×

zbMATH — the first resource for mathematics

About some generalizations of the properties of the Erdélyi-Kober operator and their application. (Russian. English summary) Zbl 1412.35069
Summary: In this work a composition of Erdélyi-Kober operator with differential operators of the high order is proved. Applying the proved theorems, explicit formulas of a solution of the analogue of the singular Cauchy problem for the iterated Klein-Gordon-Fock equation.

MSC:
35G10 Initial value problems for linear higher-order PDEs
26A33 Fractional derivatives and integrals
35R09 Integral partial differential equations
PDF BibTeX XML Cite
Full Text: DOI MNR
References:
[1] Samko S. G., Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo porjadka i ih prilozhenija, Nauka i tehnika, Minsk, 1987
[2] Sneddon I. N., Mixed Boundary Value Problems in Potential Theory, North-Holland Publ., Amsterdam, 1966 · Zbl 0139.28801
[3] Kiryakova V., Generalized Fractional Calculus and Applications, Long-man Sci. & Technical, N. York; J. Wiley & Sons, Harlow, 1994 · Zbl 0882.26003
[4] Erdélyi A., “On fractional integration and its application to the theory of Hankel transforms”, Quart. J. Math. Oxford ser., 11:44 (1940), 293–303
[5] Erdélyi A., Kober H., “Some remarks on Hankel transforms”, Quart. J. Math. Oxford ser., 11:43 (1940), 212–221
[6] Erdélyi A., “An application of fractional integrals”, J. Analyse Math., 14:44 (1965), 113–126 · Zbl 0135.33801
[7] Erdélyi A., “Axially symmetric potentials and fractional integration”, J. Soc. Industr. and Appl. Math., 13:1 (1965), 216–228 · Zbl 0158.12504
[8] Erdélyi A., “On the Euler-Poisson-Darboux equation”, J. Analyse Math., 23 (1970), 89–102 · Zbl 0206.39802
[9] Sneddon I. N., “The use in mathematical analysis of Erdélyi-Kober’ operators and of some of their applications”, Fractional Calculus and Its Applications, Proc., Lecture Notes in Mathematics, Springer-Verlag, New York, 1975, 37–79
[10] Lowndes J. S., “A generalization of the Erdélyi-Kober operators”, Proc. Edinb. Math. Soc., 17:2 (1970), 139–148 · Zbl 0224.44001
[11] Lowndes J. S., “An application of some fractional integrals”, Proc. Edinb. Math. Soc., 20:1 (1979), 35–41 · Zbl 0409.35003
[12] Lowndes J. S., “Cauchy problems for second order hyperbolic differential equations with constant coefficients”, Proc. Edinb. Math. Soc., 26:3 (1983), 307–311 · Zbl 0526.35047
[13] Heywood P., “Improved boundedness conditions for Lowndes’ operators”, Proc. Roy. Soc. Edinburgh, A, 73:9 (1975), 291–299 · Zbl 0336.44005
[14] Heywood P. , Rooney P. G., “On the boundedness of Lowndes’ operators”, J. London Math. Soc., 10:2 (1975), 241–248 · Zbl 0304.44002
[15] Weinstein A., “The generalized radiation problem and the Euler-Poisson-Darboux equation”, Summa Brasil Math., 3 (1955), 125–147 · Zbl 0068.07801
[16] Weinstein A., “On a singular differential operator”, Ann.mat. pura ed appl., 49:4 (1960), 359–365 · Zbl 0094.06101
[17] Poljanin A. D., Spravochnik po linejnym uravnenijam matematicheskoj fiziki, FIZMATLIT, M., 2001
[18] Prudnikov A. P., Brychkov Ju. A., Marichev O. I., Integraly i rjady. Jelementarnye funkcii, 2-e izd., isprav., v. 1, FIZMATLIT, M., 2002
[19] Il’in V. A., Poznjak Je. G, Osnovy matematicheskogo analiza, Ch.1. V 2-h ch.: Ucheb.: Dlja vuzov, 7-e izd., FIZMATLIT, Moskva, 2005
[20] Bejtmen G., Jerdeji A., Vysshie transcendentnye funkcii, v. 2, Nauka, M., 1973
[21] Bicadze A. V., Izbrannye trudy, Izdatel’stvo Uchrezhdenija RAN Kabardino-Balkarskogo nauchnogo centra RAN, Nal’chik, 2012
[22] Zhegalov V. I., “Kraevaja zadacha dlja uravnenija smeshannogo tipa vysshego porjadka”, DAN SSSR, 136:2 (1962), 274-276
[23] Smirnov M. M., Model’noe uravnenie smeshannogo tipa chetvertogo porjadka, L., 1972
[24] Meredov M. M., “O edinstvennosti reshenija kraevyh zadach dlja uravnenija smeshannogo tip chetvertogo porjadka”, Izvestija AN Turkm. SSR. Serija fiz.-tehn., him. i geol. nauk, 1967, no. 4, 11–16
[25] Zhegalov V. I., “Ob odnom napravlenii v teorii uravnenij s chastnymi proizvodnymi”, Materialy Mezhdunarodn. nauchn. konf. “Kraevye zadachi dlja differencial’nyh uravnenij i analiticheskih funkcij” (Kazan’, 29 sentjabrja - 1 oktjabrja, 2014), Izd-vo Kazansk. matem. o-va, 2014, 13–15
[26] Sabitov K. B., “O polozhitel’nosti reshenija neodnorodnogo uravnenija smeshannogo tipa vysshego porjadka”, Materialy Mezhdunarodn. nauchn. konf. “Kraevye zadachi dlja differencial’nyh uravnenij i analiticheskih funkcij” (Kazan’, 29 sentjabrja - 1 oktjabrja, 2014), 2014, 64–67
[27] Gal’perin S. A., Kondrashov V. E., “Zadacha Koshi dlja differencial’nyh operatorov, raspadajushhihsja na volnovye mnozhiteli”, Trudy Moskovskogo mat. obshhestva, 16, 1967, 109–136
[28] Aldashev S. A., “O zadache Koshi dlja operatorov raspadajushhihsja na mnozhiteli s osobennostjami”, Differencial’nye uravnenija, 17:2 (1981), 247–255 · Zbl 0455.35117
[29] Ivanov L. A., “Zadacha Koshi dlja nekotoryh operatorov s osobennostjami”, Differencial’nye uravnenija, 18:6 (1982), 1020–1028 · Zbl 0501.35009
[30] Glushak A. V., “Iterirovannye zadachi Koshi i Dirihle s operatorom Besselja v banahovom prostranstve”, Izv. vuzov. Matem, 1999, no. 8, 3–10 · Zbl 0988.34046
[31] Bogoljubov N. N., Logunov A. A., Oksak A. I., Todorov I. T., Obshhie principy kvantovoj teorii polja, Nauka, M., 1987
[32] Karimov Sh. T., “Ob odnom metode reshenija zadachi Koshi dlja odnomernogo polivolnovogo uravnenija s singuljarnym operatorom Besselja”, Izv. vuzov. Matem, 2017, no. 8, 27–41
[33] Kapilevich M. B., “Ob odnom uravnenii smeshannogo jelliptiko-giperbolicheskogo tipa”, Matematicheskij sbornik, 30(72):1 (1952), 11–38 · Zbl 0046.32103
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.