THERMAL SCIENCE

International Scientific Journal

Xiao-Jun YangORCID

MATHEMATICAL CHALLENGES OF HEAT EQUATIONS IN THE TWENTY-FIRST CENTURY

ABSTRACT
In the invited review paper we report the unsolved problems for the heat equations in theory of the Riemann zeta function. The Fourier cosine integral is derived from the Riemann xi function associated with the Riemann zeta function. The tempered xi function is one of the entire functions in analytic number theory. The Fourier sine integral is derived from the tempered xi function. The Fourier cosine and sine integrals can be considered as the initial value conditions of the heat equation. The deformed Fourier cosine and sine integrals related to the exponential function can be considered as the solutions of the heat equation. There exists a great many of mathematics conjectures related to the heat equations containing Riemann hypothesis, Jensen's conjecture, and boldly asserts. They are still challenges for heat equations in the twenty-first century.
KEYWORDS
heat equation, Fourier cosine integral, Fourier sine integral, Riemann zeta function, Riemann hypothesis, analytic number theory
PAPER SUBMITTED: 2026-06-01
PAPER REVISED: 2026-06-10
PAPER ACCEPTED: 2026-06-11
PUBLISHED ONLINE: 2026-06-27
DOI REFERENCE: https://doi.org/10.2298/TSCI260601092Y
REFERENCES
[1] Widder, D. V., The Heat Equation, Academic Press
[2] Cannon, J. R., The One-dimensional Heat Equation, Cambridge University Press
[3] Ouhabaz, E. M., Analysis of Heat Equations on Domains, Princeton University Press
[4] Nellis, G., and Klein, S., Heat Transfer, Cambridge University Press
[5] Cengel, Y. A., Klein, S., and Beckman, W., Heat Transfer: a Practical Approach, WBC McGraw-Hill, Boston
[6] Bergman, T. L., Lavine, A. S., Incropera, F. P., et al., Introduction to Heat Transfer, John Wiley & Sons
[7] Rohsenow, W. M., Hartnett, J. P., and Cho, Y. I., Handbook of Heat Transfer, Mcgraw-hill, New York
[8] Bejan, A., and Kraus, A. D. (Eds.), Heat Transfer Handbook, John Wiley & Sons
[9] Fourier, J., Théorie analytique de la chaleur, Firmin Didot, Paris
[10] Fourier, J., The Analytical Theory of Heat, Cambridge University Press
[11] Narasimhan, T. N., Fourier's Heat Conduction Equation: History, Influence, and Connections, Reviews of Geophysics, 37(1999), 1, pp.151-172
[12] Kasner, E., and De Cicco, J., Geometry of the Fourier Heat Equation, Transactions of the American Mathematical Society, 60(1946), 1, pp.119-132
[13] Kasner, E., and De Cicco, J., The Fourier Heat Equation in Riemannian Space, Proceedings of the National Academy of Sciences, 38(1952), 9, pp.822-825
[14] Auriault, J. L., The Paradox of Fourier Heat Equation: A Theoretical Refutation, International Journal of Engineering Science, 118(2017), 1, pp.82-88
[15] Thibault, J., Comparison of Nine Three-dimensional Numerical Methods for the Solution of the Heat Diffusion Equation, Numerical Heat Transfer, 8(1985), 3, pp.281-298
[16] Luikov, A., Analytical Heat Diffusion Theory, Elsevier
[17] Zhukovsky, K. V., and Srivastava, H. M., Analytical Solutions for Heat Diffusion beyond Fourier Law, Applied Mathematics and Computation, 293(2017), 2, pp.423-437
[18] Yang, X. J., A New Integral Transform Operator for Solving the Heat-diffusion Problem, Applied Mathematics Letters, 64(2017), 1, pp.193-197
[19] Cannon, J. R., A Cauchy Problem for the Heat Equation, Annali di Matematica Pura Ed Applicata, 66(1964), 1, pp.155-165
[20] Ginsberg, F., On the Cauchy Problem for the One-dimensional Heat Equation, Mathematics of Computation, 17(1963), 83, pp.257-269
[21] Cannon, J. R., and Douglas, Jr, J., The Cauchy Problem for the Heat Equation, SIAM Journal on 24 Numerical Analysis, 4(1967), 3, pp.317-336
