# Analytical or approximate analytical solution to "Third Order Partial Differential Equation"

How should I approach to obtain an analytical or approximate analytical solution of an equation of the form $$a_1f_{xxy} + a_2f_{xx} + a_3f_{yy} + a_4f_y + a_5f + a_6 = 0$$ with $a_1, a_2, a_3, a_4, a_5, a_6 \in\Bbb{R}$ ?

Please note that I do not require a numerical solution. Thank you

Edit: Boundary conditions $$y=0,f=0$$ $$y=0,\frac{\partial f}{\partial y}=0$$ $$x=0, f=1$$ $$x=1, f=0$$

• What are the boundary conditions? Aug 2, 2017 at 18:54

• With the specified boundary conditions, the problem becomes even more interesting since the conditions $f(x,0)=0$ and $f(0,y)=1$ introduce a special point $f(0,0)=0=1$. To overcome the apparent contradiction, one have to consider complex values of $\lambda$ so that sinusoidal terms appear into the solutions. This allows to express them in term of Fourier series, likely to fit the boundary conditions on a limited range. I let to someone else the pleasure to solve such a so interesting problem. Good luck. Aug 4, 2017 at 4:30