
# 3.1: Explicit Boundary Conditions

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For the problems of interest here we shall only consider linear boundary conditions, which express a linear relation between the function and its partial derivatives, e.g., $u(x,y=0) + x \frac{\partial u}{\partial x}(x,y=0)=0.$ As before the maximal order of the derivative in the boundary condition is one order lower than the order of the PDE. For a second order differential equation we have three possible types of boundary condition

#### Dirichlet Boundary Condition

When we specify the value of $$u$$ on the boundary, we speak of Dirichlet boundary conditions. An example for a vibrating string with its ends, at $$x=0$$ and $$x=L$$, fixed would be

$u(0,t) = u(L,t) = 0.$

#### von Neumann Boundary Conditions

In multidimensional problems the derivative of a function w.r.t. to each of the variables forms a vector field (i.e., a function that takes a vector value at each point of space), usually called the gradient. For three variables this takes the form $\mbox{grad} \space f(x,y,z) = {\nabla} f(x,y,z) = \left (\frac{\partial f}{\partial x}(x,y,z),\frac{\partial f}{\partial y}(x,y,z),\frac{\partial f}{\partial z}(x,y,z)\right ).$

Typically we cannot specify the gradient at the boundary since that is too restrictive to allow for solutions. We can – and in physical problems often need to – specify the component normal to the boundary, see Figure $$\PageIndex{1}$$ for an example. When this normal derivative is specified we speak of von Neumann boundary conditions.

In the case of an insulated (infinitely thin) rod of length $$a$$, we can not have a heat-flux beyond the ends so that the gradient of the temperature must vanish (heat can only flow where a difference in temperature exists). This leads to the BC

$\frac{\partial u}{\partial x}(0,t) = \frac{\partial u}{\partial x}(a,t) = 0.$

#### Mixed (Robin’s) Boundary Conditions

We can of course mix Dirichlet and von Neumann boundary conditions. For the thin rod example given above we could require

$u(0,t) + \frac{\partial u}{\partial x}(0,t) = u(a,t) + \frac{\partial u}{\partial x}(a,t) = 0.$