On a problem of heat equation with fractional load. (English) Zbl 1452.35245

Summary: In the paper, the solvability problems of an nonhomogeneous boundary value problem in the first quadrant for a fractionally loaded heat equation are studied. Feature of this problem is that, firstly, the loaded term is presented in the form of the Caputo fractional derivative with respect to the spatial variable, secondly, the order of the derivative in the loaded term is less than the order of the differential part and, thirdly, the point of load is moving. The problem is reduced to the Volterra integral equation of the second kind, the kernel of which contains the generalized hypergeometric series. The kernel of the obtained integral equation is estimated and it is shown that the kernel of the equation has a weak singularity (under certain restrictions on the load), this is the basis for the statement that the loaded term in the equation is a weak perturbation of its differential part. In addition, the limiting cases of the order of the fractional derivative are considered. It is proved that there is continuity in the order of the fractional derivative.


35R11 Fractional partial differential equations
35K20 Initial-boundary value problems for second-order parabolic equations
Full Text: DOI


[1] Nakhushev, A. M., Equations of Mathematical Biology (1995), Moscow: Vyssh. Shkola, Moscow · Zbl 0991.35500
[2] A. M. Nakhushev, ‘‘Loaded equations and their applications,’’ Differ. Uravn. 19, 86-94 (1983). http://www.mathnet.ru/links/76a85511435ea469331b50dca3a7faf6/de4747.pdf. · Zbl 0536.35080
[3] A. M. Nakhushev, ‘‘The Darboux problem for a certain degenerate second order loaded integrodifferential equation,’’ Differ. Uravn. 12, 103-108 (1976). http://www.mathnet.ru/links/14e4e915dbd2bb926a7d4099e8479a04/de2654.pdf. · Zbl 0349.45011
[4] Dzhenaliev, M. T., On the Theory of Linear Boundary Value Problems for Loaded Differential Equations (1995), Almaty: ITPM Computer Center, Almaty · Zbl 0868.34047
[5] M. T. Dzhenaliev, ‘‘Loaded equations with periodic boundary conditions,’’ Differ. Equat. 37, 51-57 (2001). https://link.springer.com/article/10.1023/A · Zbl 1017.35115
[6] M. T. Dzhenaliev, ‘‘About boundary value problem for linear loaded parabolic equation with non-local boundary conditions,’’ Differ. Uravn. 27, 1825-1827 (1991). http://www.mathnet.ru/links/6bed8d594d7f9e8718cfa6180af5bc5e/de7631.pdf. · Zbl 0768.35036
[7] Dzhenaliev, M. T.; Ramazanov, M. I., Loaded Equations as Perturbations of Differential Equations (2010), Almaty: Gylym, Almaty
[8] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), New York, London: Academic, New York, London · Zbl 0428.26004
[9] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications (Nauka Tehnika, Minsk, 1987; Gordon and Breach, New York, 1993). · Zbl 0617.26004
[10] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives. Theory and Applications (1993), New York: Gordon and Breach, New York · Zbl 0818.26003
[11] Nakhushev, A. M., Elements of Fractional Calculus and their Applications (2000)
[12] Le Mehaute, A.; Tenreiro Machado, J. A.; Trigeassou, J. C.; Sabatier, J., Fractional Differentiation and its Applications (2005), Bordeaux: Bordeaux Univ. Press, Bordeaux
[13] Pskhu, A. V., Partial Differential Equations of Fractional Order (2005), Moscow: Nauka, Moscow · Zbl 1193.35245
[14] S. Kh. Gekkieva, Cand. Sci. (Phys.-Math.) Dissertation (Kab.-Balk. Sci. Center of RAS, Nal’chik, 2003).
[15] A. A. Kerefov, M. Kh. Shkhanukov-Lafishev, and R. S. Kuliev, ‘‘Boundary value problems for the loaded heat equation with non-local conditions of Steklov type,’’ in Non-Classical Equations of Mathematical Physics: Proceedings of a Seminar Dedicated to the 60th Anniversary of Professor V. N. Vragov (IM, Novosibirsk, 2005), pp. 152-159. · Zbl 1096.35056
[16] Caputo, M., Lineal model of dissipation whose Q is almost frequancy independent-II, Geophys. J. Astron. Soc., 13, 529-539 (1967)
[17] Caputo, M., Elasticita e Dissipazione (1969), Bologna: Zanichelli, Bologna
[18] A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists (Fizmatlit, Moscow, 2001; Chapman and Hall/CRC, New York, 2001). · Zbl 1027.35001
[19] Gradshteyn, I. S.; Ryzhik, I. M., Table of Integrals, Series, and Products (2007), New York: Academic, New York · Zbl 1208.65001
[20] Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I., Integrals and Series: Elementary Functions (1991), New York, London: Gordon and Breach, New York, London · Zbl 0725.44001
[21] Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I., Integrals and Series: Special Functions (1998), London: Taylor and Francis, London · Zbl 0626.00033
[22] Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I., Integrals and Series: More Special Functions (1989), New York, London: Gordon and Breach, New York, London · Zbl 0728.26001
[23] Tikhonov, A. N.; Samarskii, A. A., Equations of Mathematical Physics (2011)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.