Research Article

In this paper, Cattaneo-Hristov heat diffusion is discussed in the half plane for the first time, and solved under two different boundary conditions. For the solution purpose, the Laplace, and the sine- and exponential- Fourier transforms with respect to time and space variables are applied, respectively. Since the fractional term in the problem is the Caputo-Fabrizio derivative with the exponential kernel, the solutions are in terms of time-dependent exponential and spatial-dependent Bessel functions. Behaviors of the temperature functions due to the change of different parameters of the problem are interpreted by giving 2D and 3D graphics.

Two-dimensional Cattaneo-Hristov equation, Laplace transform, sine-Fourier transform, exponential Fourier transform, Caputo-Fabrizio derivative

- Yavuz, M. and Sene, N. Approximate solutions of the model describing fluid flow using generalized $\rho$-Laplace transform method and heat balance integral method. Axioms, 9(4), 123, (2020).
- Hristov, J. Magnetic field diffusion in ferromagnetic materials: fractional calculus approaches. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 11(3), 1–15, (2021).
- Joshi, H. and Jha, B.K. Chaos of calcium diffusion in Parkinson’s infectious disease model and treatment mechanism via Hilfer fractional derivative. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 84-94, (2021).
- Martinez-Farias F.J., Alvarado-Sanchez, A., Rangel-Cortes, E. and Hernandez-Hernandez, A. Bi-dimensional crime model based on anomalous diffusion with law enforcement effect. Mathematical Modelling and Numerical Simulation with Applications, 2(1), 26-40, (2022).
- Sene, N. Second-grade fluid with Newtonian heating under Caputo fractional derivative: analytical investigations via Laplace transforms. Mathematical Modelling and Numerical Simulation with Applications, 2(1), 13-25, (2022).
- Joshi, H., Yavuz M. and Stamova, I. Analysis of the disturbance effect in intracellular calcium dynamic on fibroblast cells with an exponential kernel law. Bulletin of Biomathematics, 1(1), 24-39, (2023).
- Gurtin, M.E. and Pipkin, A.C. A general theory of heat conduction with finite wave speeds. Archive for Rational Mechanics and Analysis, 31, 113-126, (1968).
- Nigmatullin, R.R. On the theory of relaxation for systems with “remnant” memory. Physica Status Solidi (b), 124(1), 389-393, (1984).
- Green, A.E. and Naghdi, P.M. Thermoelasticity without energy dissipation. Journal of Elasticity, 31, 189-208, (1993).
- Gorenflo, R., Mainardi, F., Moretti D. and Paradisi, P. Time fractional diffusion: a discrete random walk approach. Nonlinear Dynamics, 29, 129-143, (2002).
- Cattaneo, C. Sulla Conduzione del Calore. Atti del Seminario Matematico e Fisico dell’Universita di Modena e Reggio Emilia, 3, 83-101, (1948).
- Cattaneo, C. Sur une forme de l’equation de la chaleur eliminant la paradoxe d’une propagation instantantee. Comptes Rendus de l’Académie des Sciences, 247, 431-433, (1958).
- Povstenko, Y.Z. Fractional heat conduction equation and associated thermal stress. Journal Thermal Stresses, 28(1), 83-102, (2004).
- Povstenko, Y.Z. Thermoelasticity that uses fractional heat conduction equation. Journal of Mathematical Sciences, 162, 296-305, (2009).
