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I want to find a function (or a set of functions) such that $u(x,t)$ satisfies the following partial integro-differential equation with singular kernel \begin{eqnarray} &&u_x(0,t) = \int_0^t \frac{u_\lambda(0,\lambda)}{\sqrt{t-\lambda}} d\lambda, \\ &&u(0,0) = 0. \end{eqnarray} I tried the Laplace transform method but i couldn't find the solution.
Some help would be greatly appreciated.

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