FRACTIONAL CATTANEO EQUATION WITH A HARMONIC SOURCE AND ASSOCIATED THERMAL STRESSES IN AXISYMMETRIC AND CENTRAL SYMMETRIC CASES

Vinayak Kulkarni; Sagar Sankeshwari

FRACTIONAL CATTANEO EQUATION WITH A HARMONIC SOURCE AND ASSOCIATED THERMAL STRESSES IN AXISYMMETRIC AND CENTRAL SYMMETRIC CASES

Abstract

Integer and fractional order Cattaneo equations with a source varying harmonically in time under zero initial conditions are studied in the axisymmetric case and the central symmetric case. The integral transform techniques are used to find the fundamental solutions. The displacement potential is used to find the associated thermal stresses in both cases. The impact of the fractional order parameters and time-harmonic source on the temperature as well as stress distributions has been examined. The outcomes of numerical computations are represented graphically for various values of the order of fractional derivatives. The main objective of the article is to examine the role of the order of the fractional derivatives in the rate of heat transfer and related thermal stresses. Moreover, it has been observed that the angular frequency controls the oscillatory behavior of solutions and also affects the amplitude of the oscillations. This analysis has a wide scope of applications in the study of viscoelastic materials, thermal energy storage systems, biological systems, etc.

Keywords

References

Angstrom, A. J., (1863), New method of determining the thermal conductibility of bodies, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 25(166), pp. 130-142.
Avazzadeh, Z., Nikan, O., Nguyen, A. T. and Nguyen, V. T., (2023), A localized hybrid kernel meshless technique for solving the fractional Rayleigh–Stokes problem for an edge in a viscoelastic fluid, Engineering Analysis with Boundary Elements, 146, pp. 695-705.
Beyer, H. and Kempfle, S., (1995), Definition of physically consistent damping laws with fractional derivatives, ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 75(8), pp. 623-635.
Carlsaw, H. S. and Jaeger, J. C., (1959), Conduction of heat in solids, Clarendon, Oxford.
Cattaneo, C., (1958), Sur une forme de l’equation de la chaleur eliminant la paradoxe d’une propagation instantantee, Compt. Rendu, 247, pp. 431-433.
Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., (2006), Theory and applications of fractional differential equations, elsevier.
Luo, M., Qiu, W., Nikan, O. and Avazzadeh, Z., (2023), Second-order accurate, robust and efficient ADI Galerkin technique for the three-dimensional nonlocal heat model arising in viscoelasticity, Applied Mathematics and Computation, 440, p. 127655.
Mandelis, A., (2013), Diffusion-wave fields: mathematical methods and Green functions, Springer Science and Business Media.

Morse, P. M. and Feshbach, H., (1946), Methods of theoretical physics, Technology Press.
Nikan, O., Avazzadeh, Z. and Machado, J. T., (2021), Numerical approach for modeling fractional heat conduction in porous medium with the generalized Cattaneo model, Applied Mathematical Modelling, 100, pp. 107-124.
Nikan, O., Rashidinia, J. and Jafari, H., (2025), Numerically pricing American and European options using a time fractional Black–Scholes model in financial decision-making, Alexandria Engineering Journal, 112, pp. 235-245.
Nowacki, W., (1957), State of stress in an elastic space due to a source of heat varying harmonically as function of time, Bull. Acad. Polon. Sci. Sér. Sci. Techn, 5(3), pp. 145-154.
Nowacki, W., (1986), Thermoelasticity, 2nd edn, PWN.
Parkus, H., (2013), Instationare warmespannungen, Springer-Verlag.
Povstenko, Y. Z., (2011), Fractional Cattaneo-type equations and generalized thermoelasticity, Journal of thermal Stresses, 34(2), pp. 97-114.
Povstenko, Y., (2016), Time-fractional heat conduction in a half-line domain due to boundary value of temperature varying harmonically in time, Mathematical Problems in Engineering, p. 2016.
Povstenko, Y., (2016), Fractional heat conduction in a space with a source varying harmonically in time and associated thermal stresses, Journal of Thermal Stresses, 39(11), pp. 1442-1450.
Povstenko, Y., (2024), Fractional Thermoelasticity, Second edition, Springer, Cham.
Povstenko, Y. and Ostoja-Starzewski, M., (2021), Doppler effect described by the solutions of the Cattaneo telegraph equation, Acta mechanica, 232, pp. 725-740.
Povstenko, Y. and Ostoja-Starzewski, M., (2022), Fractional telegraph equation under moving time-harmonic impact, International journal of heat and mass transfer, 182, p. 121958.
Povstenko, Y., Ostoja-Starzewski, M. and Kyrylych, T., (2023), Telegraph equation in polar coordinates: Unbounded domain with moving time-harmonic source, International Journal of Heat and Mass Transfer, 207, p. 124013.
Podlubny, I., (1999), Fractional differential equations (san diego, ca: Academic).
Vernotte, P., (1961), Some possible complications in the phenomena of thermal conduction, Compte Rendus, 252(1), pp. 2190-2191.
Vrentas, J. S. and Vrentas, C. M., (2016), Diffusion and mass transfer, CRC Press.
Zhu, Y., (2016), Heat-loss modified Angstrom method for simultaneous measurements of thermal diffusivity and conductivity of graphite sheets: The origins of heat loss in Angstrom method, International Journal of Heat and Mass Transfer, 92, pp. 784-791.

