Recovering boundary data in planar heat conduction using boundary integral equation method

We consider a Cauchy problem for the heat equation, where the temperature field is to be reconstructed from the temperature and heat flux given on a part of the boundary of the solution domain. We employ a Landweber type method proposed in [2], where a sequence of mixed well-posed problems are solved at each iteration step to obtain a stable approximation to the original Cauchy problem. We develop an efficient boundary integral equation method for the numerical solution of these mixed problems, based on the method of Rothe. Numerical examples are presented both with exact and noisy data, showing the efficiency and stability of the proposed procedure and approximations.

An iterative method for a Cauchy problem for the heat equation

An iterative method for reconstruction of the solution to a parabolic initial boundary value problem of second order from Cauchy data is presented. The data are given on a part of the boundary. At each iteration step, a series of well-posed mixed boundary value problems are solved for the parabolic operator and its adjoint. The convergence proof of this method in a weighted L2-space is included.

Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations

A numerical method for the Dirichlet initial boundary value problem for the heat equation in the exterior and unbounded region of a smooth closed simply connected 3-dimensional domain is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and an integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the parabolic problem to a sequence of stationary elliptic problems which are solved by a boundary layer approach giving a sequence of boundary integral equations of the first kind to solve. Under the assumption that the boundary surface of the solution domain has a one-to-one mapping onto the unit sphere, these integral equations are transformed and rewritten over this sphere. The numerical discretisation and solution are obtained by a discrete projection method involving spherical harmonic functions. Numerical results are included.

A desalination system with efficiency approaching the theoretical limits

The objective of this project is to design a new desalination system with energy efficiency approaching the theoretical thermodynamic limit—even at high recovery ratio. The system uses reverse osmosis (RO) and a batch principle of operation to overcome the problem of concentration factor which prevents continuous-flow RO systems from ever reaching this limit and thus achieving the minimum possible specific energy consumption, SEC. Batch operation comprises a cycle in three phases: pressurisation, purge, and refill. Energy recovery is inherent to the design. Unlike in closed-circuit desalination (CCD), no feedwater is added to the pressure circuit during the pressurisation phase. The batch configuration is compared to standard configurations such as continuous single-stage RO (with energy recovery) and CCD. Theoretical analysis has shown that the new system is able to use 33% less energy than CCD at a recovery ratio of 80%. A prototype has been constructed using readily available parts and tested with feedwater salinities and recovery ratios ranging from 2,000 to 5,000 ppm and 17.2–70.6%, respectively. Results compare very well against the standard configurations. For example, with feedwater containing 5,000 ppm NaCl and recovery ratio of 69%, a hydraulic SEC of 0.31 kWh/m3 was obtained—better than the minimum theoretically possible with a single-stage continuous flow system with energy recovery device.