Abstract
In this paper, we consider a time-fractional inverse diffusion problem, where the data is given at $x=1$ and the solution is sought in the interval $0\leq x<1$ . Such a problem is obtained from the classical diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order $\alpha\in(0,1)$ . We show that a time-fractional inverse diffusion problem is severely ill-posed and we further apply a modified kernel method to solve it based on the solution in the frequency domain. The corresponding convergence estimates are provided. Finally, an example is constructed to show the feasibility and efficiency of the proposed method.
Highlights
1 Introduction In the past decades, studies on the problems of the partial differential equation mainly focused on direct problems and inverse problems of integer order differential equation, and some numerical techniques have been proposed to solve integer order differential equation [ – ]
Fractional derivatives calculus and fractional differential equations have been used recently to solve a range of problems in mechanical engineering [ ], viscoelasticity [ ], electron transport [ ], dissipation [ ], heat conduction [, ], and high-frequency financial data [ ]
The time-fractional diffusion equation arises by replacing the standard time partial derivative in the diffusion equation with a time-fractional partial derivative
Summary
Studies on the problems of the partial differential equation mainly focused on direct problems and inverse problems of integer order differential equation, and some numerical techniques have been proposed to solve integer order differential equation [ – ]. We only know the noisy data on a part of the boundary or at some interior points of the concerned domain This leads to an inverse and ill-posed problem of the fractional diffusion equation, which means the solution does not depend continuously on the given known conditions. We investigate an inverse problem of the time-fractional diffusion equation. This kind of ill-posed problem is important in many branches of engineering sciences [ , ]. Due to the difficulty of the fractional derivative and the ill-posedness, to the authors’ knowledge, the results on inverse problem of the time-fractional diffusion equation are very few. Zheng and Wei [ , ] investigated a time-fractional inverse diffusion problem by using a spectral regularization method and a modified equation method.
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