eq=(∂_t) c[x,t]==(∂_(x,x))c[x,t]
bound=c[0,t]==Sin[t]
init=c[x,0]==0
der=(∂_x) c[x,t]==0/.x->0
sol=NDSolve[{eq,init,bound,der},c,{t,0,15},{x,0,1}]
cc=c/.sol[[1,1]]
Plot3D[cc[x,t],{t,0,10},{x,0,1}]

eq=(∂_t) c[x,t]==(∂_(x,x))c[x,t]
bound=c[0,t]==Sin[t]
init=c[x,0]==x(1-x)(1-Exp[-20t])
der=(∂_x) c[x,t]==0/.x->0
sol=NDSolve[{eq,init,bound,der},c,{t,0,15},{x,0,1}]
cc=c/.sol[[1,1]]