- Povstenko, Y.Z. Time-fractional heat conduction in an infinite medium with a spherical hole under Robin boundary condition. Fractional Calculus and Applied Analysis, 16(2), 354-369, (2013).
- Povstenko, Y.Z. and Klekot, J. The fundamental solutions to the central symmetric time-fractional heat conduction equation with heat absorption. Journal of Applied Mathematics and Computational Mechanics, 16(2), 101-112, (2017).
- Povstenko Y.Z. and Kyrylych, T. Time-fractional heat conduction in an infinite plane containing an external crack under heat flux loading. Computers and Mathematics with Applications, 78(5), 1386-1395, (2019).
- Povstenko, Y.Z. Theory of thermoelasticity based on the space-time-fractional heat conduction equation. Physica Scripta, 2009(T136), 014017, (2009).
- Sherief, H.H., El-Sayed, A.M.A. and Abd El-Latief, A.M. Fractional order theory of thermoelasticity. International Journal of Solids and Structures, 47(2), 269-275, (2010).
- El-Karamany, A.S. and Ezzat, M.A. On fractional thermoelasticity. Mathematics and Mechanics of Solids, 16(3), 334-346, (2011).
- Povstenko, Y.Z. Time-fractional radial heat conduction in a cylinder and associated thermal stresses. Archive of Applied Mechanics, 82, 345-362, (2012).
- Povstenko, Y. Fractional heat conduction and related theories of thermoelasticity. In Fractional Thermoelasticity (pp. 13-33). Cham: Springer, (2015).
- Özdemir, N., Povstenko, Y.Z., Avcı, D. and Iskender, B.B. Optimal boundary control of thermal stresses in a plate based on time-fractional heat conduction equation. Journal of Thermal Stresses, 37(8), 969-980, (2014).
- Povstenko, Y., Avcı, D., Ero˘glu, B.B.I. and Özdemir, N. Control of thermal stresses in axis-symmetric problems of fractional thermoelasticity for an infinite cylindrical domain. Thermal Science, 21(1), 19-28, (2017).
- Hristov, J. Transient heat diffusion with a non-singular fading memory: from the Cattaneo constitutive equation with Jeffrey’s kernel to the Caputo-Fabrizio time-fractional derivative. Thermal Science, 20(2), 757-762, (2016).
- Hristov, J. Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: appraising analysis with emphasis on diffusion models. In Frontiers in Fractional Calculus (pp. 270-342). Bentham Science Publishers, (2017).
- Hristov, J. Derivation of the fractional Dodson equation and beyond: transient diffusion with a non-singular memory and exponentially fading-out diffusivity. Progress in Fractional Differentiation and Applications, 3(4), 255-270, (2017).
- Hristov, J. Linear viscoelastic responses and constitutive equations in terms of fractional operators with non-singular kernels-Pragmatic approach, memory kernel correspondence requirement and analyses. The European Physical Journal Plus, 134(6), 283, (2019).
- Sene, N. Analytical solutions of Hristov diffusion equations with non-singular fractional derivatives. Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(2), 023112, (2019).
- Avci, D. and Eroglu, B.B.I. Semi-Analytical Solution of Hristov Diffusion Equation with Source. In A Closer Look at the Diffusion Equation (pp. 117-132). Nova Science Publishers (2020).
- Alkahtani, B.S.T. and Atangana, A. A note on Cattaneo-Hristov model with non-singular fading memory. Thermal Science, 21(1), 1-7, (2017).
- Koca, I. and Atangana, A. Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo-Fabrizio and Atangana-Baleanu fractional derivatives. Thermal Science, 21(6), 2299-2305, (2017).
