×

Explicit solutions of fractional integral and differential equations involving Erdélyi-Kober operators. (English) Zbl 0942.45001

Summary: By means of fractional calculus techniques we find explicit solutions to Volterra integral equations of second kind and fractional differential equations, involving Erdélyi-Kober fractional integrals or derivatives. We use the transmutation method to reduce the solutions of these equations to known solutions of simpler (Riemann-Liouville) equations of the same type. Some examples are given.

MSC:

45D05 Volterra integral equations
26A33 Fractional derivatives and integrals
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hille, E.; Tamarkin, J., On the theory of linear integral equations, Ann. Math., 31, 479-528 (1930) · JFM 56.0337.01
[2] Ross, B.; Sachdeva, B. K., The solution of certain integral equation by means of operators of arbitrary order, Amer. Math. Monthly, 97, 6, 498-502 (1990) · Zbl 0723.45002
[3] Gorenflo, R.; Luchko, Yu., An operational method for solving generalized Abel integral equations of second kind, ((1995), Freie University: Freie University Berlin), 14, Fachber. Math. und Inf., Ser. A- Math.
[4] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives (Theory and Applications) (1993), Gordon and Breach: Gordon and Breach Switzerland · Zbl 0818.26003
[5] Al-Saqabi, B. N., Solution of a class of differential equations by means of Riemann-Liouville operator, J. Fractional Calculus, 8, 95-102 (1995) · Zbl 0838.34077
[6] Al-Saqabi, B.; Tuan, V. K., Solution of a fractional differintegral equation, Integral Transforms Special Functions, 4, 1, 1-6 (1996)
[7] Gorenflo, R.; Vessella, R. S., Abel Integral Equations (1991), Springer: Springer Berlin · Zbl 0717.45002
[8] Kiryakova, V., Generalized Fractional Calculus and Applications (1994), Longman, Ser. Pitman Res. Notes in Math., no. 301, Harlow · Zbl 0882.26003
[9] Luchko, Y.; Srivastava, H. M., The exact solution of certain differential equations of fractional order by using operational calculus, Comput. Math. Appl., 29, 8, 73-85 (1995) · Zbl 0824.44011
[10] Mainardi, F.; Tomirotti, M., On a special function arising in the time fractional diffusion wave equation, (Rusey, P.; Dimovski, I.; Kiryakova, V., Transform Methods Special Functions’94 (1995), SCTP: SCTP Singapore), 171-183 · Zbl 0921.33010
[11] Podlubny, I., Fractional-order systems and fractional-order controllers, (UEF-03-94 (1994), Slovak Akad. Sci.-Inst. Exper. Phys), 18, Preprint
[12] Srivastava, H. M.; Buschman, R. G., Theory and Applications of Convolution Integral Equations (1992), Kluwer Academic Publishers, Series Mathematics and Its Applications, no. 79, Dordrecht · Zbl 0755.45002
[13] Dzrbashjan, M. M., Harmonic Analysis and Boundary Value Problems in the Complex Plain (1993), Birkhauser, Series Operator Theory: Advances and Applications, no. 65, Basel
[14] (Erdélyi, A.; etal., Higher Transcendental Functions (1953), McGraw-Hill: McGraw-Hill New York) · Zbl 0051.30303
[15] Sneddon, I. N., The use in Mathematical analysis of Erdélyi-Kober operators and some of their applications, (Ross, B., Fractional Calculus and Applications, L.N.M.,no. 457 (1975), Springer: Springer New York), 37-79
[16] Hearsh, R., The method of transmutations, (Lecture Notes in Math.,no. 446 (1975), Springer: Springer New York), 264-282
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.