Editing Initial value problem

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

In [[multivariable calculus]], an '''initial value problem'''{{efn|Also called a '''[[Cauchy problem]]''' by some authors.{{cn|date=December 2018}}}} ('''IVP''') 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 ==
An '''initial value problem''' is a differential equation
:<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]].

Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. <math>y''(t)=f(t,y(t),y'(t))</math>.

== Existence and uniqueness of solutions ==

The [[Picard–Lindelöf theorem]] guarantees a unique solution on some interval containing ''t''<sub>0</sub> if ''f'' is continuous on a region containing ''t''<sub>0</sub> and ''y''<sub>0</sub> 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.

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". This version is essentially a special case of the Banach fixed point theorem.

[[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 ''f'' is not of [[Smooth function|class ''C''<sup>1</sup>]], 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 ''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 ''f''.

==Examples==
A simple example is to solve <math>y'(t) = 0.85 y(t)</math> and <math>y(0) = 19</math>.  We are trying to find a formula for <math>y(t)</math> that satisfies these two equations.

Rearrange the equation so that <math>y</math> is on the left hand side

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

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

: <math>\int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt</math>
: <math>\ln |y(t)| = 0.85t + B </math>

Eliminate the logarithm with exponentiation on both sides

: <math> | y(t) | = 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>

: <math> 19 = C e^{0.85 \cdot 0}</math>
: <math> C = 19 </math>

this gives the final solution of <math> y(t) = 19e^{0.85t}</math>.

;Second example
The solution of

: <math>y'+3y=6t+5,\qquad y(0)=3</math>

can be found to be

: <math>y(t)=2e^{-3t}+2t+1. \,</math>

Indeed,

: <math> \begin{align}
y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\
      &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\
      &= 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==
* [[Boundary value problem]]
* [[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 & 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 }}
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[[Category:Boundary conditions]]

[[it:Problema ai valori iniziali]]