- Sene, N. Solutions of fractional diffusion equations and Cattaneo-Hristov diffusion model. International Journal of Analysis and Applications, 17(2), 191-207, (2019).
- Singh, Y., Kumar, D., Modi, K. and Gill, V. A new approach to solve Cattaneo-Hristov diffusion model and fractional diffusion equations with Hilfer-Prabhakar derivative. AIMS Mathematics, 5(2), 843-855, (2020).
- Eroglu, B.B.I. and Avci, D. Separable solutions of Cattaneo-Hristov heat diffusion equation in a line segment: Cauchy and source problems. Alexandria Engineering Journal, 60(2), 2347–2353, (2021).
- Avci, D. and Eroglu, B.B.I. Optimal control of the Cattaneo–Hristov heat diffusion model. Acta Mechanica, 232, 3529–3538, (2021).
- Avci, D. and Eroglu, B.B.I. Oscillatory heat transfer due to the Cattaneo-Hristov Model on the real line. In Fractional Calculus: New Applications in Understanding Nonlinear Phenomena (pp. 108-123). Singapore: Bentham Science Publishers, (2022).
- Avci, D. Temperature profiles and thermal stresses due to heat conduction under fading memory effect. The European Physical Journal Plus, 136, 356, (2021).
- Ångström, A.J. Neue Methode, das Wärmeleitungsvermöogen der Köorper zu bestimmen. Annalen der Physik und Chemie, 190(12), 513–530, (1861).
- Nowacki, W. State of stress in an elastic space due to a source of heat varying harmonically as function of time. Bulletin of the Polish Academy of Sciences Technical Sciences, 5(3), 145–154, (1957).
- Nowacki, W. Thermoelasticity. Pergamon Press: Oxford, UK, (1986).
- Baehr, H.D. and Stephan, K. Heat and Mass Transfer. Springer: Berlin/Heidelberg, Germany, (2006).
- Povstenko, Y.Z. and Ostoja-Starzewski, M. Doppler effect described by the solutions of the Cattaneo telegraph equation. Acta Mechanica, 232, 725-740, (2021).
- Datsko, B., Podlubny, I. and Povstenko, Y. Time-Fractional diffusion-wave equation with mass absorption in a sphere under harmonic impact. Mathematics, 7(5), 433, (2019).
- Povstenko, Y.Z. Fractional thermoelasticity problem for an infinite solid with a cylindrical hole under harmonic heat flux boundary condition. Acta Mechanica, 230, 2137-2144, (2019).
- Povstenko, Y.Z. and Kyrylych, T. Time-fractional diffusion with mass absorption in a half-line domain due to boundary value of concentration varying harmonically in time. Entropy, 20(5), 346, (2018).
- Povstenko, Y.Z. and Kyrylych, T. Time-fractional diffusion with mass absorption under harmonic impact. Fractional Calculus and Applied Analysis, 21(1), 118-133, (2018).
- Podlubny, I., Magin, R.L. and Trymorush, I. Niels Henrik Abel and the birth of fractional calculus. Fractional Calculus and Applied Analysis, 20(5), 1068-1075, (2017).
- Caputo, M. and Fabrizio, M. Applications of new time and spatial fractional derivatives with exponential kernels. Progress in Fractional Differentiation and Applications, 2(1), 1-11, (2016).
- Povstenko, Y.Z. Linear fractional diffusion-wave equation for scientists and engineers. Switzerland: Springer International Publishing, (2015).
- Prudnikov, A.P., Brychkov, Y.A., Maricheva, O.I., Romer, R.H. Integrals and series, (1988).
- Povstenko, Y.Z. Fundamental solutions to time-fractional advection diffusion equation in a case of two space variables. Mathematical Problems in Engineering, 2014(3), 1-7, (2014).

