Initial value problem: Difference between revisions

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{{Short description|Type of calculus problem}}{{Inline citations|date=May 2024}}

In the field of [[differential equation]]s, an '''initial value problem''' (also called a '''[[Cauchy problem]]''' by some authors{{cn|date=December 2018}}) is an [[ordinary differential equation]] together with a specified value, called the [[initial condition]], of the unknown function at a given point in the ___domain of the solution. In [[physics]] or other sciences, modeling a system frequently amounts to solving an initial value problem; in this context, the differential initial value is an equation that is an evolution equation specifying how, given initial conditions, the system will [[time evolution|evolve with time]].

In the field of [[differentialmultivariable equationcalculus]]s, an '''initial value problem''' (also{{efn|Also called a '''[[Cauchy problem]]''' by some authors.{{cn|date=December 2018}}}} ('''IVP''') is an [[ordinary differential equation]] together with a specified value, called thean [[initial condition]], which specifies the value of the unknown [[function (mathematics)|function]] at a given point in the [[___domain of thea solutionfunction|___domain]]. Modeling Ina system in [[physics]] or other sciences, modeling a system frequently amounts to solving an initial value problem;. inIn thisthat context, the differential initial value is an equation thatwhich is an evolution equation specifyingspecifies how, given initial conditions, the system will [[time evolution|evolveevolves with time]] given the initial conditions of the problem.

== Definition ==

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:<math>y'(t) = f(t, y(t))</math> with <math>f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n</math> where <math>\Omega</math> is an open set of <math>\mathbb{R} \times \mathbb{R}^n</math>,

together with a point in the ___domain of <math>f</math>

:<math>(t_0, y_0) \in \Omega,</math>,

called the [[initial condition]].

A '''solution''' to an initial value problem is a function <math>y</math> that is a solution to the differential equation and satisfies

:<math>y(t_0) = y_0.</math>.

In higher dimensions, the differential equation is replaced with a family of equations <math>y_i'(t)=f_i(t, y_1(t), y_2(t), \dotsc)</math>, and <math>y(t)</math> is viewed as the vector <math>(y_1(t), \dotsc, y_n(t))</math>, most commonly associated with the position in space. More generally, the unknown function <math>y</math> can take values on infinite dimensional spaces, such as [[Banach space]]s or spaces of [[distribution (mathematics)|distributions]].

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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 ƒ''f'' is continuous on a region containing ''t''₀ and ''y''₀ and satisfies the [[Lipschitz continuity|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 (mathematics)|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.

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[[Hiroshi Okamura]] obtained a [[necessary and sufficient condition]] for the solution of an initial value problem to be unique. This condition has to do with the existence of a [[Lyapunov function]] for the system.

In some situations, the function ƒ''f'' is not of [[Smooth function|class ''C''¹]], or even [[Lipschitz continuity|Lipschitz]], so the usual result guaranteeing the local existence of a unique solution does not apply. The [[Peano existence theorem]] however proves that even for ƒ''f'' merely continuous, solutions are guaranteed to exist locally in time; the problem is that there is no guarantee of uniqueness. The result may be found in Coddington & Levinson (1955, Theorem 1.3) or Robinson (2001, Theorem 2.6). An even more general result is the [[Carathéodory existence theorem]], which proves existence for some discontinuous functions ƒ''f''.

==Examples==

A simple example is to solve <math>y'(t) = cos0.85 y(xt)</math> and <math>y(0) = 119</math>. We are trying to find a formula for <math>y(t)</math> that satisfies these two equations.

StartRearrange bythe notingequation so that <math>y' = \frac{dy}{dt}</math>, sois on the left hand side

: <math>\frac{dyy'(t)}{dty(t)} = 0.85 y</math>

Now rearrangeintegrate theboth equationsides sowith thatrespect to <math>yt</math> is(this onintroduces thean leftunknown andconstant <math>tB</math> on the right).

: <math>\int \frac{dyy'(t)}{y(t)}\,dt = \int 0.85dt85\,dt</math>

: <math>\ln | y (t)| = e^Be^{0.85t} + B </math>

Eliminate the logarithm with exponentiation on both sides

Now integrate both sides (this introduces an unknown constant <math>B</math>).

: <math>\ln | y(t) | = e^Be^{0.85t + B} </math>

Eliminate the <math>\ln</math>

: <math> | y | = e^Be^{0.85t} </math>

Let <math>C</math> be a new unknown constant, <math>C = \pm e^B</math>, so

: <math> y(t) = Ce^{0.85t} </math>

Now we need to find a value for <math>C</math>. Use <math>y(0) = 19</math> as given at the start and substitute 0 for <math>t</math> and 19 for <math>y</math>

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&= 6t+5.

\end{align} </math>

:

'''Third example'''

The solution of

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

</math>

<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}}

==See also==

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* [[Constant of integration]]

* [[Integral curve]]

* [[Norton's dome]]

== References ==

{{refbegin}}

* {{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]]