[22] Fokas, A. S., A Unified Approach to Boundary Value Problems, Society for Industrial and Applied Mathematics
[23] Özışık, M. N., Boundary Value Problems of Heat Conduction, Courier Corporation
[24] Caffarelli, L. A., and Vázquez, J. L., A Free-boundary Problem for the Heat Equation Arising in Flame Propagation, Transactions of the American Mathematical Society, 347(1995), 2, pp.411-441
[25] Kozhanov, A. I., On a Nonlocal Boundary Value Problem with Variable Coefficients for the Heat Equation and the Aller Equation, Differential Equations, 40(2004), 6, pp.815-826
[26] Doob, J. L., A Probability Approach to the Heat Equation, Transactions of the American Mathematical Society, 80(1955), 1, pp.216-280
[27] Bluman, G. W., and Cole, J. D., The General Similarity Solution of the Heat Equation, Journal of Mathematics and Mechanics, 18(1969), 11, pp.1025-1042
[28] Aziz, A. K., and Monk, P., Continuous Finite Elements in Space and Time for the Heat Equation, Mathematics of Computation, 52(1989), 186, pp.255-274
[29] Cole, K., Beck, J., Haji-Sheikh, A., et al., Heat Conduction Using Greens Functions, CRC Press
[30] Adomian, G., A New Approach to the Heat Equation—An Application of the Decomposition Method, Journal of Mathematical Analysis and Applications, 113(1986), 1, pp.202-209
[31] Sneddon, I. N., Fourier Transforms, Courier Corporation
[32] Kilbas, A. A., Srivastava, H. M., and Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Vol. 204, Elsevier,2006
[33] Debnath, L., and Bhatta, D., Integral Transforms and Their Applications, Chapman and Hall/CRC
[34] Osgood, B. G., Lectures on Fourier Transform and Its Applications, IAS/Park City Mathematics Series
[35] Iwaniec, H., Lectures on the Riemann Zeta Function, Vol. 62, American Mathematical Society
[36] Titchmarsh, E. C., The theory of the Riemann Zeta-function, Heath-Brown, D. R., ed., Oxford University Press
[37] Ivic, A., The Riemann Zeta-function: Theory and Applications, Courier Corporation
[38] Montgomery, H., Nikeghbali, A., and Rassias, M. T., Exploring the Riemann Zeta Function, Springer, Cham
[39] Riemann, G. F. B., Über die Anzahl der Primzahlen unter einer gegebenen Grösse (in German), Monatsberichte der Berliner Akademie, 2 (1859), Nov., pp. 671-680
[40] Yang, X. J., Alshehri, A., and Cui, P., The Method for the Heat-diffusion Equation in Analytic Number Theory, Thermal Science, 27(2023), 5A, pp.3837-3844
[41] Edwards, H. M., Riemanns'Zeta Function, Academic press, New York, USA
[42] Jensen, J. L., Recherches sur la théorie des équations (in French), Acta Mathematica, 36 (1913), 1, pp. 181-195
[43] Griffin, M. J., Ono, K., Rolen, L., et al., Jensen Polynomials for the Riemann Xi-function, Advances in Mathematics, 397(2022), 1, ID 108186
[44] Griffin, M., Ono, K., Rolen, L., et al., Jensen Polynomials for the Riemann Zeta Function and Other Sequences, Proceedings of the National Academy of Sciences, 116(2019), 23, 25 pp.11103-11110
[45] O'Sullivan, C., Zeros of Jensen Polynomials and Asymptotics for the Riemann Xi Function, Research in the Mathematical Sciences, 8(2021), 3, ID
[46] Yang, X. J., and Abdel-Aty, M., Letter to Editor: 2024 ICMAACS with News and Awards, Alexandria Engineering Journal, 133(2025), 1, pp.517-526
[47] Csordas, G., and Williamson, J., The Zeros of Jensen Polynomials are Simple, Proceedings of the American Mathematical Society, 49(1975), 1, pp.263-264
[48] Csordas, G., and Varga, R. S., Necessary and Sufficient Conditions and the Riemann Hypothesis, Advances in Applied Mathematics, 11(1990), 3, pp.328-357
[49] Craven, T., and Csordas, G., Jensen Polynomials and the Turán and Laguerre Inequalities, Pacific Journal of Mathematics, 136(1989), 2, pp.241-260
[50] Dimitrov, D., Higher order Turán Inequalities, Proceedings of the American Mathematical Society, 126(1998), 7, pp.2033-2037
