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I know how to solve second order heat equation.

But how to solve $$u_t+au_x-bu_{xx}-cu_{xxx}=0$$ Initial condition: $u(x,0)=\cos (kx)$, $k \in R$.

I think it may could separation of variables?

Sam
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  • What have you tried? – Snoop Mar 11 '22 at 08:51
  • @Snoop Oh, I already tried Fourier transform. I obtained solution but I still think solution is too complex. So I want to know if I could use other methods to obtain easier solution. – Sam Mar 11 '22 at 08:59

1 Answers1

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Use Fourier transforms. Let $\varphi(t,\omega):=\mathcal{F}_x(u(t,x))(\omega)$, we get $$\varphi_t+(ia\omega+b\omega^2+ic\omega^3) \varphi=0$$ By using the initial condition: $$\varphi(t,\omega)=\varphi(0,\omega)e^{-(ia\omega+b\omega^2+ic\omega^3)t}=\pi(\delta(\omega-k)+\delta(\omega+k))e^{-(ia\omega+b\omega^2+ic\omega^3)t}$$ Transform back to the $x$ domain: $$u(t,x)=\frac{1}{2}(e^{ixk}e^{-(iak+bk^2+ick^3)t}+e^{-ixk}e^{-(-iak+bk^2-ick^3)t})$$ $$\frac{e^{-tbk^2}}{2}(e^{i(xk-atk-ctk^3)}+e^{-i(xk-atk-ctk^3)})=e^{-tbk^2}\cos(xk-atk-ctk^3)$$

Snoop
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