A Two Dimensional Discrete Mollification Operator and the Numerical Solution of an Inverse Source Problem

Abstract

We consider a two-dimensional time fractional diffusion equation and address the important inverse problem consisting of the identification of an ingredient in the source term. The fractional derivative is in the sense of Caputo. The necessary regularization procedure is provided by a two-dimensional discrete mollification operator. Convergence results and illustrative numerical examples are included.

A finite difference approximation of a two dimensional time fractional advection-dispersion problem

“…A useful feature of the paper is the computational framework based on matrices. Our scheme was successfully implemented for the solution of an inverse source problem in [3]. We certainly expect to develop other applications of this scheme in the near future.…”

“…Among the authors who face two dimensional time-fractional differential equations, we mention [1], in which the problem is similar to ours but the coefficients are constant and [3], which deals with a two dimensional inverse source problem and introduces a particular case of our numerical scheme for the necessary solution of the direct problem.…”

Determination of unknown functions in a mathematical model of ductal carcinoma in situ

“…In sequence a numerical approach based on space marching method and mollification approach is developed to solve the problem (39)-(45). Space marching finite difference algorithms in conjunction with discrete mollification approach have been widely used to solve inverse parabolic problems in literature [19][20][21][22][23][24][25][26][27]. Using the space marching approach yields straightforward discretization of nonconstant coefficients and nonlinear problems.…”

“…Mollification approach and implementation of GCV approach to determine the parameters have discussed widely in literature. For more details we refer to [19][20][21][22][23] and the references therein. Without lose of generality, suppose T = 1.…”

Una convolución muy útil y unas derivadas ilustres

“…En la segunda gráfica de la Figura 1 aparecen f ε y Jf ε . El código MATLAB utilizado para generar las dos funciones f y f ε para ε = 0.1, es el siguiente: fx = f(x); epsil = 0.1; fep = fx + (2*rand(size(x))-1)*epsil; Algunos ejemplos de problemas mal condicionados en los que se puede trabajar con molificación son los inversos de difusión (Mejia & Murio, 1996;Murio, 2007;Murio & Mejía, 2008a;Mejía & Piedrahita, 2017) y los inversos de identificación de modelos (Mejia & Murio, 1995;Murio & Mejía, 2008b;Mejía, et al, 2011;Acosta, et al, 2015;Echeverry & Mejía, 2018).…”