Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl
Zapraszamy na https://bibliotekanauki.pl
Zawartość
Pełne teksty:
Warianty tytułu
Abstrakty
CONTENTS
Introduction........................................................................................................................................5
I. Derivatives of noninteger order.........................................................................................................6
II. Characteristic problem for the Mangeron polyvibrating equation of noninteger order....................17
1. The problem................................................................................................................................17
2. Existence of solutions..................................................................................................................18
3. Uniqueness of the solution...........................................................................................................21
4. Continuous solutions...................................................................................................................23
5. Continuous dependence of the solution on the boundary data...................................................25
III. Noncharacteristic boundary value problem...................................................................................26
1. The problem................................................................................................................................26
2. Local solutions of the problem.....................................................................................................27
3. Extension of the local solution.....................................................................................................30
IV. Some problems for ordinary differential equations........................................................................32
1. Multipoint problem.......................................................................................................................32
;1.1. The problem..........................................................................................................................32
;1.2. Solution of the problem.........................................................................................................33
 2. Polarographic equation...............................................................................................................35
;2.1. The Cauchy problem.............................................................................................................35
;2.2. Continuous dependence of the solution on the initial data....................................................39
;2.3. The multipoint problem..........................................................................................................39
V. Further applications of the derivatives of noninteger order...........................................................40
1. An application to Mikusi/nski's operator theory............................................................................40
2. Integral representation of analytic functions................................................................................42
References........................................................................................................................................45
Introduction........................................................................................................................................5
I. Derivatives of noninteger order.........................................................................................................6
II. Characteristic problem for the Mangeron polyvibrating equation of noninteger order....................17
1. The problem................................................................................................................................17
2. Existence of solutions..................................................................................................................18
3. Uniqueness of the solution...........................................................................................................21
4. Continuous solutions...................................................................................................................23
5. Continuous dependence of the solution on the boundary data...................................................25
III. Noncharacteristic boundary value problem...................................................................................26
1. The problem................................................................................................................................26
2. Local solutions of the problem.....................................................................................................27
3. Extension of the local solution.....................................................................................................30
IV. Some problems for ordinary differential equations........................................................................32
1. Multipoint problem.......................................................................................................................32
;1.1. The problem..........................................................................................................................32
;1.2. Solution of the problem.........................................................................................................33
 2. Polarographic equation...............................................................................................................35
;2.1. The Cauchy problem.............................................................................................................35
;2.2. Continuous dependence of the solution on the initial data....................................................39
;2.3. The multipoint problem..........................................................................................................39
V. Further applications of the derivatives of noninteger order...........................................................40
1. An application to Mikusi/nski's operator theory............................................................................40
2. Integral representation of analytic functions................................................................................42
References........................................................................................................................................45
Słowa kluczowe
Tematy
Kategoryzacja MSC:
- 45P05: Integral operators
- 45E10: Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
- 45B05: Fredholm integral equations
- 47Gxx: Integral, integro-differential, and pseudodifferential operators
- 35L99: None of the above, but in this section
- 26B99: None of the above, but in this section
- 26A33: Fractional derivatives and integrals
- 45Gxx: Nonlinear integral equations
- 34A99: None of the above, but in this section
- 35D99: None of the above, but in this section
- 34B99: None of the above, but in this section
- 45D05: Volterra integral equations
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne
tom/nr w serii:
328
Liczba stron
47
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCXXVIII
Daty
wydano
1993-09-01
Twórcy
autor
- Institute of Mathematics, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warszawa, Poland
Bibliografia
- [1] M. A. Al-Bassam, Some existence theorems on differential equations of generalized order, J. Reine Angew. Math. 218 (1965), 71-78.
- [2] M. A. Al-Bassam, On fractional calculus and its applications to the theory of ordinary differential equations of non-integer order, in: Lecture Notes Pure Appl. Math. 80, Dekker, 1982, 305-331.
- [3] M. A. Al-Bassam, Some applications of fractional calculus for differential equations, in: Fractional Calculus, Res. Notes Math. 138, Pitman Adv. Publ. Prog., Boston, 1985, 1-11.
- [4] J. H. Barrett, Differential equations of non-integer order, Canad. J. Math. 6 (1954), 529-541.
- [5] R. Bellman, A note on the identification of linear systems, Proc. Amer. Math. Soc. 17 (1966), 68-71.
- [6] A. Borzymowski, A non-linear Goursat problem for a high order polyvibrating equation, Proc. Roy. Soc. Edinburgh 102A (1986), 159-172.
- [7] J. Conlan, Hyperbolic differential equations of generalized order, Appl. Anal. 14 (1983), 167-177.
- [8] K. Deimling, A Carathéodory theory for systems of integral equations, Ann. Mat. Pura Appl. 86 (1970), 217-260.
- [9] K. Deimling, Das Picard-Problem für $u_xy = f(x,y,u,u_x,u_y)$ unter Carathéodory Voraussetzungen, Math. Z. 114 (1970), 303-312.
- [10] K. Deimling, Das Goursat-Problem für $u_xy = f(x,y,u)$, Aequationes Math. 6 (1971), 206-214.
- [11] V. A. Ditkin and A. P. Prudnikov, Operational Calculus, Vysshaya Shkola, Moscow, 1975 (in Russian).
- [12] J. Dugundji and A. Granas, Fixed Point Theory, Vol. 1, PWN, Warszawa, 1982.
- [13] M. M. Dzhrbashyan, Integral Transformations and Representations of Functions in a Complex Domain, Nauka, Moscow, 1966 (in Russian).
