Geodesic
Problem without initial conditions for equation with fractional derivatives and intermediate asymptotics
Čelâbinskij fiziko-matematičeskij žurnal, Tome 8 (2023) no. 1, pp. 18-28.

Voir la notice de l'article provenant de la source Math-Net.Ru

The correct solvability of problems without initial conditions for fractional-power operator sums is shown. Solutions of problems without initial conditions are interpreted by Ya.B. Zeldovich and G.I. Barenblatt as intermediate asymptotics for problems with initial conditions. These authors point out the importance of such problems in connection with the expansion of the concept of "strict determinism" in statistical physics and quantum mechanics and raise the question of studying the properties of phenomena that do not depend on details in the initial conditions that manifest themselves after sufficient time. The present paper also provides an example of intermediate asymptotics for an equation with a fractional derivative.
Keywords: intermediate asymptotics, well-posed problem, Cauchy problem, equation without initial conditions, strongly continuous semigroup, fractional power of operator. 
@article{CHFMJ_2023_8_1_a1,
     author = {V. A. Kostin and D. V. Kostin and H. Alkady},
     title = {Problem without initial conditions for equation with fractional derivatives and intermediate asymptotics},
     journal = {\v{C}el\^abinskij fiziko-matemati\v{c}eskij \v{z}urnal},
     pages = {18--28},
     publisher = {mathdoc},
     volume = {8},
     number = {1},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHFMJ_2023_8_1_a1/}
}
TY  - JOUR
AU  - V. A. Kostin
AU  - D. V. Kostin
AU  - H. Alkady
TI  - Problem without initial conditions for equation with fractional derivatives and intermediate asymptotics
JO  - Čelâbinskij fiziko-matematičeskij žurnal
PY  - 2023
SP  - 18
EP  - 28
VL  - 8
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHFMJ_2023_8_1_a1/
LA  - ru
ID  - CHFMJ_2023_8_1_a1
ER  - 
%0 Journal Article
%A V. A. Kostin
%A D. V. Kostin
%A H. Alkady
%T Problem without initial conditions for equation with fractional derivatives and intermediate asymptotics
%J Čelâbinskij fiziko-matematičeskij žurnal
%D 2023
%P 18-28
%V 8
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHFMJ_2023_8_1_a1/
%G ru
%F CHFMJ_2023_8_1_a1
V. A. Kostin; D. V. Kostin; H. Alkady. Problem without initial conditions for equation with fractional derivatives and intermediate asymptotics. Čelâbinskij fiziko-matematičeskij žurnal, Tome 8 (2023) no. 1, pp. 18-28. http://geodesic.mathdoc.fr/item/CHFMJ_2023_8_1_a1/

[1] Tikhonov A.N., Samarskii A.A., Equations of mathematical physics, Nauka, Moscow, 1966 (In Russ.)

[2] Barenblatt G.I., Zeldovich Ya.B., “Intermediate asymptotics in mathematical physics”, Russian Matematiccal Surveys, 26:2 (1971), 45–61

[3] Yosida K., Functional Analysis, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1965

[4] Krein S.G., Linear Differential Equations in Banach Space, American Mathematical Society, 1971

[5] Kostin A.V., Kostin V.A., On the theory of Stepanov's functional spaces, Publishing and Printing Center of Voronezh State University, Voronezh, 2007 (In Russ.)

[6] Zeldovich Ya.B., Sokolov D.D., “Fractals, similarity, intermediate asymptotics”, Soviet Physics Uspekhi, 28:7 (1985), 608-616

[7] Samko S.G., Kilbas A.A., Marichev O.I., “Fractional Integrals and Derivatives”, Yverdon, Gordon and Breach, 1987

[8] Fedorov V.E., Turov M.M., “The defect of a Cauchy type problem for linear equations with several Riemann — Liouville derivatives”, Siberian Mathematical Journal, 62:5 (2021), 925–942

[9] Uchaikin V.V., Methods of fractional derivatives, Artishok, Ulyanovsk, 2008 (In Russ.)

[10] Kostin V.A., Alkadi H., “On the solvability of the problem without initial conditions for a generalized equation with a fractional-power sum”, Bulletin of Voronezh State University. Series: Mathematics. Physics, 2022, no. 3, 82–90 (In Russ.)

[11] Da Prato G., Grisvard P., “Sommes d'operateurs lineaires et equations differentielles operationelles”, Journal de Mathématiques Pures et Appliquées, 54 (1975), 305–387

[12] Dwight H.B., Tables of Integrals and Other Mathematical Data, The Macmillan Company, New York, 1947