Initial value problem: Difference between revisions

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{{Short description|Type of calculus problem}}{{Inline involvingcitations|date=May ordinary differential equations2024}}

In [[multivariable calculus]], an '''initial value problem'''^{[[#Notes{{efn|[Also called a] '''[[Cauchy problem]]}''' by some authors.{{cn|date=December 2018}}}} ('''ivpIVP''') is an [[ordinary differential equation]] together with an [[initial condition]] which specifies the value of the unknown [[function (mathematics)|function]] at a given point in the [[___domain of a function|___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 [[time evolution|evolves with time]] given the initial conditions of the problem.

== Definition ==

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== Existence and uniqueness of solutions ==

For a large class of initial value problems, the existence and uniqueness of a solution can be illustrated through the use of a calculator.

The [[Picard–Lindelöf theorem]] guarantees a unique solution on some interval containing ''t''₀ if ''f'' is continuous on a region containing ''t''₀ and ''y''₀ and satisfies the [[Lipschitz continuity|Lipschitz condition]] on the variable ''y''.

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&= 6t+5.

\end{align} </math>

:

'''Third example'''

The solution of

<math>y'=y^{\frac 2 3},\qquad y(0)=0

</divmath>

<math>\int \frac{y'}{y^{\frac 2 3}}\,dt = \int y^{-\frac 2 3}\,dy =\int 1\,dt</math>

<math>3 (y(t))^{\frac 1 3}=t+B

</math>

Applying initial conditions we get <math> B=0 </math>, hence the solution:

<math>y(t)= \frac {t^3} {27}

</math>.

However, the following function is also a solution of the initial value problem:

<math>f(t) = \left\{ \begin{array}{lll} \frac{(t-t_1)^3}{27} & \text{if} & t \leq t_1 \\ 0 & \text{if} & t_1 \leq x \leq t_2 \\ \frac{(t-t_2)^3}{27} & \text{if} & t_2 \leq t \\ \end{array} \right.</math>

The function is differentiable everywhere and continuous, while satisfying the differential equation as well as the initial value problem. Thus, this is an example of such a problem with infinite number of solutions.

==Notes==

{{notelist}}

<div style="font-size:89%">

: <small>[a]</small> Also called a '''[[Cauchy problem]]''' by some authors{{cn|date=December 2018}}

</div>

==See also==

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* [[Constant of integration]]

* [[Integral curve]]

* [[Norton's dome]]

== References ==

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* {{cite book |author1=Coddington, Earl A. |author2=Levinson, Norman | title=Theory of ordinary differential equations |url=https://archive.org/details/theoryofordinary00codd |url-access=registration | publisher=McGraw-Hill Book Company, Inc. | ___location=New York-Toronto-London | year=1955 }}

* {{cite book | author=[[Morris W. Hirsch|Hirsch, Morris W.]] and [[Stephen Smale|Smale, Stephen]] | title=Differential equations, dynamical systems, and linear algebra | publisher=Academic Press | ___location=New York-London | year=1974 }}

* {{cite journal | last=Okamura | first=Hirosi | title=Condition nécessaire et suffisante remplie par les équations différentielles ordinaires sans points de Peano | journal=Mem. Coll. Sci. Univ. Kyoto Ser. A. | volume=24 | year=1942 | language=French | pages=21&ndash;28 |mr=0031614 }}

* {{cite book |author1=Agarwal, Ravi P. |author2=Lakshmikantham, V. | title=Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations | url=https://books.google.com/books?id=q4OkW4H8BCUC | series=Series in real analysis | volume=6 | year=1993 | publisher=World Scientific | isbn=978-981-02-1357-2}}

* {{cite book |author1=Polyanin, Andrei D. |author2=Zaitsev, Valentin F. | title=Handbook of exact solutions for ordinary differential equations | edition=2nd | publisher=Chapman &amp; Hall/CRC | ___location=Boca Raton, Florida | year=2003 | isbn=1-58488-297-2 }}

* {{cite book | last=Robinson | first=James C. | title=Infinite-dimensional dynamical systems: An introduction to dissipative parabolic PDEs and the theory of global attractors | publisher=Cambridge University Press | ___location=Cambridge | year=2001 | isbn=0-521-63204-8 }}

{{refend}}

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[[Category:Boundary conditions]]

[[el:Αρχική τιμή]]

[[it:Problema ai valori iniziali]]

[[sv:Begynnelsevärdesproblem]]