partInt

partInt - Do partial integrations

Calling sequence:

partInt(expr,var):

Parameters

expr - The expression that is to be partially integrated.

var - variable or list of variables appearing inside ElementInt() in expr.

Synopsis:

The expression in expr is integrated partially once, so that the order of diffferentiation of the variable(s) in var is reduced by one. expr is expected to contain ElementInt() integrals, only such are affected.
var can be either a variable appearing in expression or a list of variables. If var is a list, one partial integration is performed for each variable, so that partInt(expr,[u(x),v(x)]) is equivalent to partInt(partInt(expr,u(x)),v(x)).
The partial integrations produce boundary integrals, BoundaryInt(), and components of normal vectors, called BdNx, BdNy etc. The normal vector is expected to be outwardly directed and normal to the boundary.
When expressions generated by partInt are passed to the code generation commands, such as mkFem, the normal vector components BdNx, BdNy etc., are replaced by their appropriate definitions which should appear in the *_bsf files specifying the elements.

Examples:

> partInt(ElementInt(f(x)*diff(g(x),x)),g(x)); /d \ -ElementInt(g(x) |-- f(x)|) + BoundaryInt(BdNx g(x) f(x)) \dx /

> partInt( 0 = ElementInt(f(x)*diff(g(x),x)) ,g(x)); /d \ 0 = -ElementInt(g(x) |-- f(x)|) + BoundaryInt(BdNx g(x) f(x)) \dx /

> Lapl:= nab(nab(u(x,y))[i])[j] &t delta[i,j]; / 2 \ / 2 \ |d | |d | Lapl := |--- u(x, y)| + |--- u(x, y)| | 2 | | 2 | \dx / \dy / > partInt( ElementInt(test(x,y)*Lapl) ,u(x,y)); /d \ /d \ -ElementInt(|-- u(x, y)| |-- test(x, y)|) \dx / \dx / /d \ + BoundaryInt(BdNx |-- u(x, y)| test(x, y)) \dx / /d \ /d \ - ElementInt(|-- u(x, y)| |-- test(x, y)|) \dy / \dy / /d \ + BoundaryInt(BdNy |-- u(x, y)| test(x, y)) \dy /