example of eliminating higher-order derivatives

To show how a partial differential equationMathworldPlanetmath involving partial derivativesMathworldPlanetmath of order higher than the first can be re-expressed as a system involving only first derivativesMathworldPlanetmath to which the simple form of the Cauchy-Kowalewski theorem can be applied, consider the following example:


First, the second derivative with respect to t can be eliminated by introducing a new variable ut and a adding a differential equation which sets ut equal to the derivativePlanetmathPlanetmath of u.


To eliminate the second derivatives involving x, we first introduce variables uxx, utx and utt. To obtain an equation with utt/t on the left-hand side, differentiate the original equation with respect to t


The notation f1 and f2 denotes the partial derivatives of the function f with respect to its first and second argumentsMathworldPlanetmath. Consider the following system of equations:


These equations will be satisfied if u satisfies the orignal equation and the new variables are set equal to appropriate partial derivatives of u. Conversely, by eliminating variables, one can show that, for any solution of the above system of equations which satisfies the boundary conditionsMathworldPlanetmath


on the surface t=0, the variable u will satisfy the original equation.

The same approach can be used to rewrite systems of any number of equations in any number of variables involving derivatives of arbitrary order as a system involving only first derivatives such that the left hand side is the time derivative of a variable and the right hand side only involves spatial derivatives. However, trying to write out the solution in general can lead to awkward and complicated notation. There is no particularly good reason to do so because, upon seeing the example presented here, it becomes obvious that it is possible.

Title example of eliminating higher-order derivatives
Canonical name ExampleOfEliminatingHigherorderDerivatives
Date of creation 2013-03-22 14:37:10
Last modified on 2013-03-22 14:37:10
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 11
Author rspuzio (6075)
Entry type Example
Classification msc 35A10