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I am attempting to solve the equation below, which requires v[y,0]==0 for all y greater than 0. I have followed the /; approach to enforce this definition, as outlined in the DSolve/Delay Differential Equations page in the language documentation.

DSolve[{Derivative[2,0][v][y,t]==Derivative[0,1][v][y,t],v[0,t]==1,v[y/;y>0]==0},v[y,t],{y,t}]

Wolfram simply outputs the cell back without an error message. I believe that my inequality restriction on y in the second boundary condition is the issue, but I am not sure how to proceed.

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  • $\begingroup$ I want to solve on {y, 0, Infinity}. The problem is startup of Couette flow. $\endgroup$ – eoncarlyle Feb 21 at 16:29
  • $\begingroup$ Do you want an analytical solution or a numerical one? $\endgroup$ – Alex Trounev Feb 21 at 16:31
  • $\begingroup$ I want an analytical solution, which I am confident exists $\endgroup$ – eoncarlyle Feb 21 at 16:34
  • $\begingroup$ The solution exists on the half-line. See answer. $\endgroup$ – Alex Trounev Feb 21 at 17:13
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Use

sol= DSolve[{Derivative[2, 0][v][y, t] == Derivative[0, 1][v][y, t], 
       v[0, t] == 1, v[y, 0] == 0}, v, {y, 0, Infinity}, {t, 0, Infinity}]
    Plot3D[Evaluate[v[y, t] /. First[sol]], {y, 0, 5}, {t, 0, 10}, 
     PlotRange -> All]

fig1

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