# Question:How to integrate the indefinite integral for PDE solve

## Question:How to integrate the indefinite integral for PDE solve

Maple 18

Dear maple users,

A fine day wishes to all.

I have solved the PDE via PDsolve. Here I need to calculate the Psi function. How to calculate the indefinite integral and how to find the constant-coefficient (C1).

Here Psi=0 at x=0

int_c.mw

 > restart:
 > with(PDEtools):
 > with(plots):
 > fcns := {f(x,t),theta(x,t)};
 (1)
 > d:=0.5:xi:=0.1:
 > R:=z->piecewise(d<=z and z<=d+1,1-2*xi*(cos((2*3.14)*((z-d)*(1/2))-1/4)-(7/100)*cos((32*3.14)*(z-d-1/2))),1);
 (2)
 > PDE1 :=(diff(f(x,t),t))=1+(1-2*theta((x,t)))*(1/(R(z)^2))*((diff(f(x,t),x,x))+(1/x)*diff(f(x,t),x))+theta((x,t));
 (3)
 > PDE2 :=2*(diff(theta(x,t),t))=(1/(R(z)^2))*((diff(theta(x,t),x,x))+(1/x)*diff(theta(x,t),x));
 (4)
 > IBC := {D[1](f)(0,t)=0,f(1,t)=0,f(x,0)=0,D[1](theta)(0,t)=0,theta(1,t)=1,theta(x,0)=0};
 (5)
 > z:=0.98:
 >
 > sol:=pdsolve(eval([PDE1,PDE2]),IBC ,numeric, time = t): sol:-value(f(x,t), output=listprocedure); fN:=eval( f(x,t), sol:-value(f(x,t), output=listprocedure)):
 (6)
 > t := 1;
 (7)
 > A1:=x*R(z)*R(z)*(fN)(x, t);
 (8)
 > A2:=eval(int(A1, x))+C1;
 (9)
 > W11:=eval(subs(x=0,A2));
 > Find_c1:=solve(W11,C1);
 (10)
 >