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Monogenic functions with values in algebras of the second rank over the complex field and a generalized biharmonic equation with a triple characteristic

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Abstract

The statement that any two-dimensional algebra 𝔹* of the second rank with unity over the field of complex numbers contains such a basis {e1; e2} that 𝔹*-valued “analytic” functions Φ(xe1 + ye2) (x, y are real variables) satisfy such a fourth-order homogeneous partial differential equation with complex coefficients that its characteristic equation has a triple root is proved. A set of all triples (𝔹*; {e1; e2}; Φ) is described in the explicit form. A particular solution of this fourth-order partial differential equation is found by use of these “analytic” functions.

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References

  1. S. V. Gryshchuk, “Monogenic Functions with Values in Commutative Complex Algebras of the Second Rank with Unit and a Generalized Biharmonic Equation with Simple Nonzero Characteristics,” Ukr. Mat. Zh., 73(4), 474–487 (2021); transl. in: Ukr. Math. J., 73 (4), 556–571 (2021).

  2. N. E. Tovmasyan, Non-Regular Differential Equations and Calculations of Electromagnetic Fields. World Scientific Publ., Singapore, 1998.

    Book  Google Scholar 

  3. E. A. Buryachenko, “On the Dimension of the kernel of the Dirichlet problem for fourth-order equations,” Differents. uravn., 51(4), 472–480 (2015); transl. in: Differential Equations, 51(4), 477–486 (2015).

  4. S. V. Gryshchuk, “Commutative Complex Algebras of the Second Rank with Unity and Some Cases of Plane Orthotropy I,” Ukr. Mat. Zh., 70(8), 1058–1071 (2018); transl. in: Ukr. Math. J., 70(8), 1221–1236 (2019).

  5. S. V. Gryshchuk, “Commutative Complex Algebras of the Second Rank with Unity and Some Cases of Plane Orthotropy II,” Ukr. Mat. Zh., 70(10), 1382–1389 (2018); transl. in: Ukr. Math. J., 70(10), 1594–1603 (2019).

  6. S. V. Gryshchuk, B0-valued monogenic functions and their applications to the theory of anisotropic plane media. In: Analytic Methods of Analysis and Differential Equations: AMADE 2018, Cambridge Scientic Publishers Ltd, UK, 33–48 (2020).

  7. S. V. Gryshchuk, “Monogenic functions in two dimensional commutative algebras to equations of plane orthotropy”, Proceedings of the Institute of Applied Mathematics and Mechanics of NAS of Ukraine, 32, 18–29 (2018).

    Article  Google Scholar 

  8. S. V. Gryshchuk, “Monogenic functions in commutative complex algebras of the second rank and the Lamé equilibrium system for some class of plane orthotropy,” Ukrainian Math. Bull., 16(3), 345–356 (2019); transl. in: J. Math, Sci., 246(1), 30–38 (2020).

  9. S. V. Gryshchuk, “Monogenic functions with values in commutative complex algebras of the second rank with unity and generalized biharmonic equation with a double characteristic,” Ukr. Math. Jh., 74(1), 14–23 (2022).

    Google Scholar 

  10. I. P. Mel’nichencko, “Biharmonic bases in algebras of the second rank,” Ukr. Math. Jh., 38(2), 252–254 (1986); transl. in: Ukr. Math. J., 38(2), 224–226 (1986).

  11. P. W. Ketchum, “Solution of partial differential equations by means of hypervariables,” American Journal of Mathematics, 54(2), 253–264 (1932).

    Article  MathSciNet  Google Scholar 

  12. R. Z. Yeh, “Hyperholomorphic functions and higher order partial differential equations in the plane,” Pacific J. Math., 142(2), 379–399 (1990).

    Article  MathSciNet  Google Scholar 

  13. A. P. Soldatov, To elliptic theory for domains with piecewise smooth boundary in the plane. In: Partial Differential and Integral Equations (G. W. Begehr et al. (eds.)), Kluwer Academic Publishers, 177–186 (1999).

  14. V. S. Shpakivskyi, “Hypercomplex method of solving linerar patial differential equations,” Proceedings of the Institute of Applied Mathematics and Mechanics of NAS of Ukraine, 32, 147–168 (2018).

  15. V. S. Shpakivskyi, “Monogenic functions in finite-dimensional commutative associative algebras.” Zb. Pr. Inst. Mat. NAN Ukr., 12(3), 251–268 (2015).

    MathSciNet  MATH  Google Scholar 

  16. S. A. Plaksa and R. P. Pukhtaievych, “Monogenic functions in a finite-dimensional semi-simple commutative algebra.” An. St. Univ. Ovidiuhs Constanta, 22(1), 221–235 (2014).

    MathSciNet  MATH  Google Scholar 

  17. E. Study, “Über systeme complexer zahlen und ihre anwendungen in der theorie der transformationsgruppen,” Monatshefte f¨ur Mathematik, 1(1), 283–354 (1890).

  18. V. F. Kovalev and I. P. Mel’nichenko, “Biharmonic functions on the biharmonic plane,” Reports Acad. Sci. USSR, ser. A., (8), 25–27 (1981).

  19. A. Douglis, “A function-theoretic approach to elliptic systems of equations in two variables,” Communications on Pure and Applied Mathematics, 6(2), 259–289 (1953).

    Article  MathSciNet  Google Scholar 

  20. L. Sobrero, “Nuovo metodo per lo studio dei problemi di elasticità, con applicazione al problema della piastra forata,” Ricerche di Ingegneria, 13(2), 255–264 (1934).

    MathSciNet  MATH  Google Scholar 

  21. S. V. Gryshchuk and S. A. Plaksa, “On logarithmic residue of monogenic functions of biharmonic variable,” Zb. Pr. Inst. Mat. NAN Ukr., 7(2), 227–234 (2010).

    MATH  Google Scholar 

  22. W. E. Baylis, (Edt.) Clifford (Geometric) Algebras: with applications to physics, mathematics, and engineering. Birkhäuser, Boston etc., 1996.

  23. S. V. Gryshchuk and S. A. Plaksa, “Monogenic functions in a biharmonic algebra,” Ukr. Mat. Zh., 61(12), 1587–1596 (2009); transl. in: Ukr. Math. J., 61(12), 1865–1876 (2009).

  24. I. P. Mel’nichenko and S. A. Plaksa, Commutative Algebras and Space Potential Fields [in Russian]. Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, 2008.

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Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    Serhii V. Gryshchuk

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  1. Serhii V. Gryshchuk
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Correspondence to Serhii V. Gryshchuk.

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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 19, No. 1, pp. 35–48, January–March, 2022.

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Gryshchuk, S.V. Monogenic functions with values in algebras of the second rank over the complex field and a generalized biharmonic equation with a triple characteristic. J Math Sci 262, 154–164 (2022). https://doi.org/10.1007/s10958-022-05807-x

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