Initial value problem

In mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with 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 equation is an evolution equation specifying how, given initial conditions, the system will evolve with time.

Definition

An initial value problem is a differential equation

$y'(t)=f(t,y(t))\quad {\text{with}}\quad f:\mathbb {R} \times \mathbb {R} \to \mathbb {R}$

together with a point in the domain of ƒ

$(t_{0},y_{0})\in \mathbb {R} \times \mathbb {R} ,$

called the initial condition.

A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies

$y(t_{0})=y_{0}.\,$

This statement subsumes problems of higher order, by interpreting y as a vector. For derivatives of second or higher order, new variables (elements of the vector y) are introduced.

More generally, the unknown function y can take values on infinite dimensional spaces, such as Banach spaces or spaces of distributions.

Existence and uniqueness of solutions

For a large class of initial value problems, the existence and uniqueness of a solution can be demonstrated.

The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t₀ if ƒ and its partial derivative ∂ƒ / ∂y are continuous on a region containing t₀ and y₀. 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 ƒ 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 ƒ 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.

Example

The solution of

$y'+3y=6t+5,\qquad y(0)=3$

can be found to be

$y(t)=2e^{-3t}+2t+1.\,$

Indeed,

${\begin{aligned}y'+3y&={\tfrac {d}{dt}}(2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1)\\&=(-6e^{-3t}+2)+(6e^{-3t}+6t+3)\\&=6t+5.\end{aligned}}$

See also

• Boundary value problem
• Constant of integration
• Integral curve

References

• Coddington, Earl A. and Levinson, Norman (1955). Theory of ordinary differential equations. New York-Toronto-London: McGraw-Hill Book Company, Inc.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Hirsch, Morris W. and Smale, Stephen (1974). Differential equations, dynamical systems, and linear algebra. New York-London: Academic Press.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Okamura, Hirosi (1942). "Condition nécessaire et suffisante remplie par les équations différentielles ordinaires sans points de Peano". Mem. Coll. Sci. Univ. Kyoto Ser. A. (in French). 24: 21–28.
• Polyanin, Andrei D. and Zaitsev, Valentin F. (2003). Handbook of exact solutions for ordinary differential equations (2nd ed.). Boca Raton, FL: Chapman & Hall/CRC. ISBN 1-58488-297-2.{{cite book}}: CS1 maint: multiple names: authors list (link)
• 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.

External links

• MIT Video Lecture on Initial Value Problems
• Introduction to modeling via differential equations, with critical remarks and special emphasis on the role of initial conditions.