I'm a bit lost in all the numerous methods for solving differential equations and I would be very grateful if someone could point me to some direction.
I want to solve the following boundary conditioned differential equation:
$\vec a_1+a_2\overrightarrow{\nabla f(x,y)}+a_3\Delta f(x,y)\cdot\overrightarrow{\nabla f(x,y)} +\left[a_4+a_5\Delta f(x,y)\right]^\frac{2}{3}\cdot\overrightarrow{\nabla(\Delta f(x,y))}=0$
$a_i$ - are constants
$\nabla$ - is the gradient
$\Delta$ - is the Laplace operator
This equation is solved on a rectangular domain. let say for a start that the boundaries are: $f(0,y)=0$ ; $f(x,0)=0$ ; $f(x,a)=C_1$ ; $\nabla f(b,y)=C_2$
$(a,b)$ - Is the corner of the rectangle and $C_1,C_2$ are known constants.
- $f$ - represents electric potential
$\nabla f$ - represents electric field
$\nabla^2 f$ - represents small excess charge density - To simplify the equation I can perform a first order expansion to $\nabla^2f(x,y)$ because it is very small. Than I can solve it rather easily using finite difference method, but I really prefer not to neglect this term $\left[\nabla^2f(x,y)\right]$.
Thanks again