Abstract. We propose and study a numerical method for time discretization of linear and semilinear integro-partial differential equations that are intermediate between diffusion and wave equations, or are subdiffusive. The method uses convolution quadrature based on the second-order backward differentiation formula. Second-order error bounds of the time discretization and regularity estimates for the solution are shown in a unified way under weak assumptions on the data in a Banach space framework. Numerical experiments illustrate the theoretical results.

A generalization of the linear fractional integral equation u(t) = u 0 + ∂ −α Au(t), 1 < α < 2, which is written as a Volterra matrix-valued equation when applied as a pixel-by-pixel technique, has been proposed for image denoising (restoration, smoothing,...). Since the fractional integral equation interpolates a linear parabolic equation and a hyperbolic equation, the solution enjoys intermediate properties. The Volterra equation we propose is well-posed for all t > 0, and allows us to handle the diffusion by means of a viscosity parameter instead of introducing non linearities in the equation as in the Perona-Malik and alike approaches. Several experiments showing the improvements achieved by our approach are provided.

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Runge-Kutta based convolution quadrature methods for abstract, well-posed, linear, and homogeneous Volterra equations, non necessarily of sectorial type, are developed. A general representation of the numerical solution in terms of the continuous one is given. The error and stability analysis is based on this representation, which, for the particular case of the backward Euler method, also shows that the numerical solution inherits some interesting qualitative properties, such as positivity, of the exact solution. Numerical illustrations are provided. Mathematics Subject Classification (2000)65J15 · 65M12 · 65L05 · 65M20

The use of cross-diffusion problems as mathematical models of different image processes is investigated. Here the image is represented by two real-valued functions which evolve in a coupled way, generalizing the approaches based on real and complex diffusion. The present paper is concerned with linear filtering. First, based on principles of scale invariance, a scalespace axiomatic is built. Then some properties of linear complex diffusion are generalized, with particular emphasis on the use of one of the components of the crossdiffusion problem for edge detection. The performance of the cross-diffusion approach is analyzed by numerical means in some one-and two-dimensional examples.