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

Abstract

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.

Keywords:

• Cauchy problem
• Inverse problem
• Ill-posed
• Singular value decomposition
• Volterra integral operator
• Convection--diffusion equation

Notes

¹ Explicit knowledge of the kernel function is useful also in a situation, where it is necessary to impose a constraint (e.g., monotonicity) on the solution.

² The case when there is a double root does not give a solution of (3).

³ We choose the parameters in the intervals 0.05 ≤ a ≤ 1 and −2 ≤ b ≤ 2.

⁴ The eigenvalues corresponding to the hyperbolic equation u_t=2bu_x are purely imaginary.

⁵ This is the case treated in 6.