Thank you.  Here are the two worksheets as described in the original question.  The c(H) I attached is more abrupt between the segments than my model but it does demonstrate the difficulties I encounter.

pdsolve-c_variable_of_H.mw

pdsolve-c_variable_of_t.mw

@dharr nice. Thank you.  It certainly is on the help page.  The help page with the wealth of information is a bit overwhelming when you are new to Maple; went right past it.

Thank you for providing the answer.  I also find another approach that seems to work:

psol := pds:-value(output = listprocedure);
psol(0.0001, 0.1);

and to get the u(x,t) value:

solxt := eval(u(x, t), psol);
solxt(0.0001, 0.1);

Could have never dreamt this up.

Thank you for the quick response.  I am new to Maple.  There is quite a bit to digest and learn there.