Some Inverse Source Problems of Determining a Space Dependent Source in Fractional-Dual-Phase-Lag Type Equations

Abstract

The dual-phase-lag heat transfer models attract a lot of interest of researchers in the last few decades. These are used in problems arising from non-classical thermal models, which are based on a non-Fourier type law. We study uniqueness of solutions to some inverse source problems for fractional partial differential equations of the Dual-Phase-Lag type. The source term is supposed to be of the form h(t)f(x) with a known function h(t). The unknown space dependent source f(x) is determined from the final time observation. New uniqueness results are formulated in Theorem 1 (for a general fractional Jeffrey-type model). Here, the variational approach was used. Theorem 2 derives uniqueness results under weaker assumptions on h(t) (monotonically increasing character of h(t) was removed) in a case of dominant parabolic behavior. The proof technique was based on spectral analysis. Section 4 shows that an analogy of Theorem 2 for dominant hyperbolic behavior (fractional Cattaneo–Vernotte equation) is not possible.