Geodesic
The Cauchy problem for the degenerated partial differential equation of the high even order
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 853-862.

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In this paper we develop a method for investigating the Cauchy problem for a degenerate differential equation of high even order. Applying the generalized Erdélyi–Kober operator, the formulated problem reduces to a problem for an equation without degeneracy. Further, necessary and sufficient conditions for reducing the order of the equation are proved. Two examples demonstrate the application of the developed method.
Keywords: Fractional integrals and derivatives, generalized Erdélyi–Kober operator, Bessel operator, degenerate differential equations. 
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Sh. T. Karimov. The Cauchy problem for the degenerated partial differential equation of the high even order. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 853-862. http://geodesic.mathdoc.fr/item/SEMR_2018_15_a94/

[1] A.V. Bitsadze, Some classes of partial differential equations, Gordon and Breach Science Publishers, Amsterdam, 1988 | MR | Zbl

[2] M.M. Smirnov, Degenerating elliptic and hyperbolic equations, Izd. Nauka, Glav. Red. Fiz-Mat. Lit., M., 1966 (Russian) | MR

[3] R.W. Carroll, R. E. Showalter, Singular and Degenerate Cauchy Problems, Academic Press, New York–London, 1976 | MR

[4] A. D. Baev, “On general boundary problem in bar for degenerating high order elliptic equation”, Vestnik SamGU –- Estestvennonauchnaya seriya, 3(62) (2008), 27–39 (Russian) | MR | Zbl

[5] Y. Akdim, E. Azroul, M. Rhoudaf, “Higher order nonlinear degenerate elliptic problems with weak monotonicity”, Conference, Electron. J. Diff. Eqns., 14, 2006, 53–71 | MR | Zbl

[6] B. Andreianov, M. Bendahmane, K. H. Karlsen, “Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations”, J. Hyperbolic Differ. Equ., 7:1 (2010), 1–67 | DOI | MR | Zbl

[7] Jin Liang, Ti-Jun Xiao, “Higher-Order Degenerate Cauchy Problems in Locally Convex Spaces”, Mathematical and Computer Modelling, 41:6–7 (2005), 837–847 | MR | Zbl

[8] R.E. Showalter, “Nonlinear degenerate evolution equations in mixed formulation”, SIAM Journal on Mathematical Analysis, 42:5 (2010), 2114–2131 | DOI | MR | Zbl

[9] C.O. Alves, V. Rǎdulescu, “Editorial. Degenerate and Singular Differential Operators with Applications to Boundary Value Problems”, Boundary Value Problems, 2010, Hindawi Publishing Corporation, 2010, 521070, 2 pp. | DOI | MR

[10] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon Breach Sci. Publ., Yverdon, 1993 | MR | Zbl

[11] J. S. Lowndes, “A generalisation of the Erdélyi–Kober operators”, Proc. Edinburgh Math. Soc. Ser. (2), 17 (1970/1971), 139–148 | DOI | MR | Zbl

[12] V. Kiryakova, Generalized Fractional Calculus and Applications, Longman Sci. Technical, Harlow; J. Wiley Sons, N. York, 1994 | MR | Zbl

[13] A. Erdélyi, et al. (eds.), Higher Transcendental Functions, v. 1, McGraw-Hill, New York, 1953

[14] Sh.T. Karimov, “New properties of the generalized Erdélyi–Kober operator and their applications”, Dokl. Academy of Sciences of Uzbekistan, 5 (2014), 11–13 (Russian)

[15] Sh. T. Karimov, “About some generalizations of the properties of the Erdélyi–Kober operator and their application”, Vestnik KRAUNC. Fiz.-mat. nauki, 2(18) (2017), 20–40 (Russian) | DOI | MR

[16] R.W. Carroll, Transmutation and Operator Differential Equations, North-Holland Mathematics Studies, 67, Amsterdam–New York, 1979 | MR | Zbl

[17] V. Kiryakova, “Transmutation method for solving hyper-Bessel differential equations based on the Poisson-Dimovski transformation”, Fract. Calc. Appl. Anal., 11:3 (2008), 299–-316 | MR | Zbl

[18] S. M. Sitnik, Transmutations and applications: a survey, 2010/12/16, 141 pp., arXiv: 1012.3741

[19] A.K. Urinov, S.T. Karimov, “Solution of the Cauchy Problem for Generalized Euler-Poisson-Darboux Equation by the Method of Fractional Integrals”, Progress in Partial Differential Equations, Springer Proceedings in Mathematics Statistics, 44, eds. Reissig M., Ruzhansky M., Springer, Heidelberg, 2013, 321–337 | DOI | MR | Zbl

[20] Sh.T. Karimov, “Multidimensional generalized Erdélyi-Kober operator and its application to solving Cauchy problems for differential equations with singular coefficients”, Fract. Calc. Appl. Anal., 18:4 (2015), 845–861 | DOI | MR | Zbl

[21] A.D. Polianin, Handbook of linear partial differential equations for engineers and scientists, Chapman Hall/CRC Press, 2002 | MR