Initial value problem

Short description: Type of calculus problem involving ordinary differential equations

In multivariable calculus, an initial value problem^{[lower-alpha 1]} (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the 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 evolves with time given the initial conditions of the problem.

Definition

An initial value problem is a differential equation

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

together with a point in the domain of [math]\displaystyle{ f }[/math]

[math]\displaystyle{ (t_0, y_0) \in \Omega, }[/math]

called the initial condition.

A solution to an initial value problem is a function [math]\displaystyle{ y }[/math] that is a solution to the differential equation and satisfies

[math]\displaystyle{ y(t_0) = y_0. }[/math]

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

Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. [math]\displaystyle{ 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 t₀ if f is continuous on a region containing t₀ and y₀ and satisfies the 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 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 class C¹, or even 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]\displaystyle{ y'(t) = 0.85 y(t) }[/math] and [math]\displaystyle{ y(0) = 19 }[/math]. We are trying to find a formula for [math]\displaystyle{ y(t) }[/math] that satisfies these two equations.

Rearrange the equation so that [math]\displaystyle{ y }[/math] is on the left hand side

[math]\displaystyle{ \frac{y'(t)}{y(t)} = 0.85 }[/math]

Now integrate both sides with respect to [math]\displaystyle{ t }[/math] (this introduces an unknown constant [math]\displaystyle{ B }[/math]).

[math]\displaystyle{ \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt }[/math]

[math]\displaystyle{ \ln |y(t)| = 0.85t + B }[/math]

Eliminate the logarithm with exponentiation on both sides

[math]\displaystyle{ | y(t) | = e^Be^{0.85t} }[/math]

Let [math]\displaystyle{ C }[/math] be a new unknown constant, [math]\displaystyle{ C = \pm e^B }[/math], so

[math]\displaystyle{ y(t) = Ce^{0.85t} }[/math]

Now we need to find a value for [math]\displaystyle{ C }[/math]. Use [math]\displaystyle{ y(0) = 19 }[/math] as given at the start and substitute 0 for [math]\displaystyle{ t }[/math] and 19 for [math]\displaystyle{ y }[/math]

[math]\displaystyle{ 19 = C e^{0.85 \cdot 0} }[/math]

[math]\displaystyle{ C = 19 }[/math]

this gives the final solution of [math]\displaystyle{ y(t) = 19e^{0.85t} }[/math].

Second example

The solution of

[math]\displaystyle{ y'+3y=6t+5,\qquad y(0)=3 }[/math]

can be found to be

[math]\displaystyle{ y(t)=2e^{-3t}+2t+1. \, }[/math]

Indeed,

[math]\displaystyle{ \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]\displaystyle{ y'=y^{\frac 2 3},\qquad y(0)=0 }[/math]

[math]\displaystyle{ \int \frac{y'}{y^{\frac 2 3}}\,dt = \int y^{-\frac 2 3}\,dy =\int 1\,dt }[/math]

[math]\displaystyle{ 3 (y(t))^{\frac 1 3}=t+B }[/math]

Applying initial conditions we get [math]\displaystyle{ B=0 }[/math], hence the solution:

[math]\displaystyle{ y(t)= \frac {t^3} {27} }[/math].

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

[math]\displaystyle{ 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

↑ Also called a Cauchy problem by some authors.^{[citation needed]}

References

Coddington, Earl A.; Levinson, Norman (1955). Theory of ordinary differential equations. New York-Toronto-London: McGraw-Hill Book Company, Inc.. https://archive.org/details/theoryofordinary00codd.
Hirsch, Morris W. and Smale, Stephen (1974). Differential equations, dynamical systems, and linear algebra. New York-London: Academic Press.
Okamura, Hirosi (1942). "Condition nécessaire et suffisante remplie par les équations différentielles ordinaires sans points de Peano" (in French). Mem. Coll. Sci. Univ. Kyoto Ser. A 24: 21–28.
Agarwal, Ravi P.; Lakshmikantham, V. (1993). Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. Series in real analysis. 6. World Scientific. ISBN 978-981-02-1357-2. https://books.google.com/books?id=q4OkW4H8BCUC.
Polyanin, Andrei D.; Zaitsev, Valentin F. (2003). Handbook of exact solutions for ordinary differential equations (2nd ed.). Boca Raton, Florida: Chapman & Hall/CRC. ISBN 1-58488-297-2.
Robinson, James C. (2001). Infinite-dimensional dynamical systems: An introduction to dissipative parabolic PDEs and the theory of global attractors. Cambridge: Cambridge University Press. ISBN 0-521-63204-8.

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[1] Also called a Cauchy problem by some authors.^{[citation needed]}

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