I have ran into the following Heat equation IBVP, but I am not quite sure how to solve it as it has these time dependent boundary conditions
$$ v_t = kv_{xx} \ \ \ \ \ \ ( 0 \le x \le \infty, \ \ 0 < t < \infty) ,$$ $$ v(x,0) = \delta (x - x_0) \ \ \ \ \ \ \mathrm{for} \ \ t = 0, \ \ x_0 > 0 $$ $$ v(x = ((2kt) ^{1 /2k}), ((2kt) ^{1 /2k})) = 0 $$ $$ \lim_{x\to\infty} v(x, ((2kt) ^{1 /2k})) = 0 . $$
How to solve this problem ? Thanks.
Edit: How I got to this problem.
Consider the IBVP $$v_p = v_q + kp^{2k - 1}v_{qq}$$ $$ v(q,0) = \delta (q - q_0) \ \ \ \ \ \ \mathrm{for} \ \ p = 0, \ \ q_0 > 0 $$ $$ v(0, p) = 0 $$ $$ \lim_{q\to\infty} v(q, p) = 0 . $$
Let us make transformation to eliminate the first term in the right-hand side $$ x = q + p.% $$ After that Eq. reads: $$ v_p = k p^{2k - 1}v_{xx}.% $$ Introduce now new variable $t= p^{2k}/2k$, then our Eqn reduces to the conventional heat equation $$ v_t = kv_{xx}$$
The initial condition $v(q,0) = \delta(q - q_0)$ in new variables reads $v(x,0) = \delta(x - q_0)$. Then, the boundary condition $v(0,p) = 0$ in the new variables becomes: $$ v\left[x = (2kt)^{1/2k},(2kt)^{1/2k}\right] = 0. $$ I hope this helps in resolving any ambiguities, in my OP.
Let $Z(x,t)=(4\pi k t)^{-1/2}e^{-x^2/(4kt)}$ be the fundamental solution of the heat equation and denote $$G(x,x_0,t)=Z(x-x_0,t)-Z(x+x_0,t)$$ the Green's function of the first BVP in the domain $x>0$. Then $$ v(q,p)= e^{\large q_0-q-\frac{p^{2 k}}{4 k}} G\left(q,q_0,\frac{p^{2 k}}{2 k}\right). $$ is the required solution.