Applications of Transmutations of Vekua – Erd´elyi - Lowndes Type to Differential Equations
DOI:
https://doi.org/10.52575/2687-0959-2024-56-1-27-34Keywords:
Transmutations, Vekua – Erd´elyi – Lowndes Transmutations, Composition Method, Telegraph Equation, Bessel FunctionsAbstract
In the paper we study an important class of transmutations. Transmutation theory is a well-known field of differential equations, by its methods many remarkable results on differential equations were received. We consider an important class of transmutations - Vekua-Erdelyi-Lowndes operators. These transmutations, which were introduced and studied by above mentioned mathematicians, transmute differential operators of different nature with a spectral parameter to similar operators without a spectral parameter. In particular, by this method explicit connection formulas are obtained for solutions of perturbed and unperturbed differential equations. To illustrate our results some special differential equations are considered.
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Carroll RW, Showalter RE. Singular and Degenerate Cauchy problems. Academic Press, New York; 1976. 343 p.
Katrakhov VV., Sitnik SM. The Transmutation Method and Boundary-Value Problems for Singular Elliptic Equations. Contemporary Mathematics. Fundamental Directions. 2018;64(2):211–426. (in Russian)
Kravchenko VV. , Sitnik SM. (Eds.) Some recent developments in the transmutation operator approach. Springer International Publishing, Cham; 2020. 686 p.
Shishkina EL., Sitnik SM. Transmutations, singular and fractional differential equations with applications to mathematical physics. Elsevier, Amsterdam; 2020. 592 p.
Erd’elyi A. On fractional integration and its application to the Hankel transforms. Quart. J. Math. Oxford. 1940;11:293–303.
Erd’elyi A. An integral equation involving Legendre functions. SIAM Rev. 1964;12:1:15–30.
Erd’elyi A. An application of fractional integrals. J. Analyse Math. 1965;14:113–126.
Erd’elyi A. Some integral equations involving finite parts of divergent integrals. Glasgow Math. J. 1967;8:1:50–54.
Vekua IN. New Methods for Solving Elliptic Equations. North-Holland Series in Applied Mathematics & Mechanics), North–Holland Publishing Company, 1967. 358 p.
Lowndes JS. An application of some fractional integrals. Glasg. Math. J. 1979;20:1:35–41.
Lowndes JS. On some generalizations of Riemann—Liouville and Weil fractional integrals and their applications. Glasg. Math. J. 1981;22:2:73–80.
Lowndes JS. Cauchy problems for second order hyperbolic differential equations with constant coefficients. Proc. Edinb. Math. Soc. 1983;26:3:97–105.
Sitnik SM., Lyakhovetskii GV. The Vekua-Erdelyi-Lowndes transmutations. Preprint. Institute of automation and control processes of the Far East Branch of the Russian Academy of Sciences, Vladivostok, 1994. (in Russian).
Sitnik SM., Lyakhovetskii GV. Construction of Vekua-Erdelyi-Laundes transformation operators. "Differential equations, theory of functions and applications". International conference dedicated to the 100th anniversary of the birth of Academician Ilya Nestorovich Vekua. Abstracts of reports. Novosibirsk, 2007, 469–470. (in Russian).
Karimov ShT. On some generalizations of properties of the Lowndes operator and their applications to partial differential equations of high order, Filomat 2018;32:3:873–883.
Beerends RJ. A transmutation property of the generalized Abel transform associated with root system
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