@Rouben Rostamian  Sorry for the confusion.  Since I ran out of my trial license, I cannot open to see exactly what I have written. 

For the first file "pdsolve-c_variable_of_H.mw," if I remember correctly (counting on my memory is a risky business these days), I think I have tried c(H(t)) insteat of c(H) and get the same error (Error, (in pdsolve/numeric/process_PDEs) variable(s) {H} are in the PDE system but are not dependent or independent variables). 

The second file "pdsolve-c_variable_of_t.mw," I made the c as a function of t and replace all H with Ho, and specify Ho:=Hb+H2*sin(2*Pi*t).  The Hb is a dc bias to make c transition from one segment to another as H cycle through.  The logic of using Ho and not t is that I do not have to figure out the corresponding t's for transition which is based on H and it becomes impractical to calculate and specify all t's of one hundred cycles, even if Maple does not object.

You are right, I can maintain H and write H:=Hb+H2*sin(2*Pi*t) which should also do.  But that still would not overcome Maple's objection that the piecewise has to be based on t or x and not an expression of t.  I will try things out when I get my license back and report back.  

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.