Details

Primary Language

English

Subjects

Numerical Solution of Differential and Integral Equations, Partial Differential Equations, Real and Complex Functions (Incl. Several Variables), Theoretical and Applied Mechanics in Mathematics

Journal Section

Research Article

Authors

Vinayak Kulkarni ^*
0000-0002-2507-4458
India

Sagar Sankeshwari
0000-0002-7482-5684
India

Publication Date

November 3, 2025

Submission Date

September 19, 2024

Acceptance Date

January 15, 2025

Published in Issue

Year 2025 Volume: 15 Number: 11

IZ

https://izlik.org/JA88MS54YL

Cite

RIS / Bibtex

APA

Kulkarni, V., & Sankeshwari, S. (2025). FRACTIONAL CATTANEO EQUATION WITH A HARMONIC SOURCE AND ASSOCIATED THERMAL STRESSES IN AXISYMMETRIC AND CENTRAL SYMMETRIC CASES. TWMS Journal of Applied and Engineering Mathematics, 15(11), 2599-2612. https://izlik.org/JA88MS54YL

AMA

1.Kulkarni V, Sankeshwari S. FRACTIONAL CATTANEO EQUATION WITH A HARMONIC SOURCE AND ASSOCIATED THERMAL STRESSES IN AXISYMMETRIC AND CENTRAL SYMMETRIC CASES. JAEM. 2025;15(11):2599-2612. https://izlik.org/JA88MS54YL

Chicago

Kulkarni, Vinayak, and Sagar Sankeshwari. 2025. “FRACTIONAL CATTANEO EQUATION WITH A HARMONIC SOURCE AND ASSOCIATED THERMAL STRESSES IN AXISYMMETRIC AND CENTRAL SYMMETRIC CASES”. TWMS Journal of Applied and Engineering Mathematics 15 (11): 2599-2612. https://izlik.org/JA88MS54YL.

EndNote

Kulkarni V, Sankeshwari S (November 1, 2025) FRACTIONAL CATTANEO EQUATION WITH A HARMONIC SOURCE AND ASSOCIATED THERMAL STRESSES IN AXISYMMETRIC AND CENTRAL SYMMETRIC CASES. TWMS Journal of Applied and Engineering Mathematics 15 11 2599–2612.

IEEE

[1]V. Kulkarni and S. Sankeshwari, “FRACTIONAL CATTANEO EQUATION WITH A HARMONIC SOURCE AND ASSOCIATED THERMAL STRESSES IN AXISYMMETRIC AND CENTRAL SYMMETRIC CASES”, JAEM, vol. 15, no. 11, pp. 2599–2612, Nov. 2025, [Online]. Available: https://izlik.org/JA88MS54YL

ISNAD

Kulkarni, Vinayak - Sankeshwari, Sagar. “FRACTIONAL CATTANEO EQUATION WITH A HARMONIC SOURCE AND ASSOCIATED THERMAL STRESSES IN AXISYMMETRIC AND CENTRAL SYMMETRIC CASES”. TWMS Journal of Applied and Engineering Mathematics 15/11 (November 1, 2025): 2599-2612. https://izlik.org/JA88MS54YL.

JAMA

1.Kulkarni V, Sankeshwari S. FRACTIONAL CATTANEO EQUATION WITH A HARMONIC SOURCE AND ASSOCIATED THERMAL STRESSES IN AXISYMMETRIC AND CENTRAL SYMMETRIC CASES. JAEM. 2025;15:2599–2612.

MLA

Kulkarni, Vinayak, and Sagar Sankeshwari. “FRACTIONAL CATTANEO EQUATION WITH A HARMONIC SOURCE AND ASSOCIATED THERMAL STRESSES IN AXISYMMETRIC AND CENTRAL SYMMETRIC CASES”. TWMS Journal of Applied and Engineering Mathematics, vol. 15, no. 11, Nov. 2025, pp. 2599-12, https://izlik.org/JA88MS54YL.

Vancouver

1.Vinayak Kulkarni, Sagar Sankeshwari. FRACTIONAL CATTANEO EQUATION WITH A HARMONIC SOURCE AND ASSOCIATED THERMAL STRESSES IN AXISYMMETRIC AND CENTRAL SYMMETRIC CASES. JAEM [Internet]. 2025 Nov. 1;15(11):2599-612. Available from: https://izlik.org/JA88MS54YL