How do I solve a PDE with space derivatives of order higher than two?

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 + Qyf

ux,0 P

0,uy Q

For boundary conditions, consider the following examples:

1. u, u_xx, and u_yy 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 u_x and u_y 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, u_xx = u_yy = 0
Boundary 3: u = x, u_xx = 0, u_yy = -x
Boundary 4: u = sin(y), u_x = sin(y), u_y = 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.

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