[Solution] Partial differential equations Assume a solution of the form u(x, t)=X(x) T(t) to the modified diffusion equation u_t-D u_x x-α u=0. First

Question: Partial differential equations

Assume a solution of the form $u(x, t)=X(x) T(t)$ to the modified diffusion equation $u_{t}-D u_{x x}-\alpha u=0$. First show that the equation separates and find the general solution for $X(x)$ and $T(t)$. Next, assuming that $D>0, \alpha \geq$ $0, L>0$, solve the boundary value problem

\[\begin{array}{ll} u_{t}-D u_{x x}-\alpha u=0 & \text { for all } 0 \leq x \leq L, t \geq 0 \\ u(0, t)=u(L, t)=0, & \text { for all } t \geq 0 \\ u(x, 0)=\sin (\pi x / L)+\sin (2 \pi x / L) . & \end{array}\]

Finally characterise the difference between the solutions for $\alpha>D \pi^{2} / L^{2}$ and \(\alpha

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[Solution] Partial differential equations Assume a solution of the form u(x, t)=X(x) T(t) to the modified diffusion equation u_t-D u_x x-α u=0. First

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