# How should I numerically solve this PDE?

I am hoping to figure out the function $$u(x,y,t)$$ for some integer arguments when $$u(x,y,0)$$ is given (by figuring out I mean generating some images in MatLab), also time $$t \ge 0$$.

$$\frac{\partial u}{\partial t} = -(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}) - (\frac{\partial^4 u}{\partial x^4} + \frac{\partial^4 u}{\partial y^4}) - u \cdot (\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y})$$

What should I do? I believe it is reasonable to express $$\partial u / \partial t$$ and relate to numerical integration which means something like

$$u(x,y,t+1) = u(x,y,t) + \ldots$$

The sum of second derivatives is known as Laplacian and approximated in this Wikipedia page so

$$u(x,y,t+1) = u(x,y,t) - (u(x-1,y,t) + u(x+1,y,t) + u(x,y-1,t) + u(x,y+1,t) - 4u(x,y,t)) + \ldots$$

I do not know the approximation of summed fourth and first derivatives, also how to take care of that directly used $$u$$ multiplier (although I guess it may be something like $$u(x,y,t)+u(x,y,t-1)+u(x,y,t-2)+\ldots$$).

• Thanks. However I am really less experienced than you might think and I am checking out the visual nature of emerging patters rather than seeking a highly precise solution (by the way youtu.be/TgOnwK0i4Dw is a one-dimensional case of the equation in question). So I can now approximate all the derivatives (by finite difference), but how about $u(x,y,t)$ itself in terms of values at time $t-1$? Jun 9, 2013 at 9:21