Elliptic differential equation
[latex]
\[
\textbf{Explanation of the Elliptic Differential Equation:}
\]
\[
a(x,y) \frac{\partial^2 u}{\partial x^2} + 2b(x,y) \frac{\partial^2 u}{\partial x \partial y} + c(x,y) \frac{\partial^2 u}{\partial y^2} + d(x,y) \frac{\partial u}{\partial x} + e(x,y) \frac{\partial u}{\partial y} + f(x,y) u = g(x,y)
\]
1. The function \( u(x, y) \) is the unknown solution we are solving for.
2. The terms \( a(x,y), b(x,y), c(x,y), d(x,y), e(x,y), f(x,y), \text{ and } g(x,y) \) are coefficients and functions of the independent variables \( x \) and \( y \).
3. The equation includes:
– Second-order partial derivatives of \( u \):
– \( \frac{\partial^2 u}{\partial x^2} \): the second derivative with respect to \( x \).
– \( \frac{\partial^2 u}{\partial y^2} \): the second derivative with respect to \( y \).
– \( \frac{\partial^2 u}{\partial x \partial y} \): the mixed partial derivative.
– First-order partial derivatives of \( u \):
– \( \frac{\partial u}{\partial x} \): the derivative with respect to \( x \).
– \( \frac{\partial u}{\partial y} \): the derivative with respect to \( y \).
– The term \( f(x,y)u \): represents the coefficient multiplying the unknown function \( u(x, y) \).
– \( g(x, y) \): the source term or forcing function.
4. For this equation to be classified as elliptic, the coefficients \( a(x, y), b(x, y), \text{ and } c(x, y) \) must satisfy the discriminant condition:
\[
b^2(x,y) – a(x,y)c(x,y) < 0
\]
This ensures that the equation behaves like an elliptic equation, which typically models steady-state phenomena, such as heat conduction or incompressible fluid flow.
\]