The previous post looked at using maximum principles to bound the solution to a boundary value problem. This is a similar post, focusing on an **initial value problem**. As before we start with the differential operator

*L*[*u*] = *u*” + *g*(*x*)*u’ + **h*(*x*)

where *g *and *h* are bounded and *h* is non-positive. We are interested in solutions to the equation *L*[*u*] = *f*, but this time we want to specify the *initial* conditions *u*(0) = *u*‘_{0} and *u*(0) = *u*‘_{0}. We will use the same maximum principle as before, but a slightly different approach to constructing upper and lower bounds on solutions. Our conclusion will also be a little different: this time we’ll get upper and lower bounds not only on the solution *u* but also on its derivative *u*‘.

We look for a function *z*(*x*) satisfying the differential *inequality *

*L*[*u*] ≥ *f*

and the two initial *inequalities **z*(0) ≥ *u*‘_{0} and *z*(0) ≥ *u*‘_{0}. We also look for a function *w*(*x*) satisfying the same inequalities in the opposite direction, i.e. replacing ≥ with ≤. (Note that replacing equality with inequality in the differential equation and initial conditions makes it much easier to find solutions.)

The maximum principle tells us that z and *w* provide upper and lower bounds respectively on *u*, and their derivatives provide bounds on the derivative of *u*. That is,

*w*(*x*) ≤ *u*(*x*) ≤ *z*(*x*)

and

*w*‘(*x*) ≤ *u*‘(*x*) ≤ *z*‘(*x*).

We illustrate this technique on a variation on Bessel’s equation:

*u*”(*x*) + *u*‘(*x*)/*x* − *u*(*x*) = 0

with initial conditions *u*(0) = 1 and *u*‘(0) = 0. We wish to bound the solution *u* on the interval [0, 1].

As candidates, we try *z* = *c*_{1}*x*^{2} + 1 and *w* = *c*_{2}*x*^{2} + 1. (Why not a general quadratic polynomial? The initial conditions tell us that the linear coefficient will be 0 and the constant term will be 1.)

Since

*L*[*cx*^{2} + 1] = *c*(4 − *x*^{2}) − 1

it suffices to pick *c*_{1} = 1/3 and *c*_{2} = 1/4. This tells us that

*x*/4 + 1 ≤ *u*(*x*) ≤ *x*/3 + 1.

To see how well these bounds work, we compare with the exact solution *I*_{0}, the so-called modified Bessel function of the first kind.

The lower bound is very good, though a comparably good upper bound would take a more careful choice of *z*.

Source: Maximum Principles in Differential Equations by Protter and Weinberger

Comments are closed.