Diego A. Murio

Implicit finite difference approximation for time fractional diffusion equations

Time fractional diffusion equations are used when attempting to describe transport processes with long memory where the rate of diffusion is inconsistent with the classical Brownian motion model. In this paper we develop an implicit unconditionally stable numerical method to solve the one-dimensional linear time fractional diffusion equation, formulated with Caputos fractional derivative, on a finite slab. Several numerical examples of interest are also included.

Stable numerical solution of a fractional-diffusion inverse heat conduction problem

The ill-posed problem of attempting to recover the boundary temperature and the heat flux functions from one measured transient data temperature at some interior point of a one-dimensional semi-infinite conductor when the governing linear diffusion equation is of fractional type is discussed. A simple algorithm based on space marching mollification techniques is introduced for the numerical solution of the discrete problem. Stability bounds, error estimates and numerical examples of interest are also presented.

Discrete mollification and automatic numerical differentiation

Abstract An automatic method for numerical differentiation, based on discrete mollification and the principle of generalized cross validation is presented. With data measured at a discrete set of points of a given interval, the method allows for the approximate recovery of the derivative function on the entire domain. No information about the noise is assumed. Error estimates are included together with several numerical examples of interest.

Time fractional IHCP with Caputo fractional derivatives

The numerical solution of the time fractional inverse heat conduction problem (TFIHCP) on a finite slab is investigated in the presence of measured (noisy) data when the time fractional derivative is interpreted in the sense of Caputo. A finite difference space marching scheme with adaptive regularization, using mollification techniques, is introduced. Error estimates are derived for the numerical solution of the mollified problem and several numerical examples of interest are provided.

The mollification method and the numerical solution of the inverse heat conduction problem by finite differences

Abstract The inverse heat conduction problem involves the calculation of surface heat flux and/or temperature histories from transient, measured temperatures inside solids. We consider the one dimensional semi-infinite linear case and present a new solution algorithm based on a data filtering interpretation of the mollification method that automatically determines the radius of mollification depending on the amount of noise in the data and finite differences. A fully explicit and stable space marching scheme is developed. We describe several numerical experiments of interest showing that the new procedure is accurate and stable with respect to perturbations in the data even for small dimensionless time steps.

Automatic numerical differentiation by discrete mollification

Abstract A new, very simple, totally automated and powerful technique for numerical differentiation based on the computation of the derivative of a suitable filtered version of the noisy data by discrete mollification is presented. Several numerical examples of interest are also analyzed.

Stable numerical evaluation of Grünwald–Letnikov fractional derivatives applied to a fractional IHCP

The computation of Grünwald–Letnikov fractional derivatives from noisy data is considered as an ill-posed problem and treated by mollification techniques. It is shown that, with the appropriate choice of the radius of mollification, the method is a regularizing algorithm. Next, the recovery of the boundary temperature and heat flux functions from one measured transient temperature data at some interior point of a one-dimensional semi-infinite conductor when the governing diffusion equation is of fractional type is discussed. A simple algorithm based on space marching mollification techniques and Grünwald–Letnikov fractional derivatives is introduced for the numerical solution of this inverse ill-posed problem. In all cases, stability and error estimates are included together with numerical examples of interest.

On the stable numerical evaluation of caputo fractional derivatives

The computation of Caputos fractional derivatives in the presence of measured data is considered as an ill-posed problem and treated by mollification techniques. It is shown that, with the appropriate choice of the radius of mollification, the method is a regularizing algorithm, and the order of convergence is derived. Error estimates are included together with numerical examples of interest.

Identification of parameters in one-dimensional IHCP

Abstract We present a new numerical method based on discrete mollification for identification of parameters in one-dimensional inverse heat conduction problems (IHCP). With the approximate noisy data functions (initial temperature on the boundary t = 0, 0 ≤ x ≤ 1, temperature and space derivative of temperature on the boundary x = 0, 0 ≤ t ≤ 1) measured at a discrete set of points, the diffusivity coefficient, the heat flux, and the temperature functions are approximately recovered in the unit square of the (x, t) plane. In contrast to other related results, the method does not require any information on the amount and/or characteristics of the noise in the data and the mollification parameters are chosen automatically. Another important feature of the algorithm is that it allows for the recovery of much more general diffusivity parameters, including discontinuous coefficients. Error bounds and numerical examples are provided.

A mollified space marching finite differences algorithm for the inverse heat conduction problem with slab symmetry

Abstract The one-dimensional inverse heat conduction problem (IHCP) for a slab is considered. A new solution algorithm based on a data filtering interpretation of the mollification method is presented and a fully explicit space marching finite differences scheme is developed. After showing the numerical stability of the algorithm, the efficiency of the method is demonstrated by means of several examples. The numerical results show the procedure to be very useful in the presence of noisy data, even for some nonlinear cases.