Fractional derivative is a promising mathematical tool which has been found to successfully simulate a variety of heat/mass transfer phenomena. In this work, an asymptotic analysis for a thermoelastic problem in the semi-infinite space (known as Danilovskaya problem) is presented for temperature and stresses when the fractional Fourier law with Riemann-Liouville fractional derivative replaces the classical version. Exact formulas, in both the short-time and long-time domains, are derived in terms of the Fox H-function. All closed-form expressions are validated by comparing them with numerical results. The anomalous diffusion limit is a special case of the derived asymptotic solutions whenever the thermoelastic coupling vanishes (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varepsilon \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>). Whilst the temperature solution shows two different behaviors in the short-time and the long-time domains in the presence of thermoelastic coupling, this transitional behavior disappears when (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varepsilon \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>), where both the short-time and long-time solutions coincide with the one presented by the ancestors (Schneider and Wyss, 1989). The anomalous thermal conduction relationship adapted from the Compte and Metzler suggestion for anomalous diffusion coefficient, (Compte and Metzler, 1997), <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\kappa }_{\alpha }=\kappa {\tau }^{1-\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>κ</mi> <mi>α</mi> </msub> <mo>=</mo> <mi>κ</mi> <msup> <mrow> <mi>τ</mi> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mi>α</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation> is the classical Fourier thermal conductivity, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation> is a characteristic time constant, and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$0<\alpha <1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, is shown to present different anomalous thermal conductivities (low thermal conduction <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${\kappa }_{\alpha }<\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>κ</mi> <mi>α</mi> </msub> <mo><</mo> <mi>κ</mi> </mrow> </math></EquationSource> </InlineEquation> or high “divergent” thermal conduction <InlineEquation ID="IEq8"> <EquationSource Format="TEX">${\kappa }_{\alpha }>\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>κ</mi> <mi>α</mi> </msub> <mo>></mo> <mi>κ</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\alpha \ <\ 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mspace width="4pt" /> <mo><</mo> <mspace width="4pt" /> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>) depending on certain choices for the constants <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>.

Asymptotic analysis for Cauchy stresses in a semi-infinite space with anomalous thermal conduction

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2026年3月18日

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Emad Awad

摘要

Fractional derivative is a promising mathematical tool which has been found to successfully simulate a variety of heat/mass transfer phenomena. In this work, an asymptotic analysis for a thermoelastic problem in the semi-infinite space (known as Danilovskaya problem) is presented for temperature and stresses when the fractional Fourier law with Riemann-Liouville fractional derivative replaces the classical version. Exact formulas, in both the short-time and long-time domains, are derived in terms of the Fox H-function. All closed-form expressions are validated by comparing them with numerical results. The anomalous diffusion limit is a special case of the derived asymptotic solutions whenever the thermoelastic coupling vanishes ( $\varepsilon \rightarrow 0$ $ε \to 0$ ). Whilst the temperature solution shows two different behaviors in the short-time and the long-time domains in the presence of thermoelastic coupling, this transitional behavior disappears when ( $\varepsilon \rightarrow 0$ $ε \to 0$ ), where both the short-time and long-time solutions coincide with the one presented by the ancestors (Schneider and Wyss, 1989). The anomalous thermal conduction relationship adapted from the Compte and Metzler suggestion for anomalous diffusion coefficient, (Compte and Metzler, 1997), ${\kappa }_{\alpha }=\kappa {\tau }^{1-\alpha }$ $κ_{α} = κ τ^{1 - α}$ , where $\kappa $ $κ$ is the classical Fourier thermal conductivity, $\tau $ $τ$ is a characteristic time constant, and $0<\alpha <1$ $0 < α < 1$ , is shown to present different anomalous thermal conductivities (low thermal conduction ${\kappa }_{\alpha }<\kappa $ $κ_{α} < κ$ or high “divergent” thermal conduction ${\kappa }_{\alpha }>\kappa $ $κ_{α} > κ$ , $\alpha \ <\ 1$ $α < 1$ ) depending on certain choices for the constants $\tau $ $τ$ and $\alpha $ $α$ .