I thought equation below can be solved using fourier transform but I could not apply the boundary conditions. Is there anybody who can solve this equation?
∂²U/∂ε² + ∂U/∂ε + ∂U/∂T =0
where U(ε,T) is (dimensionless) temperature T and ε dimensionless is boundary position respectively.
The boundary conditions are:
u(ε,0) = 0, ε > 0 u(0,T) = 1, T ≥ 0 u(∞,T) =0, T ≥ 0
On
The semi-infinite domain is different between the Cauchy problem.
Semi-infinite heat/diffusion equation with time-dependent B.C. at x=0
It might have the same question with you.
It is not difficult to see why the problem $$ \begin{cases} u_{xx}+u_x+u_t=0, \quad x>0,\;t>0,\\ u|_{t=0}=0,\quad x>0,\\ u|_{x=0}=1,\quad t>0,\\ \lim\limits_{x\to\infty}u(x,t)=0, \quad t>0, \end{cases}\tag{1} $$ cannot be solved. Indeed, let a weak solution to problem $(1)$ exist in $Q_{\tau}=(0,\infty)\times (0,\tau)$ for some $\tau>0$. Even if it be a weak solution of the class $L^2(Q_{\tau})$, solution $u\in C^{\infty}(Q_{\tau})$ while $u(\cdot,s)\in C[0,\infty)$ and $u(x,s)\to 0\,$ as $\,x\to\infty$ with any fixed $s\in (0,\tau)$. Solving problem $(1)$, well-posed backwards with the initial data $u|_{t=s}=u(\cdot,s)$, one gets a classical backward solution $$ u\in C^{\infty}\bigl((0,\infty)\times (-\delta,s)\bigr)\cap C\bigl ([0,\infty)\times (-\delta,s]\bigr),\quad \delta>0. $$ The latter must coincide with the original solution to $(1)$, while the backward solution is continuous at the origin and $u(0,0)=1$, which is in contradiction with the original initial condition $$ u|_{t=0}=0, \quad t>0 $$ in $(1)$, and which implies that problem $(1)$ cannot possess even a weak solution.
But there is no need in solving the ill-posed problem $(1)$. Google indicates that its formulation is copy-pasted from the paper http://nmcabuja.org/proceedings/workshop_Oyelami_Asere.pdf that is unbelievably bad-written, with its wrong citations and wrong references turning a well-posed laser-drilling problem to the ill-posed nonsense $(1)$. The unperturbed original source for $(1)$ is: http://www.zentralblatt-math.org/MIRROR/zmath/en/advanced/?q=an:00762217&type=pdf&format=complete with further references therein. A detailed introduction into a laser-drilling equation can be found in Ch. 4 of "Industrial Mathematics. Case Studies in the Diffusion of Heat and Matter" by Glenn Fulford and Philip Broadbridge.