Initial value problem
In the field of differential equations, an initial value problem (also called a Cauchy problem by some authors) 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 evolve with time.
An initial value problem is a differential equation
- with where is an open set of ,
together with a point in the domain of
called the initial condition.
A solution to an initial value problem is a function that is a solution to the differential equation and satisfies
In higher dimensions, the differential equation is replaced with a family of equations , and is viewed as the vector , most commonly associated with the position in space. More generally, the unknown function 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. .
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 t0 if ƒ 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 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.
A simple example is to solve and . We are trying to find a formula for that satisfies these two equations.
Start by noting that , so
Now rearrange the equation so that is on the left and on the right
Now integrate both sides (this introduces an unknown constant ).
Let be a new unknown constant, , so
Now we need to find a value for . Use as given at the start and substitute 0 for and 19 for
this gives the final solution of .
- Second example
The solution of
can be found to be
- Coddington, Earl A.; Levinson, Norman (1955). Theory of ordinary differential equations. New York-Toronto-London: McGraw-Hill Book Company, Inc.
- Hirsch, Morris W. and Smale, Stephen (1974). Differential equations, dynamical systems, and linear algebra. New York-London: Academic Press.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. MR 0031614.
- 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.
- 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.