How to solve the Cauchy problem?

Question

I have a task that looks like $$ \frac{\partial u}{\partial t} = 2\frac{\partial^2 u}{\partial x^2} + \frac{\partial u}{\partial x} + u + 3e^{t},\, u\left(x, 0\right)=\sin{x}$$

Please help, thanks :)

Answer 1

We continue Matthew Pilling solution: Using Laplace transform method we get ode $$2v_{xx}+v_{x}+(1-s)v=-\sin(x)+\frac{3}{1-s}$$ Particular solution is $$v=\frac{s\sin x+\sin x+\cos x}{s^2+2s+2}+\frac{3}{(s-1)^2}$$ Final solution of Cauchy problem is $$u(x,t)=\mathcal{L}^{-1}v=e^{-t}\sin(x+t)+3te^t$$

Answer 2

Some details: $$\mathcal{L}^{-1}\left(\ \frac{3}{(s-1)^2}\right)=3te^t$$ $$\mathcal{L}^{-1}\left(\ \frac{1}{s^2+2s+2}\right)=e^{-t}\sin t$$ $$\mathcal{L}^{-1}\left(\ \frac{s}{s^2+2s+2}\right)=e^{-t}(\cos t-\sin t)$$