Some Inverse Source Problems of Determining a Space Dependent Source in Fractional-Dual-Phase-Lag Type Equations
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 p...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2020-08-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/8/8/1291 |
| Summary: | 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 <inline-formula><math display="inline"><semantics><mrow><mi>h</mi><mo>(</mo><mi>t</mi><mo>)</mo><mi>f</mi><mo>(</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></semantics></math></inline-formula> with a known function <inline-formula><math display="inline"><semantics><mrow><mi>h</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>. The unknown space dependent source <inline-formula><math display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi mathvariant="bold">x</mi><mo>)</mo></mrow></semantics></math></inline-formula> 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 <inline-formula><math display="inline"><semantics><mrow><mi>h</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> (monotonically increasing character of <inline-formula><math display="inline"><semantics><mrow><mi>h</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> was removed) in a case of <i>dominant parabolic</i> behavior. The proof technique was based on spectral analysis. Section Modified Model for <inline-formula><math display="inline"><semantics><mrow><msub><mi>τ</mi><mi>q</mi></msub><mo>></mo><msub><mi>τ</mi><mi>T</mi></msub></mrow></semantics></math></inline-formula> shows that an analogy of Theorem 2 for <i>dominant hyperbolic</i> behavior (fractional Cattaneo–Vernotte equation) is not possible. |
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| ISSN: | 2227-7390 |