- Yavuz, M. and Sene, N. Approximate solutions of the model describing fluid flow using generalized $\rho$-Laplace transform method and heat balance integral method. Axioms, 9(4), 123, (2020).
- Hristov, J. Magnetic field diffusion in ferromagnetic materials: fractional calculus approaches. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 11(3), 1–15, (2021).
- Joshi, H. and Jha, B.K. Chaos of calcium diffusion in Parkinson’s infectious disease model and treatment mechanism via Hilfer fractional derivative. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 84-94, (2021).
- Martinez-Farias F.J., Alvarado-Sanchez, A., Rangel-Cortes, E. and Hernandez-Hernandez, A. Bi-dimensional crime model based on anomalous diffusion with law enforcement effect. Mathematical Modelling and Numerical Simulation with Applications, 2(1), 26-40, (2022).
- Sene, N. Second-grade fluid with Newtonian heating under Caputo fractional derivative: analytical investigations via Laplace transforms. Mathematical Modelling and Numerical Simulation with Applications, 2(1), 13-25, (2022).
- Joshi, H., Yavuz M. and Stamova, I. Analysis of the disturbance effect in intracellular calcium dynamic on fibroblast cells with an exponential kernel law. Bulletin of Biomathematics, 1(1), 24-39, (2023).
- Gurtin, M.E. and Pipkin, A.C. A general theory of heat conduction with finite wave speeds. Archive for Rational Mechanics and Analysis, 31, 113-126, (1968).
- Nigmatullin, R.R. On the theory of relaxation for systems with “remnant” memory. Physica Status Solidi (b), 124(1), 389-393, (1984).
- Green, A.E. and Naghdi, P.M. Thermoelasticity without energy dissipation. Journal of Elasticity, 31, 189-208, (1993).
- Gorenflo, R., Mainardi, F., Moretti D. and Paradisi, P. Time fractional diffusion: a discrete random walk approach. Nonlinear Dynamics, 29, 129-143, (2002).
- Cattaneo, C. Sulla Conduzione del Calore. Atti del Seminario Matematico e Fisico dell’Universita di Modena e Reggio Emilia, 3, 83-101, (1948).
- Cattaneo, C. Sur une forme de l’equation de la chaleur eliminant la paradoxe d’une propagation instantantee. Comptes Rendus de l’Académie des Sciences, 247, 431-433, (1958).
- Povstenko, Y.Z. Fractional heat conduction equation and associated thermal stress. Journal Thermal Stresses, 28(1), 83-102, (2004).
- Povstenko, Y.Z. Thermoelasticity that uses fractional heat conduction equation. Journal of Mathematical Sciences, 162, 296-305, (2009).
- Povstenko, Y.Z. Time-fractional heat conduction in an infinite medium with a spherical hole under Robin boundary condition. Fractional Calculus and Applied Analysis, 16(2), 354-369, (2013).
- Povstenko, Y.Z. and Klekot, J. The fundamental solutions to the central symmetric time-fractional heat conduction equation with heat absorption. Journal of Applied Mathematics and Computational Mechanics, 16(2), 101-112, (2017).
- Povstenko Y.Z. and Kyrylych, T. Time-fractional heat conduction in an infinite plane containing an external crack under heat flux loading. Computers and Mathematics with Applications, 78(5), 1386-1395, (2019).
- Povstenko, Y.Z. Theory of thermoelasticity based on the space-time-fractional heat conduction equation. Physica Scripta, 2009(T136), 014017, (2009).
- Sherief, H.H., El-Sayed, A.M.A. and Abd El-Latief, A.M. Fractional order theory of thermoelasticity. International Journal of Solids and Structures, 47(2), 269-275, (2010).
- El-Karamany, A.S. and Ezzat, M.A. On fractional thermoelasticity. Mathematics and Mechanics of Solids, 16(3), 334-346, (2011).
- Povstenko, Y.Z. Time-fractional radial heat conduction in a cylinder and associated thermal stresses. Archive of Applied Mechanics, 82, 345-362, (2012).
- Povstenko, Y. Fractional heat conduction and related theories of thermoelasticity. In Fractional Thermoelasticity (pp. 13-33). Cham: Springer, (2015).
- Özdemir, N., Povstenko, Y.Z., Avcı, D. and Iskender, B.B. Optimal boundary control of thermal stresses in a plate based on time-fractional heat conduction equation. Journal of Thermal Stresses, 37(8), 969-980, (2014).
- Povstenko, Y., Avcı, D., Ero˘glu, B.B.I. and Özdemir, N. Control of thermal stresses in axis-symmetric problems of fractional thermoelasticity for an infinite cylindrical domain. Thermal Science, 21(1), 19-28, (2017).
- Hristov, J. Transient heat diffusion with a non-singular fading memory: from the Cattaneo constitutive equation with Jeffrey’s kernel to the Caputo-Fabrizio time-fractional derivative. Thermal Science, 20(2), 757-762, (2016).
- Hristov, J. Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: appraising analysis with emphasis on diffusion models. In Frontiers in Fractional Calculus (pp. 270-342). Bentham Science Publishers, (2017).
