Numerical analysis of an ill-posed Cauchy problem for a convection--diffusion equation

• Published in: Inverse problems in science and engineering Vol. 15; no. 3; pp. 191 - 211
• Main Authors: Ranjbar, Z., Eldén, L.
• Format: Journal Article
• Language: English
• Published: Taylor & Francis 01-01-2007
• Subjects: Cauchy problem; Convection - diffusion equation; Ill-posed; Inverse problem; Singular value decomposition; TECHNOLOGY; TEKNIKVETENSKAP; Volterra integral operator
• ISSN: 1741-5977; 1741-5985; 1741-5985
• DOI: 10.1080/17415970600557299

The mathematical and numerical properties of an ill-posed Cauchy problem for a convection--diffusion equation are investigated in this study. The problem is reformulated as a Volterra integral equation of the first kind with a smooth kernel. The rate of decay of the singular values of the integral operator determines the degree of ill-posedness. The purpose of this article is to study how the convection term influences the degree of ill-posedness by computing numerically the singular values. It is also shown that the sign of the coefficient in the convection term determines the rate of decay of the singular values. Some numerical examples are also given to illustrate the theory.