Inverse Problems of Fractional Wave Equations

In this chapter, we firstly concern with a backward problembackward problem (or called initial inverse problem) for a nonlinear time fractional wave equationfractional wave equation(s) in a bounded domain. By applying the properties of Mittag–Leffler functionsMittag-Leffler function(s) and the method of eigenfunction expansion, we establish some results about the existenceexistence and uniquenessuniqueness of mild solutionsmild solution(s) of the proposed problem based on the compact technique. Due to the ill-posedness of backward problembackward problem in the sense of Hadamard, a general filter regularizationregularization method is utilized to approximate the solution, and further we prove the convergence rate for the regularized solutions. In Sect. 5.2, we consider the backward problembackward problem for an inhomogeneous time fractional wave equationfractional wave equation(s) in a general bounded domain. We show that the backward problembackward problem is ill-posed, and we propose a regularizing scheme by using a fractional Landweber regularizationregularization method. We also present error estimates between the regularized solution and the exact solution under two parameter choice rules. In Sect. 5.3, we consider the terminal value problemterminal value problem(s) of determining the initial value, in a general class of time fractional wave equationfractional wave equation(s) with the Caputo derivative, from a given final value. We are concerned with the existenceexistence and regularityregularity upon the terminal value data of the mild solutionmild solution(s). Under some assumptions of the nonlinear source function, we address and show the well-posednesswell-posedness for the terminal value problemterminal value problem(s). Some regularityregularity results for the mild solutionmild solution(s) and its derivatives of first and fractional orders are also derived. The effectiveness of our methods is shown by applying the results to two interesting models: time fractional Ginzburg–Landau equation and time fractional Burgers equation, where time and spatial regularity estimates are obtained. The contents of Sect. 5.1 are taken from He and Zhou (Proc R. Soc Edinburgh Sect A 152(6):1589–1612, 2022). The results in Sect. 5.2 are adopted from Huynh et al. (Appl Anal 100(4):860–878, 2021). Section 5.3 is from Tuan et al. (Nonlinearity 34(3):1448, 2021).