Monogenic functions with values in сommutative сomplex algebras of the second rank with unit and generalized biharmonic equation with simple nonzero simple characteristics

Abstract

УДК 517.5, 539.3

Among all two-dimensional algebras of the second rank with a unit $e$ over the field of complex numbers $\mathbb{C},$ we found a semisimple algebra $\mathbb{B}_{0}:=\{c_1 e+c_2\omega\colon c_k\in\mathbb{C},k=1,2\},$ $\omega^2=e,$ containing bases $\{e_1,e_2\}$ such that $\mathbb{B}_{0}$-valued ``analytic'' functions $\Phi(xe_1+ye_2),$ where $x, y$ are real variables, satisfy a homogeneous partial differential equation of the fourth order that has only simple nonzero characteristics.
The set of pairs $(\{e_1,e_2\},\Phi)$ is described in an explicit form.

References

Mekhanika v SSSR za pyat`desyat let. T.3. Mekhanika deformiruemogo tverdogo tela, pod red. L. I. Sedova i dr., Nauka, Moskva (1972).

S. G. Lekhniczkij, Teoriya uprugosti anizotropnogo tela, Nauka, Moskva (1977).

D. I. Sherman, Ploskaya zadacha teorii uprugosti dlya anizotropnoj sredy`, Trudy` Sejsm. in-ta AN SSSR, № 86, 51 – 78 (1938).

Yu. A. Bogan, Regulyarny`e integral`ny`e uravneniya dlya vtoroj kraevoj zadachi v anizotropnoj dvumernoj teorii uprugosti, Izv. RAN. Mekhanika tverdogo tela, № 4, 17 – 26 (2005).

I. P. Mel`nichenko, Bigarmonicheskie bazisy` v algebrakh vtorogo ranga, Ukr. mat. zhurn., 38, № 2, 252 – 254 (1986).

S. V. Grishhuk, Komutativni kompleksni algebri drugogo rangu z odiniczeyu ta deyaki vipadki ploskoyi ortotropiyi, I, Ukr. mat. zhurn., 70, № 8, 1058 – 1071 (2018).

P. W. Ketchum, Solution of partial differential equations by means of hypervariables, Amer. J. Math., 54, № 2, 253 – 264 (1932), https://doi.org/10.2307/2370988 DOI: https://doi.org/10.2307/2370988

V. F. Kovalev, I. P. Mel`nichenko, Bigarmonicheskie funkczii na bigarmonicheskoj ploskosti, Dokl. AN USSR. Ser. A, № 8, 25 – 27 (1981).

H. H. Snyder, Elliptic systems in the plane associated with certain partial differential equations of deformable media, Contemp. Math., 11, 199 – 211 (1982). DOI: https://doi.org/10.1090/conm/011/00012

R. Z. Yeh, Hyperholomorphic functions and higher order partial differential equations in the plane, Pacif. J. Math., 142, № 2, 379 – 399 (1990) DOI: https://doi.org/10.2140/pjm.1990.142.379

A. P. Soldatov, E`llipticheskie sistemy` vy`sokogo poryadka, Differencz. uravneniya, 25, 136 – 144 (1989).

A. P. Soldatov, To elliptic theory for domains with piecewise smooth boundary in the plane, Partial Different. and Integral Equat., Kluwer Acad. Publ. (1999), p. 177 – 186, https://doi.org/10.1007/978-1-4613-3276-3_11 DOI: https://doi.org/10.1007/978-1-4613-3276-3_11

V. S. Shpakivskyi, Monogenic functions in finite-dimensional commutative associative algebras, Zb. pracz` In-tu matematiki NAN Ukrayini, 12, № 3, 251 – 268 (2015).

V. S. Shpakivs`kij, Giperkompleksnij metod rozv'yazuvannya linijnikh diferenczial`nikh rivnyan` z chastinnimi pokhidnimi, Praczi In-tu prikl. matematiki i mekhaniki NAN Ukrayini, 32, 147 – 168 (2018).

E. Study, U¨ ber Systeme komplexer Zahlen und ihre Anwendungen in der Theorie der Transformationsgruppen, Monatsh. Math., 1, № 1, 283 – 354 (1890), https://doi.org/10.1007/BF01692479 DOI: https://doi.org/10.1007/BF01692479

N. G. Chebotarev, Vvedenie v teoriyu algebr. 3-e izd., Fiziko-matematicheskoe nasledie: matematika (algebra), Izd-vo LKI, Moskva (2008).

W. E. Baylis, Clifford (geometric) algebras with applications to physics, mathematics, and engineering, Birkh¨auser, Boston etc. (1996), https://doi.org/10.1007/978-1-4612-4104-1 DOI: https://doi.org/10.1007/978-1-4612-4104-1

S. V. Grishhuk, S. A. Plaksa, Monogenny`e funkczii v bigarmonicheskoj algebre, Ukr. mat. zhurn., 61, № 12, 1587 – 1596 (2009). DOI: https://doi.org/10.1007/s11253-010-0319-5

S. V. Grishhuk, Monogenni funkcziyi u dvovimirnikh komutativnikh algebrakh dlya rivnyan` ploskoyi ortotropiyi, Praczi In-tu prikl. matematiki i mekhaniki NAN Ukrayini, 32, 18 – 29 (2018).

S. V. Grishhuk, Monogenny`e funkczii v kompleksny`kh kommutativny`kh algebrakh vtorogo ranga i sistema ravnovesiya Lyame dlya odnogo klassa ploskoj ortotropii, Ukr. mat. visn., 16, № 3, 345 – 356 (2019).

S. V. Grishhuk, Komutativni kompleksni algebri drugogo rangu z odiniczeyu ta deyaki vipadki ploskoyi ortotropiyi. II, Ukr. mat. zhurn., 70, № 10, 1382 – 1389 (2018).

S. G. Mikhlin, Ploskaya zadacha teorii uprugosti, Trudy` Sejsm. in-ta AN SSSR, № 65 (1934), 83 c.

S. A. Plaksa, R. P. Pukhtayevich, Konstruktivnij opis monogennikh funkczij v skinchennovimirnij napivprostij komutativnij algebri, Dop. NAN Ukrayini, № 1, 14 – 21 (2014).

S. A. Plaksa, R. P. Pukhtaievych, Monogenic functions in a finite-dimensional semi-simple commutative algebra>, An. ¸ Stiin¸t. Univ. “Ovidius” Constan¸ta Ser. Mat., 22, № 1, 221 – 235 (2014), https://doi.org/10.2478/auom-2014-0018 DOI: https://doi.org/10.2478/auom-2014-0018

S. G. Mikhlin, Ploskaya deformacziya v anizotropnoj srede, Trudy` Sejsm. in-ta AN SSSR, № 76, 1 – 19 (1936).

S. G. Mikhlin, N. F. Morozov, M. V. Paukshto, The integral equations of the theory of elasticity (Teubner-Texte zur Mathematik, 135), Springer, Stuttgart etc. (1995), https://doi.org/10.1007/978-3-663-11626-4 DOI: https://doi.org/10.1007/978-3-663-11626-4

Published
21.04.2021
How to Cite
Gryshchuk , S. V. “Monogenic Functions With Values in сommutative сomplex Algebras of the Second Rank With Unit and Generalized Biharmonic Equation With Simple Nonzero Simple Characteristics”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 4, Apr. 2021, pp. 474 -487, doi:10.37863/umzh.v73i4.6199.
Section
Research articles