[51] Dimitrov, D., and Lucas, F., Higher order Turán inequalities for the Riemann ?-function, Proceedings of the American Mathematical Society, 139(2011), 3, pp.1013-1022
[52] Yang, X. J., On a Tempered Xi Function Associated with the Riemann Xi Function, Fractals, 32(2024), 04, ID 2340116
[53] Yang, X. J., Baleanu, D., Tenreiro Machado, J. A., et al.,. Preface—Special Issue on Fractals and Local Fractional Calculus: Recent Advances and Future Challenges, Fractals, 32(2024), 04, ID 2402003
[54] Yang, X. J., Letter to editor: ICMAACS and History of Its Mathematical Awards, Alexandria Engineering Journal, 109(2024), 7, pp.270-273
[55] Yang, X. J., Preface: Advances in Fractionality in Mathematical Physics, International Journal of Geometric Methods in Modern Physics, 23(2026), 01, ID 2602001
[56] Yang, X. J., Letter to Editor (or Editorial): History of ICMAACS, Qin-Jiu-Shao Prize and Yang-Hui Award, International Journal of Geometric Methods in Modern Physics, 22(2025), 02, ID 2401001
[57] Yang, X. J., 2024 ICMAACS News and Awards, Thermal Science, 29(2025), 5B, pp.3723-3728
[58] Yang, X. J., and Capozziello, S., Letter to Editor: 2024 ICMAACS and Awards, International Journal of Geometric Methods in Modern Physics, 23(2026), 01, ID 2601001
[59] Yang, X. J., Abdel-Aty, M., and Geng, L. L., A New Odd Entire Function of Order One Arising in the Heat Problem, Thermal Science, 27(2023), 1B, pp.505-512
[60] Yang, X. J., New Conjectures for the Entire Functions Associated with Fractional Calculus, Fractals, 32(2024), 04, ID 2340129
[61] Yang, X. J., A New Program for the Entire Functions in Number Theory, Fractals, 32(2024), 04, ID 2340122
[62] Pólya, G.,Über trigonometrische Integrale mit nur reellen Nullstellen, Journal für die reine und angewandte Mathematik, 156(1927), 2, pp.6-18
[63] De Bruijn, N. G., The Roots of Trigonometric Integrals, Duke Mathematical Journal, 17(1950), 3, pp.197-226
[64] Newman, C. M., Fourier Transforms with only Real Zeros, Proceedings of the American Mathematical Society, 61(1976), 2, pp.245-251
[65] Odlyzko, A. M., An Improved Bound for the de Bruijn-Newman Constant, Numerical Algorithms, 25(2000), 1, pp.293-303
[66] Rodgers, B., and Tao, T., The De Bruijn-Newman Constant is Non-negative, Forum of Mathematics, Pi, 8(2020), e6, 62 pages
[67] Ki, H., Kim, Y. O., and Lee, J., On the de Bruijn-Newman Constant, Advances in Mathematics, 222(2009), 1, pp.281-306
[68] Riele, H. J., A New Lower Bound for the de Bruijn-Newman Constant, Numerische Mathematik, 58(1990), 1, pp. 661-667
[69] Csordas, G., Norfolk, T. S., and Varga, R. S., A Low Bound for the de Bruijn-newman Constant Λ, Numerische Mathematik, 52(1987), 5, pp.483-497
[70] Saouter, Y., Gourdon, X., and Demichel, P., An Improved Lower Bound for the de Bruijn-Newman Constant, Mathematics of Computation, 80(2011), 276, pp.2281-2287
[71] Stopple, J., Notes on Low Discriminants and the Generalized Newman Conjecture, Functiones et Approximatio Commentarii Mathematici, 51(2014), 1, pp.23-41
[72] Andrade, J., Chang, A., and Miller, S. J., Newman's Conjecture in Various Settings, Journal of Number Theory, 144(2014), 2, pp.70-91
[73] Newman, C., and Wu, W., Constants of de Bruijn-Newman Type in Analytic Number Theory and Statistical Physics, Bulletin of the American Mathematical Society, 57(2020), 4, pp.595-614
[74] Yang, X. J., Analysis of Hasse-Weil L-function and Associated Entire Functions Applied in Mathematical Physics, International Journal of Geometric Methods in Modern Physics, 22(2025), 04, ID 2450307
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Mathematical challenges of heat equations in the twenty-first century

© 2026 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution 4.0 International licence