- Hristov, J. Derivation of the fractional Dodson equation and beyond: transient diffusion with a non-singular memory and exponentially fading-out diffusivity. Progress in Fractional Differentiation and Applications, 3(4), 255-270, (2017).
- Hristov, J. Linear viscoelastic responses and constitutive equations in terms of fractional operators with non-singular kernels-Pragmatic approach, memory kernel correspondence requirement and analyses. The European Physical Journal Plus, 134(6), 283, (2019).
- Sene, N. Analytical solutions of Hristov diffusion equations with non-singular fractional derivatives. Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(2), 023112, (2019).
- Avci, D. and Eroglu, B.B.I. Semi-Analytical Solution of Hristov Diffusion Equation with Source. In A Closer Look at the Diffusion Equation (pp. 117-132). Nova Science Publishers (2020).
- Alkahtani, B.S.T. and Atangana, A. A note on Cattaneo-Hristov model with non-singular fading memory. Thermal Science, 21(1), 1-7, (2017).
- Koca, I. and Atangana, A. Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo-Fabrizio and Atangana-Baleanu fractional derivatives. Thermal Science, 21(6), 2299-2305, (2017).
- Sene, N. Solutions of fractional diffusion equations and Cattaneo-Hristov diffusion model. International Journal of Analysis and Applications, 17(2), 191-207, (2019).
- Singh, Y., Kumar, D., Modi, K. and Gill, V. A new approach to solve Cattaneo-Hristov diffusion model and fractional diffusion equations with Hilfer-Prabhakar derivative. AIMS Mathematics, 5(2), 843-855, (2020).
- Eroglu, B.B.I. and Avci, D. Separable solutions of Cattaneo-Hristov heat diffusion equation in a line segment: Cauchy and source problems. Alexandria Engineering Journal, 60(2), 2347–2353, (2021).
- Avci, D. and Eroglu, B.B.I. Optimal control of the Cattaneo–Hristov heat diffusion model. Acta Mechanica, 232, 3529–3538, (2021).
- Avci, D. and Eroglu, B.B.I. Oscillatory heat transfer due to the Cattaneo-Hristov Model on the real line. In Fractional Calculus: New Applications in Understanding Nonlinear Phenomena (pp. 108-123). Singapore: Bentham Science Publishers, (2022).
- Avci, D. Temperature profiles and thermal stresses due to heat conduction under fading memory effect. The European Physical Journal Plus, 136, 356, (2021).
- Ångström, A.J. Neue Methode, das Wärmeleitungsvermöogen der Köorper zu bestimmen. Annalen der Physik und Chemie, 190(12), 513–530, (1861).
- Nowacki, W. State of stress in an elastic space due to a source of heat varying harmonically as function of time. Bulletin of the Polish Academy of Sciences Technical Sciences, 5(3), 145–154, (1957).
- Nowacki, W. Thermoelasticity. Pergamon Press: Oxford, UK, (1986).
- Baehr, H.D. and Stephan, K. Heat and Mass Transfer. Springer: Berlin/Heidelberg, Germany, (2006).
- Povstenko, Y.Z. and Ostoja-Starzewski, M. Doppler effect described by the solutions of the Cattaneo telegraph equation. Acta Mechanica, 232, 725-740, (2021).
- Datsko, B., Podlubny, I. and Povstenko, Y. Time-Fractional diffusion-wave equation with mass absorption in a sphere under harmonic impact. Mathematics, 7(5), 433, (2019).
- Povstenko, Y.Z. Fractional thermoelasticity problem for an infinite solid with a cylindrical hole under harmonic heat flux boundary condition. Acta Mechanica, 230, 2137-2144, (2019).
- Povstenko, Y.Z. and Kyrylych, T. Time-fractional diffusion with mass absorption in a half-line domain due to boundary value of concentration varying harmonically in time. Entropy, 20(5), 346, (2018).
- Povstenko, Y.Z. and Kyrylych, T. Time-fractional diffusion with mass absorption under harmonic impact. Fractional Calculus and Applied Analysis, 21(1), 118-133, (2018).
- Podlubny, I., Magin, R.L. and Trymorush, I. Niels Henrik Abel and the birth of fractional calculus. Fractional Calculus and Applied Analysis, 20(5), 1068-1075, (2017).
- Caputo, M. and Fabrizio, M. Applications of new time and spatial fractional derivatives with exponential kernels. Progress in Fractional Differentiation and Applications, 2(1), 1-11, (2016).
- Povstenko, Y.Z. Linear fractional diffusion-wave equation for scientists and engineers. Switzerland: Springer International Publishing, (2015).
- Prudnikov, A.P., Brychkov, Y.A., Maricheva, O.I., Romer, R.H. Integrals and series, (1988).
- Povstenko, Y.Z. Fundamental solutions to time-fractional advection diffusion equation in a case of two space variables. Mathematical Problems in Engineering, 2014(3), 1-7, (2014).

Primary Language | English |
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Subjects | Mathematical Physics (Other), Theoretical and Applied Mechanics in Mathematics |

Journal Section | Research Articles |

Authors | |

Publication Date | September 30, 2023 |

Submission Date | August 9, 2023 |

Published in Issue | Year 2023 Volume: 3 Issue: 3 |

The published articles in MMNSA are licensed under a Creative Commons Attribution 4.0 International License