On integral representation to solutions for a differential equation with singular coefficients
Vladikavkazskij matematičeskij žurnal, Tome 12 (2010) no. 4, pp. 73-78 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider integral representations to solutions for a differential equation with singular coefficients with Bessel operator. An existence of such integral representations is proved by successive approximations using transmutation operators method. Unlike known results we consider potentials with strong singularities at the origin and also slightly improve well-known method of B. M. Levitan by more accurate calculation of Green function in terms of Legendre function. It leads to more precise estimates without unnecessary constants.
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S. M. Sitnik. On integral representation to solutions for a differential equation with singular coefficients. Vladikavkazskij matematičeskij žurnal, Tome 12 (2010) no. 4, pp. 73-78. http://geodesic.mathdoc.fr/item/VMJ_2010_12_4_a10/

[1] Carroll R., Transmutation, scattering theory and special functions, North-Holland Publ. Company, Amsterdam–New York, 1982, 457 pp. | MR | Zbl

[2] Carroll R., Transmutation theory and applications, North-Holland Publ. Company, Amsterdam–New York, 1986, 351 pp. | MR

[3] Gilbert R., Begehr H., Transformations, transmutations and Kernel functions, v. 1–2, Longman, Harlow, 1992, 399 pp. | MR | Zbl

[4] Fage D. K., Nagnibida N. I., Problema ekvivalentnosti obyknovennykh differentsialnykh operatorov, Nauka, Novosibirsk, 1977, 280 pp.

[5] Marchenko V. A., Operatory Shturma–Liuvillya i ikh prilozheniya, Naukova Dumka, Kiev, 1977, 331 pp. | MR

[6] Sitnik S. M., “Operatory preobrazovaniya i ikh prilozheniya”, Issledovaniya po sovremennomu analizu i matematicheskomu modelirovaniyu, eds. Yu. F. Korobeinik, A. G. Kusraev, VNTs RAN, Vladikavkaz, 2008, 226–293

[7] Sitnik S. M., “Reshenie zadachi ob unitarnom obobschenii operatorov preobrazovaniya Sonina–Puassona”, Nauchnye vedomosti Belgorodskogo gosudarstvennogo universiteta, 18:5(76) (2010), 135–153

[8] Sitnik S. M., “Metod faktorizatsii operatorov preobrazovaniya v teorii differentsialnykh uravnenii”, Vestn. SamGU. Estestv. seriya, 67:8/1 (2008), 237–248

[9] Agranovich Z. S., Marchenko V. A., Obratnaya zadacha teorii rasseyaniya, izd. KhGU, Kharkov, 1960, 268 pp.

[10] Levitan B. M., Obratnye zadachi Shturma—Liuvillya, Nauka, M., 1984, 240 pp. | MR | Zbl

[11] Shadan K., Sabate P., Obratnye zadachi v kvantovoi teorii rasseyaniya, Mir, M., 1980, 408 pp. | MR

[12] Levin B. Ya., “Preobrazovaniya tipa Fure i Laplasa pri pomoschi reshenii differentsialnogo uravneniya vtorogo poryadka”, Dokl. AN SSSR, 106:2 (1956), 187–190 | Zbl

[13] Levitan B. M., “Razlozheniya po funktsiyam Besselya v ryady i integraly Fure”, UMN, 6:2 (1951), 102–143 | MR | Zbl

[14] Sokhin A. S., “Ob odnom klasse operatorov preobrazovaniya”, Tr. fiz.-tekh. in-ta nizkikh temperatur AN USSR, 1969, no. 1, 117–125

[15] Sokhin A. S., “Obratnye zadachi rasseyaniya dlya uravnenii s osobennostyu”, Tr. fiz.-tekh. in-ta nizkikh temperaturur AN USSR, 1971, no. 2, 182–233

[16] Sokhin A. S., “Obratnye zadachi rasseyaniya dlya uravnenii s osobennostyami spetsialnogo vida”, Teoriya funktsii, funktsionalnyi analiz i ikh prilozheniya, 1973, no. 17, 36–64

[17] Sokhin A. S., “O preobrazovanii operatorov dlya uravnenii s osobennostyu spetsialnogo vida”, Vestn. Kharkovskogo un-ta, 1974, no. 113, 36–42 | Zbl

[18] Stashevskaya V. V., “Ob obratnoi zadache spektralnogo analiza dlya differentsialnogo operatora s osobennostyu v nule”, Uch. zap. Kharkovskogo mat. ob-va, 1957, no. 5, 49–86

[19] Sitnik S. M., Operatory preobrazovaniya dlya differentsialnogo vyrazheniya Besselya, Dep. v VINITI, No 535–V87, Voronezh. un-t, Voronezh, 1987, 28 pp.

[20] Levitan B. M., “Primenenie operatorov obobschennogo sdviga k lineinym differentsialnym uravneniyam vtorogo poryadka”, Uspekhi mat. nauk, 4:1(29) (1949), 3–112 | MR | Zbl

[21] Levitan B. M., Operatory obobschennogo sdviga i nekotorye ikh primeneniya, GIFML, M., 1962, 324 pp. | MR

[22] Borovskikh A. V., “Formula rasprostranyayuschikhsya voln dlya odnomernoi neodnorodnoi sredy”, Dif. uravneniya, 38:6 (2002), 758–767 | MR | Zbl

[23] Borovskikh A. V., “Metod rasprostranyayuschikhsya voln”, Tr. seminara im. I. G. Petrovskogo, 24, 2004, 3–43 | MR

[24] Sitnik S. M., Operator preobrazovaniya i predstavlenie Iosta dlya uravneniya s singulyarnym potentsialom, preprint, IAPU DVO RAN, Vladivostok, 1993, 21 pp.

[25] Katrakhov V. V., Sitnik S. M., “Otsenki reshenii Iosta odnomernogo uravneniya Shredingera s singulyarnym potentsialom”, Dokl. AN SSSR, 340:1 (1995), 18–20 | MR | Zbl

[26] Marichev O. I., Kilbas A. A., Repin O. A., Kraevye zadachi dlya uravnenii v chastnykh proizvodnykh s razryvnymi koeffitsientami, Izd-vo SamGEU, Samara, 2008, 163 pp. | Zbl

[27] Marichev O. I., Metod vychisleniya integralov ot spetsialnykh funktsii, Nauka i tekhnika, Minsk, 1978, 312 pp. | MR | Zbl