Boundary Conditions of a Typical Partial Differential Equation in Two Dimensions

For the purpose of simplicity, we shall start our consideration with partial differential equations in only two variables and linear in the second derivatives. Such equations have the general form

$$A(x,y)\frac{\partial^2\psi}{\partial x^2} +2B(x,y)\frac{\partial^... ...psi ,\frac{\partial\psi}{\partial x},\frac{\partial\psi}{\partial y}\right)\,.$$

Such an equation is called a quasilinear second order partial differential equation. If the expression $\Phi$ where linear in $\psi$ , i.e., if

$$\Phi = D(x,y)\frac{\partial\psi}{\partial x} +E(x,y)\frac{\partial\psi} {\partial y} +F(x,y)\psi +G(x,y)\,,$$

then the equation would be a linear p.d.e., but this need not be the case.

The equation has a nondenumerable infinity of solution. In order to single out a unique solution, the to-be-found function $\psi (x,y)$ must satisfy additional conditions. They are usually specified at the boundary of the domain of the p.d.e.

In three dimensional space, this boundary is a surface, but in our two dimensional case, we have a boundary line which can be specified by the parametrized curve

$$x = \xi (s)$$

$$y = \eta (s)\,,$$

where $s$ is the arclength parameter

$$s=\int ds = \int\sqrt{dx^2+dy^2}\,.$$

The tangent to this curve has components

$$\left(\frac{d\xi}{ds}\,,~\frac{d\eta}{ds}\right)\,.$$

They satisfy

$$\left(\frac{d\xi}{ds}\right)^2+\left(\frac{d\eta}{ds}\right)^2=1~.$$

The normal to this boundary curve has components

$$\left(\frac{d\eta}{ds}\,,-\frac{d\xi}{ds}\right) = \vec n\,.$$

We assume that $\vec n$ points towards the interior of the domain where the solution is to be found. If this is not the case, we reverse the signs of the components of it.

The additional conditions which the to-be-found solution $\psi$ is to satisfy are imposed at this boundary curve, and they are conditions on the partial derivatives and the value of the function $\psi$ evaluated at the curve.

The boundary curve accomodates three important types of boundary conditions.

1. Dirichlet conditions: $\psi (s)$ is specified at each point of the boundary.
2. Neumann conditions: $\frac{d\psi}{dn}(s)=\vec n\cdot\nabla \psi$ , the normal componet of the graident of $\psi$ is specified at each point of the boundary.
3. Cauchy conditions: $\psi (s)$ and $\frac{d\psi}{dn}(s)$ are specified at each point of the boundary. The parameter $s$ is usually a time parameter. Consequently, Cauchy conditions are also called intial value conditions or initial value data or simply Cauchy data.

There exists also the mixed Dirichlet-Neumann conditions. They are intermediate between the Dirichlet and the Neumann boundary conditions, and they are given by

$$\alpha (s)\psi (s)+\beta (s)\frac{d\psi}{dn}(s) = f(s)\,.$$

Here $\alpha (s)$ , $\beta (s)$ , and $f(s)$ are understood to be given on the boundary.

We recall that in the theory of ordinary second order differential equations, a unique solution was obtained once the solution and its derivative were specified at a point. The generalization of this condition to partial differential equations consists of the Cauchy boundary conditions.

Consequently, we now inquire whether the solution of the partial differential equation is uniquely determined by specifying Cauchy boundary conditions on the boundary $(\xi (s),\eta (s))$ .