Two-dimensional Cattaneo-Hristov heat diffusion in the half-plane

Abstract

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.

Keywords

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

References

1. 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).
2. 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).
3. 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).
4. 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).
5. 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).
6. 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).
7. 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).
8. Nigmatullin, R.R. On the theory of relaxation for systems with “remnant” memory. Physica Status Solidi (b), 124(1), 389-393, (1984).

9. Green, A.E. and Naghdi, P.M. Thermoelasticity without energy dissipation. Journal of Elasticity, 31, 189-208, (1993).
10. Gorenflo, R., Mainardi, F., Moretti D. and Paradisi, P. Time fractional diffusion: a discrete random walk approach. Nonlinear Dynamics, 29, 129-143, (2002).
11. Cattaneo, C. Sulla Conduzione del Calore. Atti del Seminario Matematico e Fisico dell’Universita di Modena e Reggio Emilia, 3, 83-101, (1948).
12. 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).
13. Povstenko, Y.Z. Fractional heat conduction equation and associated thermal stress. Journal Thermal Stresses, 28(1), 83-102, (2004).
14. Povstenko, Y.Z. Thermoelasticity that uses fractional heat conduction equation. Journal of Mathematical Sciences, 162, 296-305, (2009).
15. 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).
16. 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).
17. 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).
18. Povstenko, Y.Z. Theory of thermoelasticity based on the space-time-fractional heat conduction equation. Physica Scripta, 2009(T136), 014017, (2009).
19. 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).
20. El-Karamany, A.S. and Ezzat, M.A. On fractional thermoelasticity. Mathematics and Mechanics of Solids, 16(3), 334-346, (2011).
21. Povstenko, Y.Z. Time-fractional radial heat conduction in a cylinder and associated thermal stresses. Archive of Applied Mechanics, 82, 345-362, (2012).
22. Povstenko, Y. Fractional heat conduction and related theories of thermoelasticity. In Fractional Thermoelasticity (pp. 13-33). Cham: Springer, (2015).
23. 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).
24. 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).
25. 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).
26. 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).
27. 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).
28. 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).
29. Sene, N. Analytical solutions of Hristov diffusion equations with non-singular fractional derivatives. Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(2), 023112, (2019).
30. 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).
31. 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).
32. 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).
33. Sene, N. Solutions of fractional diffusion equations and Cattaneo-Hristov diffusion model. International Journal of Analysis and Applications, 17(2), 191-207, (2019).
34. 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).
35. 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).
36. Avci, D. and Eroglu, B.B.I. Optimal control of the Cattaneo–Hristov heat diffusion model. Acta Mechanica, 232, 3529–3538, (2021).
37. 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).
38. Avci, D. Temperature profiles and thermal stresses due to heat conduction under fading memory effect. The European Physical Journal Plus, 136, 356, (2021).
39. Ångström, A.J. Neue Methode, das Wärmeleitungsvermöogen der Köorper zu bestimmen. Annalen der Physik und Chemie, 190(12), 513–530, (1861).
40. 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).
41. Nowacki, W. Thermoelasticity. Pergamon Press: Oxford, UK, (1986).
42. Baehr, H.D. and Stephan, K. Heat and Mass Transfer. Springer: Berlin/Heidelberg, Germany, (2006).
43. Povstenko, Y.Z. and Ostoja-Starzewski, M. Doppler effect described by the solutions of the Cattaneo telegraph equation. Acta Mechanica, 232, 725-740, (2021).
44. 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).
45. 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).
46. 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).
47. 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).
48. 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).
49. 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).
50. Povstenko, Y.Z. Linear fractional diffusion-wave equation for scientists and engineers. Switzerland: Springer International Publishing, (2015).
51. Prudnikov, A.P., Brychkov, Y.A., Maricheva, O.I., Romer, R.H. Integrals and series, (1988).
52. 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).