Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl
Zapraszamy na https://bibliotekanauki.pl
Tytuł książki
Tytuł rozdziału
Bessel-Clifford third order differential operator and corresponding Laplace type integral transform
Autorzy
Treść / Zawartość
Pełne teksty:
Ścieżka wydawnicza (wydawca, książka, część, rozdział...)
Abstrakty
EN
Abstract: The theory of hyper-Bessel differential operators of arbitrary order m > 1 has been shown to be closely related to the Meijer's G-functions ([9], [10], [18], [20]-[25], [27]). However, most of the operational calculi, integral transforms and solutions to the Bessel type differential equations developed by different authors concern special cases mainly of order m = 2 when the role of these special functions is not evident. Here, we give an example of a third order Bessel type operator and emphasize on the use of the generalized fractional calculus and G-functions. Main attention is paid to the corresponding Laplace-Obrechkoff type integral transform with examples of its applications for solving initial value problems for Bessel-Clifford differential equations of third order.
Słowa kluczowe
Kategorie tematyczne
Twórcy
autor
- Institute of Mathematics, Bulgarian Academy of Sciences, P.O. Box 373, 1090 Sofia, Bulgaria
autor
- Universidad de Las Palmas de Gran Canaria, Dept. de Matematicas, Gran Canaria, Spain
Strony
Bibliografia
[1] J. J. Betancor, A Mikusiński calculus for the Bessel operator $B_{α₁}, α₂ = x^{1-α₁-α₂} Dx^{α₁} Dx^{α₂}$ and a variant of the Meijer transform, J. Inst. Math. & Comp. Sci. (Math. Ser.) 2, No 2 (1989), 231-242.
[2] J. J. Betancor and J. A. Barrios, Operational calculus for the generalized Bessel type operator, Simon Stevin 66 (1992), 89-98.
[3] P. Delerue, Sur le calcul symbolique à n variables et fonctions hyperbesséliennes (II), Ann. Soc. Sci. Bruxelles Sér. I 3 (1953), 229-274.
[4] I. Dimovski, Operational calculus for a class of differential operators, C. R. Acad. Bulg. Sci. 19, No 12 (1966), 1111-1114.
[5] I. Dimovski, On a Bessel-type integral transform due to N. Obrechkoff, C. R. Acad. Bulgare Sci. 27 (1974), 23-26.
[6] I. Dimovski, Foundations of operational calculi for the Bessel type differential operators, Serdica 1 (1975), 51-63.
[7] I. Dimovski, A convolutional method in operational calculus, D. Sci. Thesis, Sofia, 1976.
[8] I. Dimovski, Convolutional Calculus, Kluwer Acad. Publ., East European Ser. 43, Dordrecht, 1990.
[9] I. Dimovski and V. Kiryakova, Complex inversion formulas for the Obrechkoff transform, Pliska 4 (1981), 110-116.
[10] I. Dimovski and V. Kiryakova, Transmutations, convolutions and fractional powers of Bessel-type operators via Meijer's G-function, in: Proc. Complex Anal. and Appls., Varna 1983, Sofia, 1985, 45-66.
[11] V. A. Ditkin and A. P. Prudnikov, The operational calculus of a class of generalized Bessel operators, in: Symposium on Operational Calculus and Generalized Functions, Dubrovnik, 1971, preprint.
[12] A. Erdélyi (ed.), Higher Transcendental Functions, Vol. 1, 2, McGraw-Hill, New York, 1953; Vol. 3, 1955.
[13] J. M. Gonzalez Rodriguez, Nuevas propriedades de la funcion di-Bessel de Exton, Rev. Téc. Ing. Univ. Zulia 15 (1992), 135-141.
[14] N. Hayek, Estudio de la ecuación diferencial xy'' + (ν +1) y' + y = 0 y de sus aplicaciones, Collect. Math. 18 (1967), 57-174.
[15] N. Hayek, Funciones de Bessel-Clifford de tercer orden, in: Actas XII Jornadas Luso-Esp. de Mat., Braga, 1987, 346-351.
[16] N. Hayek and V. Hernández Suárez, Sobre las funciones de Bessel-Clifford de tercer orden, in: Actas del XII C.E.D.Y.A., Oviedo, 1991, 135-140.
[17] N. Hayek and V. Hernández Suárez, Representaciones integrales de las funciones de Bessel-Clifford de tercer orden, Rev. Acad. Cienc. Zaragoza 47 (1991), 51-60.
[18] N. Hayek and V. Hernández Suárez, On a class of functions connected with the hyper-Bessel functions, Jnanabha 23 (1992), to appear.
[19] S. L. Kalla, Operators of fractional integration, in: Proc. Conf. Analytic Functions, Kozubnik 1979, Lecture Notes in Math. 798, Springer, Berlin, 1980, 258-280.
