A Diffusion Problem with Neumann Boundary Control Utilizing Total Mass · arXivDesk2007.03666Jul 7, 2020Keywords: Diffusion Problem, Neumann Boundary Conditions, Total Mass Control, Finite Difference
A Diffusion Problem with Neumann Boundary Control Utilizing Total Mass
M. Salman
Abstract
The author studies the diffusion problem ut=uxx, 0<x<1, t>0; u(x,0)=0,
and
−ux(0,t)=ux(1,t)=φ(t), where
is a control function that ensures that the total mass
∫01u(x,tk)dx stays between two predetermined values. Mathematically,
for
t2k<t<t2k+1 and
for
t2k+1<t<t2k+2, k=0,1,2,… with
and the sequence
is determined by the equations
∫01u(x,tk)dx=M, for
k=1,3,5,…, and
∫01u(x,tk)dx=m, for
k=2,4,6,… and where
0<m<M<u0 . Note that the switching times
are unknowns. Determination of switching times
and their differences
tk+1−tk are obtained. Numerical verifying examples are presented.