Utilization Of Total Mass As A Control In Diffusion Processes

As motivation for the mathematical problems considered in this work, consider a chamber in the form of a long linear transparent tube. We allow for the introduction or removal of material in a gaseous state at the ends of the tube. The material diffuses throughout the tube with or without reaction with other materials. By illuminating the tube on one side with a light source with a frequency range spanning the absorption range for the material and collecting the residual light that passes through the tube with photo-reception equipment, we can obtain a measurement of the total mass of material contained in the tube as a function of time. Using the total mass as switch points for changing the boundary conditions for introduction or removal of material. The objective is to keep the total mass of material in the tube oscillating between two set values such as m < M. The physical application for such a system is the control of reaction diffusion systems such as production of a chemical material in a reaction chamber via the introduction of reactants at the boundary of chamber. In Chapter 1, we study the diffusion problem ut = uxx, 0 < x < 1, t > 0; u(x, 0) = 0, and u(0, t) = u(1, t) = ψ(t), where ψ(t) = u0 for t2k < t < t2k+1 and ψ(t) = 0 for t2k+1 < t < t2k+2, k = 0, 1, 2, . . . with t0 = 0 and the sequence tk is determined by the equations R 1 0 u(x, tk)dx = M, for k = 1, 3, 5, . . . , and R 1 0 u(x, tk)dx = m, for k = 2, 4, 6, . . . and where 0 < m < M < u0. Note that the switching points tk, k = 1, 2, 3, . . . are unknown Existence and uniqueness are demonstrated. Theoretical estimates of the tk and tk+1 − tk are obtained and numerical verifications of the estimates are presented. In Chapter 2, we consider the problem ut = uxx − u, 0 < x < 1, t > 0; u(x, 0) = 0, and u(0, t) = u(1, t) = ψ(t), where ψ(t) = u0 for t2k < t < t2k+1 and ψ(t) = 0 for t2k+1 < t < t2k+2, k = 0, 1, 2, . . . with t0 = 0 and the sequence tk is determined by the equations R 1 0 u(x, tk)dx = M, for k = 1, 3, 5, . . . , and R 1 0 u(x, tk)dx = m, for k = 2, 4, 6, . . . and where 0 < m < M. Note that the switching points tk, k = 1, 2, 3, . . . are unknown. Existence and uniqueness are demonstrated. Theoretical estimates of the tk and tk+1 − tk are obtained and numerical verifications of the estimates are presented. The case of ux(0, t) = ux(1, t) = ψ(t) is also considered and analyzed. In Chapter 3, study the problem ut = uxx, 0 < x < 1, t > 0; u(x, 0) = 0, and −ux(0, t) = ux(1, t) = ψ t), where ψ(t) = 1 for t2k < t < t2k+1 and ψ(t) = −1 for t2k+1 < t < t2k+2, k = 0, 1, 2, . . . with t0 = 0 and the sequence tk is determined by the equations R 1 0 u(x, tk)dx = M, for k = 1, 3, 5, . . . , and R 1 0 u(x, tk)dx = m, for k = 2, 4, 6, . . . and where 0 < m < M. The sequence tk is analytically determined. A finite difference method is used to compute this sequence. Under certain restrictions on the mesh size, the answer coincides with the one found analytically. Numerical estimates are presented. In Chapter 4, we study the problem ut = uxx − au, 0 < x < 1, t > 0; u(x, 0) = 0, and −ux(0, t) = ux(1, t) = φ(t), where a = a(x, t, u), and φ(t) = 1 for t2k < t < t2k+1 and φ(t) = 0 for t2k+1 < t < t2k+2, k = 0, 1, 2, . . . with t0 = 0 and the sequence tk is determined by the equations R 1 0 u(x, tk)dx = M, for k = 1, 3, 5, . . . , and R 1 0 u(x, tk)dx = m, for k = 2, 4, 6, . . . and where 0 < m < M. Note that the switching points tk, k = 1, 2, 3, . . . are unknown. A maximum principal argument has been used to prove that the solution is positive under certain conditions. Existence and uniqueness are demonstrated. Theoretical estimates of the tk and tk+1 − tk are obtained and numerical verifications of the estimates are presented. In conclusion, the analytical and computational results of chapters 1 through 4 show that such control mechanisms are feasible.

Author dc:creator

• Salman, Mohamed

Contributors dc:contributor

• Cannon, John

• Diffusion
• Control
• Mathematics

Language dc:language

English

Identifier

CFE0000551

OAI identifier oai:identifier

oai:stars.library.ucf.edu:etd-1383