A BIE for a Second Order Elliptic Partial Differential Equation with Variable Coefficients

Abstract

The second order elliptic partial differential equation

$$ \frac{\partial }{{{\partial_x}}}\left[ {K\left( {x,y} \right)\frac{{\partial \phi }}{{\partial x}}} \right] + \frac{\partial }{{\partial y}}\left[ {K\left( {x,y} \right)\frac{{\partial \phi }}{{\partial y}}} \right] = 0,\quad \quad \phi = \phi (x,y), $$

((1))

with K(x,y) being a suitably given function of the independent variables x and y, occurs in many applications Of particular interest is the boundary value problem which requires solving equation (1) in a finite region R on the Oxy plane subject to the conditions

$$ \begin{gathered} \quad \phi = u(x,y)\quad on\quad {C_1} \hfill \\ \frac{{\partial \phi }}{{\partial n}} = v(x,y)\quad on\quad {C_2} \hfill \\ \end{gathered} $$

((2))

where C = C ₁ ∪ C ₂ is the boundary of the region R, u and υ are suitably prescribed functions of x and y, and ∂∅/∂n = $\overrightarrow {n \cdot } \overrightarrow \Delta \phi$, with $\overrightarrow n$ being a unit normal outward vector to C.