Solving PDE with Analytical Solution: Exploring LaTex Code and Techniques

[Geoffrey]

LaTex Code: \frac{\partial U}{\partial t} + ax\frac{\partial U}{\partial x} + b\frac{\partial^2 U}{\partial x^2} = 0


Can someone please tell me how to solve this PDE?

Thanks,
Geoff

 

[kosovtsov]

Assuming that

$$U(t,x)=\int_{-\infty}^\infty F(x,\tau)e^{it\tau}\,d\tau$$

we come to

$$-ib\frac{\partial^2 F(x,\tau)}{\partial x^2}-iax\frac{\partial F(x,\tau)}{\partial x}+\tau F(x,\tau)=0$$

which solution is as follows

$$F(x,\tau) = x[H1(\tau)KummerM(-\frac{-2a+i\tau}{2a},\frac{3}{2},\frac{ax^2}{2b})+H2(\tau)KummerU(-\frac{-2a+i\tau}{2a},\frac{3}{2},\frac{ax^2}{2b})]e^{-\frac{ax^2}{2b}}$$

so the general solution to your PDE is

$$U(t,x)=xe^{-\frac{ax^2}{2b}}\int_{-\infty}^\infty [H1(\tau)KummerM(-\frac{-2a+i\tau}{2a},\frac{3}{2},\frac{ax^2}{2b})+H2(\tau)KummerU(-\frac{-2a+i\tau}{2a},\frac{3}{2},\frac{ax^2}{2b})]e^{it\tau}\,d\tau$$

where H1 and H2 are arbitrary functions.