437 research outputs found
An inverse Sturm-Liouville problem with a fractional derivative
In this paper, we numerically investigate an inverse problem of recovering
the potential term in a fractional Sturm-Liouville problem from one spectrum.
The qualitative behaviors of the eigenvalues and eigenfunctions are discussed,
and numerical reconstructions of the potential with a Newton method from finite
spectral data are presented. Surprisingly, it allows very satisfactory
reconstructions for both smooth and discontinuous potentials, provided that the
order of fractional derivative is sufficiently away from 2.Comment: 16 pages, 6 figures, accepted for publication in Journal of
Computational Physic
Inverse Problems of Determining Coefficients of the Fractional Partial Differential Equations
When considering fractional diffusion equation as model equation in analyzing
anomalous diffusion processes, some important parameters in the model, for
example, the orders of the fractional derivative or the source term, are often
unknown, which requires one to discuss inverse problems to identify these
physical quantities from some additional information that can be observed or
measured practically. This chapter investigates several kinds of inverse
coefficient problems for the fractional diffusion equation
Solución de un problema inverso de advección-dispersión con derivada temporal fraccionaria por medio de molificación discreta
Consideramos un problema inverso para una ecuación de advección-dispersión con derivada temporal fraccionaria, en una configuración unidimensional. La derivada fraccionaria se interpreta en el sentido de Caputo y las coeficientes de advección y de dispersión son constantes. El problema inverso involucra la reconstruccción simultánea de la concentración de soluto y del flujo de dispersión en una de las fronteras del dominio fÃsico, a partir de lecturas de datos perturbados en un punto interior del dominio. Mostramos que el problema inverso es mal condicionado y por tanto una solución numérica del problema requiere de alguna técnica de regularización. Proponemos un esquema de diferencias finitas de marcha en el espacio, que utiliza molifocación discreta como técnica de regularización. Se incluyen estimativos de error y ejemplos numéricos ilustrativos.We consider an inverse problem for a time fractional advection-dispersion equation in a 1-D semi-infinite setting. The fractional derivative is interpreted in the sense of Caputo and advection and dispersion coefficients are constant. The inverse problem consists on the recovery of the boundary distribution of solute concentration and dispersion flux from measured (noisy) data known at an interior location. This inverse problem is ill-posed and thus the numerical solution must include some regularization technique. Our approach is a finite difference space marching scheme enhanced by adaptive discrete mollification. Error estimates and illustrative numerical examples are provided
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