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Dor
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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:
$$a_1+a_2\nabla f(x,y)+a_3\nabla f(x,y)\cdot \nabla^2 f(x,y)+\left[a_4+a_5\nabla^2f(x,y)\right]^\frac{2}{3}\cdot\nabla^3 f(x,y)=0$$ ##a_i## - are constants
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.
I want to solve the following boundary conditioned differential equation:
$$a_1+a_2\nabla f(x,y)+a_3\nabla f(x,y)\cdot \nabla^2 f(x,y)+\left[a_4+a_5\nabla^2f(x,y)\right]^\frac{2}{3}\cdot\nabla^3 f(x,y)=0$$ ##a_i## - are constants
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]##.
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