[20] V. Kiryakova, An application of Meijer's G-function to Bessel type operators, in: Proc. Constr. Function Theory, Varna '84, Sofia, 1984, 457-462.
[21] V. Kiryakova, Generalized operators of fractional integration and differentiation and applications, Ph.D. Thesis, Sofia, 1986.
[22] V. Kiryakova, Solving hyper-Bessel differential equations by means of Meijer's G-functions, I: Two alternative approaches, Reports Strathclyde Univ., Math. Dept. No 19 (1992), 36 pp.
[23] V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Res. Notes in Math. 301, Longman, Harlow, 1994.
[24] V. Kiryakova and A. McBride, Explicit solution of the nonhomogeneous hyper-Bessel differential equation, C. R. Acad. Bulgare Sci. 46 (5) (1993), 23-26.
[25] E. Koh, A Mikusiński calculus for the Bessel operator $B_ν$, in: Lecture Notes in Math. 564, Springer, Berlin, 1976, 291-300.
[26] E. Krätzel, Eine Verallgemeinerung der Laplace- und Meijer-Transformation, Wiss. Z. Friedrich-Schiller-Univ. Math.-Naturwiss. Reihe 14 (1965), 369-381.
[27] A. McBride, Fractional powers of a class of ordinary differential operators, Proc. London Math. Soc. (III) 45 (1982), 519-546.
[28] N. A. Meller, On the operational calculus for the operator $B = t^{-α} Dt^{α+1} D$, Vychisl. Mat. 6 (1960), 161-168 (in Russian).
[29] J. Mikusiński, Operational Calculus, Vols. 1, 2, PWN-Polish Scientific Publishers and Pergamon Press, Warszawa, 1983, 1987.
[30] N. Obrechkoff, On some integral representations of real functions on the real half-line, Izv. Mat. Inst. (Sofia) 3 (1958), 3-28 (in Bulgarian).
[31] D. Przeworska-Rolewicz, Algebraic Analysis, PWN-Polish Scientific Publishers and D. Reidel, Warszawa-Dordrecht, 1988.
[32] A. Prudnikov, Yu. Brychkov and O. Marichev, Integrals and Series. Elementary Functions, Nauka, Moscow, 1981 (in Russian).
[33] A. Prudnikov, Yu. Brychkov and O. Marichev, Integrals and Series. Additional Chapters, Nauka, Moscow, 1986 (in Russian).
[34] J. Rodriguez, Operational calculus for the generalized Bessel operator, Serdica 15 (1989), 179-186.
[35] J. Rodriguez, Operational calculus through the convolutional method for the Bessel operator, Pure Appl. Math. Sci. 29 (1989), 71-81.
[36] J. Rodriguez, Associated operational calculus for a Bessel-Clifford operator, Jnanabha 20 (1990), 81-91.
[37] I. Sneddon, Mixed Boundary Value Problems in Potential Theory, North-Holland, Amsterdam, 1966.
[38] I. Sneddon, The use in mathematical analysis of Erdélyi-Kober operators and some of their applications, in: Lecture Notes in Math. 457, Springer, Berlin, 1975, 37-79.
[2] J. J. Betancor and J. A. Barrios, Operational calculus for the generalized Bessel type operator, Simon Stevin 66 (1992), 89-98.
[3] P. Delerue, Sur le calcul symbolique à n variables et fonctions hyperbesséliennes (II), Ann. Soc. Sci. Bruxelles Sér. I 3 (1953), 229-274.
[4] I. Dimovski, Operational calculus for a class of differential operators, C. R. Acad. Bulg. Sci. 19, No 12 (1966), 1111-1114.
[5] I. Dimovski, On a Bessel-type integral transform due to N. Obrechkoff, C. R. Acad. Bulgare Sci. 27 (1974), 23-26.
[6] I. Dimovski, Foundations of operational calculi for the Bessel type differential operators, Serdica 1 (1975), 51-63.
[7] I. Dimovski, A convolutional method in operational calculus, D. Sci. Thesis, Sofia, 1976.
[8] I. Dimovski, Convolutional Calculus, Kluwer Acad. Publ., East European Ser. 43, Dordrecht, 1990.
[9] I. Dimovski and V. Kiryakova, Complex inversion formulas for the Obrechkoff transform, Pliska 4 (1981), 110-116.
[10] I. Dimovski and V. Kiryakova, Transmutations, convolutions and fractional powers of Bessel-type operators via Meijer's G-function, in: Proc. Complex Anal. and Appls., Varna 1983, Sofia, 1985, 45-66.
[11] V. A. Ditkin and A. P. Prudnikov, The operational calculus of a class of generalized Bessel operators, in: Symposium on Operational Calculus and Generalized Functions, Dubrovnik, 1971, preprint.
