I plotted the function given in the figure with respect to "z" .I obtained different plots for different distinct values of "t".
Now please can you help how to obtain the intersection point of the straight line and the curves.I.e how to precisely obtain the value of " z" at each intersection point. Actually I am using mathematica online.So some sorts of code to obtain the intersection point will be helpfull.
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int = Integrate[1/(x^3 - 1), x];
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Since you have a function, f[z, t], defined that gave you the plots you wanted. I suggest trying the following:
Find the root of:
f[z,t] - k
Just substitute k for the y value you are looking for the intercept.