Platform: All Platforms Versions: All versions

## Problem Description

How do I solve a PDE with space derivatives of order higher than two? For example, the equation

for a function

.

## Solution

Introduce names for the second derivatives of u, say

.

Then the equation can be written

.

You can now solve the following equivalent system of PDEs for the variables u, P, and Q with COMSOL Multiphysics:

This system can be entered as a General form PDE for the variables u, P, and Q, with the equations:

`Px, Py + Qy``f`

`ux,0` `P`

`0,uy` `Q`

For boundary conditions, consider the following examples:

1. u, uxx, and uyy are given on the boundary. This can be implemented by using Dirichlet conditions for u, P, and Q.

2. u and its normal derivative du/dn are both given on the boundary. This implies that the derivative of u in the tangential direction can also be computed. Hence, expressions for ux and uy are known on the boundary. These boundary conditions can be implemented by using a Dirichlet condition for u, and Neumann conditions for P and Q:

`u`,
,
,

where

-`nx`

and

-`ny`.

The supplied example model solves this system of equations with f = 1 and the following boundary conditions:
Boundaries 1 and 2: u = 0, uxx = uyy = 0
Boundary 3: u = x, uxx = 0, uyy = -x
Boundary 4: u = sin(y), ux = sin(y), uy = cos(y)

Other equations of order higher than two can be treated similarly, by introducing names for certain derivatives of u . Use as high an order of lagrange elements as you can afford, to make the higher order derivatives of _u as smooth as possible.

## Related Files

 high_order_derivatives_60.mph 285 KB