How do I specify mixed boundary conditions on two boudaries for a system of coupled elliptic partial differential equations in Partial Differential Equations Toolbox 1.0.18 (R2011a)?

I have a system of coupled elliptic partial differential equations. The boundary conditions are needed to be specified on two sections of the boundary such that there is one mixed boundary condition (both Dirichlet and Neumann) on each section. For the first section the Dirichlet boundary condition is applied on the first variable, and for the second section the Dirichlet boundary condition is applied on second variable. The Neumann boundary condition for both the sections is generalized. How can I do that using the PDETOOL GUI in MATLAB.

Answer 1

[MathWorks Support Team]

You can specify mixed boundary conditions in the PDE Toolbox 1.0.18 (R2011a) via the following approach. For the first section of the boundary, you can write the boundary conditions in the following form:

(h11 h12)u = r1

c*grad(u)+(q11;q21,q12;q22)u = (g1;g2)

Here, the Dirichlet boundary condition can be applied on first variable u(1).

For the second section of the boundary, you can write the boundary conditions in the following form:

(h21 h22)u = r2

c*grad(u)+(q11;q21,q12;q22)u = (g1;g2)

Here, the Dirichlet boundary condition can be applied on second variable u(2).

In the above equations:

h is a 2-by-2 matrix

r is a 2-by-1 vector

(h and r correspond to Dirichlet boundary condition of the form 'hu = r')

q is a 2-by-2 matrix

g is a 2-by-1 vector

(q and g correspond to mixed boundary condition)

In the PDETOOL GUI, choose the boundary section on which you want to specify the boundary condition, navigate to Boundary Menu --> Specify Boundary Condition and choose the 'Mixed' option for the boundary condition, and specify the h,r,q and g matrices/vectors as required.