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## 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:

*u*,*u*, and_{xx}*u*are given on the boundary. This can be implemented by using Dirichlet conditions for_{yy}*u*,*P*, and*Q*.*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*and_{x}*u*are known on the boundary. These boundary conditions can be implemented by using a Dirichlet condition for_{y}*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}= -xBoundary 4:

*u =*sin(

*y*),

*u*sin(

_{x}=*y*),

*u*cos(

_{y}=*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 |

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