Fredrik Berntsson

Wavelet and Fourier Methods for Solving the Sideways Heat Equation

We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x=1, where the solution is wanted for

Numerical solution of a Cauchy problem for the Laplace equation

0 \leq x < 1

A spectral method for solving the sideways heat equation

. The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. We consider stabilizations based on replacing the time derivative in the heat equation by wavelet-based approximations or a Fourier-based approximation. The resulting problem is an initial value problem for an ordinary differential equation, which can be solved by standard numerical methods, e.g., a Runge--Kutta method. We discuss the numerical implementation of Fourier and wavelet methods for solving the sideways heat equation. Theory predicts that the Fourier method and a method based on Meyer wavelets will give equally good results. Our numerical experiments indicate that also a method based on Daubechies wavelets gives comparable accuracy. As test problems we take model equations with constant and variable coefficients. We also solve a problem from an industrial application with actual measured data.

An alternating iterative procedure for the Cauchy problem for the Helmholtz equation

We consider a two-dimensional steady state heat conduction problem. The Laplace equation is valid in a domain with a hole. Temperature and heat-flux data are specified on the outer boundary, and we wish to compute the temperature on the inner boundary. This Cauchy problem is ill-posed, i.e. the solution does not depend continuously on the boundary data, and small errors in the data can destroy the numerical solution. We consider two numerical methods for solving this problem. A standard approach is to discretize the differential equation by finite differences, and use Tikhonov regularization on the discrete problem, which leads to a large sparse least squares problem. We propose to use a conformal mapping that maps the region onto an annulus, where the equivalent problem is solved using a technique based on the fast Fourier transform. The ill-posedness is dealt with by filtering away high frequencies in the solution. Numerical results using both methods are given.

A stability estimate for a Cauchy problem for an elliptic partial differential equation

We consider an inverse heat conduction problem, the sideways heat equation, which is the model of a problem where one wants to determine the temperature on the surface of a body, using interior mea ...

An Inverse Heat Conduction Problem and Improving Shielded Thermocouple Accuracy

Abstract We present a modification of the alternating iterative method, which was introduced by Kozlov and Maz’ya, for solving the Cauchy problem for the Helmholtz equation in a Lipschitz domain. The reason for this modification is that the standard alternating iterative algorithm does not always converge for the Cauchy problem for the Helmholtz equation. The method is then implemented numerically using the finite difference method.

An accelerated alternating procedure for the Cauchy problem for the Helmholtz equation

A two-dimensional inverse steady state heat conduction problem in the unit square is considered. Cauchy data are given for y ≤ 0, and boundary data are for x ≤ 0 and x ≤ 1. The elliptic operator is ...

A one-dimensional model of viscous blood flow in an elastic vessel

A shielded thermocouple is a measurement device used for monitoring the temperature in chemically, or mechanically, hostile environments. The sensitive parts of the thermocouple are protected by a shielding layer. In order to improve the accuracy of the measurement device, we study an inverse heat conduction problem where the temperature on the surface of the shielding layer is sought, given measured temperatures in the interior of the thermocouple. The procedure is well suited for real-time applications where newly collected data is continuously used to compute current estimates of the surface temperature. Mathematically we can formulate the problem as a Cauchy problem for the heat equation, in cylindrical coordinates, where data is given along the line r = r 1 and the solution is sought at r 1 < r ≤ r 2. The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. Thus, regularization techniques are needed. The ill–posedness of the problem is analyzed and a numerical method is proposed. Numerical experiments demonstrate that the proposed method works well.

Boundary identification for an elliptic equation

In this paper we study the Cauchy problem for the Helmholtz equation. This problem appears in various applications and is severely ill–posed. The modified alternating procedure has been proposed by ...

An inverse heat conduction problem and an application to heat treatment of aluminium

In this paper we present a one-dimensional model of blood flow in a vessel segment with an elastic wall consisting of several anisotropic layers. The model involves two variables: the radial displacement of the vessels wall and the pressure, and consists of two coupled equations of parabolic and hyperbolic type. Numerical simulations on a straight segment of a blood vessel demonstrate that the model can produce realistic flow fields that may appear under normal conditions in healthy blood vessels; as well as flow that could appear during abnormal conditions. In particular we show that weakening of the elastic properties of the wall may provoke a reverse blood flow in the vessel.