- [14] M. M. Dzhrbashyan and A. B. Nersesyan, Fractional derivatives and the Cauchy problem for differential equations of fractional order, Izv. Akad. Nauk Armyan. SSR Mat. 3 (1968), 3-29 (in Russian).
- [15] V. P. Fedosov and N. N. Yanenko, Partial differential equations of half integral order, Dokl. Akad. Nauk SSSR 276 (1984), 804-808 (in Russian).
- [16] S. Fučik and A. Kufner, Nonlinear Differential Equations, Elsevier, Amsterdam, 1980.
- [17] M. Grennes and K. B. Oldham, Semiintegral electroanalysis-theory and verification, Anal. Chem. 44 (1972), 1124-1129.
- [18] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1992.
- [19] S. Krantz, Function Theory of Several Complex Variables, Wiley, New York, 1982.
- [20] S. Łojasiewicz, An Introduction to the Theory of Real Functions, Wiley, Chichester, 1988.
- [21] D. Mangeron, Problèmes à la frontière concernant des équations polyvibrantes, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), 870-873, 976-979, 1050-1052, 1103-1106, 1121-1124.
- [22] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley, New York, 1976.
- [23] M. W. Michalski, A non-linear Goursat problem for a polyvibrating equation of Mangeron, Mem. Secţ. Ştiinţ. Acad. Repub. Soc. România Ser. IV 8 (1985), 97-112.
- [24] M. W. Michalski, On a characteristic problem for a certain differential equation of non-integer order, Appl. Anal. 28 (1988), 151-161.
- [25] M. W. Michalski, The multipoint problem for a differential equation of non-integer order, Z. Anal. Anwendungen 8 (1989), 479-483.
- [26] M. W. Michalski, On a certain differential equation of non-integer order, ibid. 10 (1990), 205-210.
- [27] M. W. Michalski, N-dimensional characteristic problem for the Mangeron equation of non-integer order, Appl. Anal. 40 (1991), 123-137.
- [28] M. W. Michalski, Some boundary value problems for partial differential equations of non-integer order, in: Constantin Carathéodory: An International Tribute, Th. M. Rassias (ed.), World Sci., 1991, 850-862.
- [29] S. M. Nikol'skiĭ, Approximation of Functions of Several Variables and Imbedding Theorems, Nauka, Moscow, 1977 (in Russian).
- [30] G. O. Okikiolu, Aspects of the Theory of Bounded Operators in L$^p$ Spaces, Academic Press, London, 1971.
- [31] K. B. Oldham, Unified treatment of electrolysis at an expanding mercury electrode, Anal. Chem. 41 (1969), 936-945.
- [32] K. B. Oldham, A new approach to the solution of electrochemical problems involving diffusion, ibid., 1904-1905.
- [33] K. B. Oldham and J. Spanier, The replacement of Fick's laws by a formulation involving semidifferentiation, J. Electroanal. Chem. 26 (1970), 331-341.
- [34] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
- [35] E. Pitcher and W. E. Sewell, Existence theorems for solutions of differential equations of nonintegral order, Bull. Amer. Math. Soc. 44 (1938), 100-107, 888.
- [36] J. Reinermann and V. Stalbohm, Eine Anwendung des Edelsteinschen Fixpunktsatzes auf Integralgleichungen vom Abel-Liouvilleschen Typ, Arch. Math. (Basel) 22 (1971), 643-647.
- [37] W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973.
- [38] S. G. Samko, A. A. Kilbas and O. I. Marichev, Integrals and Derivatives of Noninteger Order and their Applications, Nauka i Tekhnika, Minsk, 1987 (in Russian).
- [39] L. Schwartz, Analyse Mathématique, Vol. 1, Hermann, Paris, 1967.
- [40] N. P. Semenchuk, On a class of differential equations of noninteger order, Differentsial'nye Uravneniya 18 (1982), 1831-1833 (in Russian).
- [41] B. V. Shabat, Introduction to Complex Analysis, Nauka, Moscow, 1969 (in Russian).
- [42] N. V. Shragin, Conditions of measurability of superposition, Dokl. Akad. Nauk SSSR 197 (1971), 295-298 (in Russian).
- [43] V. Ya. Skorobogatko, Investigations into the Qualitative Theory of Partial Differential Equations, Naukova Dumka, Kiev, 1980 (in Russian).
- [44] Z. Szmydt, Sur un problème concernant un système d'équations différentielles hyperboliques d'ordre arbitraire à deux variables indépendantes, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 5 (1957), 577-582.
- [45] J. D. Tamarkin, On integrable solutions of Abel's integral equation, Ann. of Math. (2) 31 (1930), 219-228.
- [46] A. Z.-A. M. Tazali, Local existence theorems for ordinary differential equations of fractional order, in: Lecture Notes in Math. 964, Springer, 1982, 652-665.
- [47] W. Walter, Differential and Integral Inequalities, Springer, Berlin, 1970.
- [48] K. Wiener, Über Lösungen einer in der Theorie der Polarographie auftretenden Differentialgleichung von nichtganzzahliger Ordnung, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 32 (1983), 41-46.
- [49] K. Wiener, Lösungen einer Differentialgleichung von nichtganzzahliger Ordnung aus der Polarographie, ibid. 35 (1986), 162-167.
Języki publikacji
| EN |
Uwagi
1991 Mathematics Subject Classification: 26A33, 26B99, 34A99, 34B99, 35D99, 35L99, 45B05, 45D05, 45E10, 45Gxx, 45P05, 47Gxx.
Identyfikator YADDA
bwmeta1.element.zamlynska-a013d549-e9d1-4ed8-a35a-e8314cb93cf3
Identyfikatory
ISSN
0012-3862
Kolekcja
DML-PL
Zawartość książki
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.