Inverse Problems of Determining Sources of the Fractional Partial Differential Equations

Abstract

In this chapter, we mainly review theoretical results on inverse source problems for diffusion equations with the Caputo time-fractional derivatives of order $\alpha \in (0,1)$. Our survey covers the following types of inverse problems: 1. determination of time-dependent functions in interior source terms 2. determination of space-dependent functions in interior source terms 3. determination of time-dependent functions appearing in boundary conditions

References in corpus (14)

• Distributed Order Calculus and Equations of Ultraslow Diffusion
• Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients
• Distributed-order fractional Cauchy problems on bounded domains
• Strong maximum principle for fractional diffusion equations and an application to an inverse source problem
• Initial-boundary value problems for fractional diffusion equations with time-dependent coeffcients
• Strong Maximum Principle for Multi-Term Time-Fractional Diffusion Equations and its Application to an Inverse Source Problem
• On time-fractional diffusion equations with space-dependent variable order
• Reconstruction of the Temporal Component in the Source Term of a (Time-Fractional) Diffusion Equation
• Distributed Order Derivatives and Relaxation Patterns
• Initial-boundary value problems for multi-term time-fractional diffusion equations with x-dependent coefficients
• Recovery of a potential in a fractional diffusion equation
• An inverse problem for distributed order time-fractional diffusion equations
• Unique continuation principle for the one-dimensional time-fractional diffusion equation
• Global Lipschitz stability for a fractional inverse transport problem by Carleman estimates

Cited by in corpus (20)

• Initial-boundary value problems for coupled systems of time-fractional diffusion equations
• Uniqueness for inverse source problems for fractional diffusion-wave equations by data during not acting time
• Inverse moving source problem for time-fractional evolution equations: Determination of profiles
• Numerical reconstruction of the spatial component in the source term of a time-fractional diffusion equation
• Inverse source problem for a one-dimensional time-fractional diffusion equation and unique continuation for weak solutions
• Uniqueness in determining the orders of time and spatial fractional derivatives
• Unique determination of several coefficients in a fractional diffusion(-wave) equation by a single measurement
• Uniqueness of inverse source problems for time-fractional diffusion equations with singular functions in time
• Total variation regularization for recovering the spatial source term in a time-fractional diffusion equation
• Long-time asymptotic estimate and a related inverse source problem for time-fractional wave equations
• Recovering Multiple Fractional Orders in Time-Fractional Diffusion in an Unknown Medium
• Reconstruction of a Space-Time Dependent Source in Subdiffusion Models via a Perturbation Approach
• Numerical schemes for reconstructing profiles of moving sources in (time-fractional) evolution equations
• Uniqueness for an inverse coefficient problem for a one-dimensional time-fractional diffusion equation with non-zero boundary conditions
• Recovery of a Space-Time Dependent Diffusion Coefficient in Subdiffusion: Stability, Approximation and Error Analysis
• Backward problems in time for fractional diffusion-wave equation
• An Inverse Potential Problem for Subdiffusion: Stability and Reconstruction
• Determination of source and initial values for acoustic equations with a time-fractional attenuation
• Inverse $t$-source problem and a strict positivity property for coupled subdiffusion systems
• Inverse problems for a half-order time-fractional diffusion equation in arbitrary dimension by Carleman estimates