these solutions. Sometimes several, sometimes a single, sometimes none of the solutions satisfy the initial condition.

Anyone who adds an initial condition to a differential equation poses an initial value problem . A particularly exciting question is how an initial condition for a given differential equation must be designed so that the resulting initial value problem allows exactly one uniquely determined solution.

Practical meaning

If the differential equation describes a development over time, for example the movement of an object in space, the initial condition defines the state in which the movement begins, such as where the object is initially located.

The question of what kind of initial condition is suitable to mark out an unambiguous solution then means: What do I need to know about the present of a system in order to be able to fully recalculate its historical development and fully predict its future?

An initial condition in the mathematical sense does not necessarily have to refer to a temporal or spatial starting point. What appears colloquially as “end condition” or “intermediate state” is also called “initial condition” in mathematics.

For linear differential equations , the presence of initial conditions (not equal to zero) is equivalent to the excitation of the same system with a pulse , but here the initial conditions are zero.

example

The free fall (such as an apple from a tree) is described by the equation of motion

${\displaystyle {\begin{aligned}y''(t)&=-g\\\Rightarrow y'(t)&=-g\cdot t+v_{0}\end{aligned}}}$

with the constant ( acceleration due to gravity ). ${\displaystyle g\approx 9{,}81~\mathrm {m} /\mathrm {s} ^{2}}$

The set of solutions to this differential equation consists initially of all functions of the form

${\displaystyle \Rightarrow y(t)=-{\tfrac {1}{2}}gt^{2}+v_{0}\cdot t+y_{0}}$

with any integration constants and . ${\displaystyle y_{0}}$ ${\displaystyle v_{0}}$

A possible initial condition says e.g. For example, assume that at the beginning of the movement the apple is hanging from a branch three meters high:

${\displaystyle y(0)=y_{0}=3~\mathrm {m} }$

and is at rest:

${\displaystyle y'(0)=v_{0}=0~\mathrm {m} /\mathrm {s} }$.

This initial condition now draws the one function in the solution set of the differential equation

${\displaystyle \Rightarrow y(t)=3~\mathrm {m} -{\tfrac {1}{2}}gt^{2}}$

as the uniquely determined solution of the initial value problem.

generalization

In the case of partial differential equations , i.e. when the function you are looking for depends not only on one but on several variables, boundary conditions are often used instead of initial conditions. Sometimes the special case of a boundary condition, the domain of which forms a hyperplane in the full domain of the differential equation, is called the initial condition.

annotation

1. ↑ In the case of functions of time, the specific point can be the point in time, which can be found as the beginning of time in the terms start condition and start value.

literature

• Hans Heiner Storrer: Introduction to the mathematical treatment of the natural sciences. Volume 2, 1st edition, Birkhäuser Verlag, 1995, ISBN 978-37-6435-325-4 .
• Klaus D. Schmidt: Mathematics. Fundamentals for economists, 2nd revised edition, Springer Verlag Berlin - Heidelberg, Berlin 2000, ISBN 978-3-540-66521-2 .

Web links

• Solve differential equations (accessed October 15, 2015)
• Dependence on initial conditions (accessed October 15, 2015)
• Parabolic boundary initial value problems with random initial condition (accessed October 15, 2015)