[12] A. Erdélyi (ed.), Higher Transcendental Functions, Vol. 1, 2, McGraw-Hill, New York, 1953; Vol. 3, 1955.
[13] J. M. Gonzalez Rodriguez, Nuevas propriedades de la funcion di-Bessel de Exton, Rev. Téc. Ing. Univ. Zulia 15 (1992), 135-141.
[14] N. Hayek, Estudio de la ecuación diferencial xy'' + (ν +1) y' + y = 0 y de sus aplicaciones, Collect. Math. 18 (1967), 57-174.
[15] N. Hayek, Funciones de Bessel-Clifford de tercer orden, in: Actas XII Jornadas Luso-Esp. de Mat., Braga, 1987, 346-351.
[16] N. Hayek and V. Hernández Suárez, Sobre las funciones de Bessel-Clifford de tercer orden, in: Actas del XII C.E.D.Y.A., Oviedo, 1991, 135-140.
[17] N. Hayek and V. Hernández Suárez, Representaciones integrales de las funciones de Bessel-Clifford de tercer orden, Rev. Acad. Cienc. Zaragoza 47 (1991), 51-60.
[18] N. Hayek and V. Hernández Suárez, On a class of functions connected with the hyper-Bessel functions, Jnanabha 23 (1992), to appear.
[19] S. L. Kalla, Operators of fractional integration, in: Proc. Conf. Analytic Functions, Kozubnik 1979, Lecture Notes in Math. 798, Springer, Berlin, 1980, 258-280.
[20] V. Kiryakova, An application of Meijer's G-function to Bessel type operators, in: Proc. Constr. Function Theory, Varna '84, Sofia, 1984, 457-462.
[21] V. Kiryakova, Generalized operators of fractional integration and differentiation and applications, Ph.D. Thesis, Sofia, 1986.
[22] V. Kiryakova, Solving hyper-Bessel differential equations by means of Meijer's G-functions, I: Two alternative approaches, Reports Strathclyde Univ., Math. Dept. No 19 (1992), 36 pp.
[23] V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Res. Notes in Math. 301, Longman, Harlow, 1994.
[24] V. Kiryakova and A. McBride, Explicit solution of the nonhomogeneous hyper-Bessel differential equation, C. R. Acad. Bulgare Sci. 46 (5) (1993), 23-26.
[25] E. Koh, A Mikusiński calculus for the Bessel operator $B_ν$, in: Lecture Notes in Math. 564, Springer, Berlin, 1976, 291-300.
[26] E. Krätzel, Eine Verallgemeinerung der Laplace- und Meijer-Transformation, Wiss. Z. Friedrich-Schiller-Univ. Math.-Naturwiss. Reihe 14 (1965), 369-381.
[27] A. McBride, Fractional powers of a class of ordinary differential operators, Proc. London Math. Soc. (III) 45 (1982), 519-546.
[28] N. A. Meller, On the operational calculus for the operator $B = t^{-α} Dt^{α+1} D$, Vychisl. Mat. 6 (1960), 161-168 (in Russian).
[29] J. Mikusiński, Operational Calculus, Vols. 1, 2, PWN-Polish Scientific Publishers and Pergamon Press, Warszawa, 1983, 1987.
[30] N. Obrechkoff, On some integral representations of real functions on the real half-line, Izv. Mat. Inst. (Sofia) 3 (1958), 3-28 (in Bulgarian).
[31] D. Przeworska-Rolewicz, Algebraic Analysis, PWN-Polish Scientific Publishers and D. Reidel, Warszawa-Dordrecht, 1988.
[32] A. Prudnikov, Yu. Brychkov and O. Marichev, Integrals and Series. Elementary Functions, Nauka, Moscow, 1981 (in Russian).
[33] A. Prudnikov, Yu. Brychkov and O. Marichev, Integrals and Series. Additional Chapters, Nauka, Moscow, 1986 (in Russian).
[34] J. Rodriguez, Operational calculus for the generalized Bessel operator, Serdica 15 (1989), 179-186.
[35] J. Rodriguez, Operational calculus through the convolutional method for the Bessel operator, Pure Appl. Math. Sci. 29 (1989), 71-81.
[36] J. Rodriguez, Associated operational calculus for a Bessel-Clifford operator, Jnanabha 20 (1990), 81-91.
[37] I. Sneddon, Mixed Boundary Value Problems in Potential Theory, North-Holland, Amsterdam, 1966.
[38] I. Sneddon, The use in mathematical analysis of Erdélyi-Kober operators and some of their applications, in: Lecture Notes in Math. 457, Springer, Berlin, 1975, 37-79.
Kolekcja
DML-PL
Identyfikator YADDA
bwmeta1.element.zamlynska-1a611516-2947-4fe1-980c-26defa42ccc4
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.