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

Frederick Maes and Marián Slodička
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Frederick Maes: Research Group NaM 2 , Department of Electronics and Information Systems, Ghent University, Krijgslaan 281, 9000 Ghent, Belgium
Marián Slodička: Research Group NaM 2 , Department of Electronics and Information Systems, Ghent University, Krijgslaan 281, 9000 Ghent, Belgium

Mathematics, 2020, vol. 8, issue 8, 1-17

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 Modified Model for τ q > τ T shows that an analogy of Theorem 2 for dominant hyperbolic behavior (fractional Cattaneo–Vernotte equation) is not possible.

Keywords: fractional dual-phase-lag equation; inverse source problem; uniqueness (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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