We prove that any commutative and associative algebra 𝔹* of the second rank with unit over the field of complex numbers ℂ contains bases {e1, e2} for which 𝔹*-valued “analytic” functions Φ(xe1 + ye2), where x and y are real variables, satisfy a homogeneous partial differentional equation of the fourth order with complex coefficients whose characteristic equation has a single multiple root and the remaining roots are simple. We present a complete description of the set of all triples (𝔹*, {e1, e2}, Φ).
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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, No. 4, 474–487 (2021); English translation: Ukr. Math. J., 73, No. 4, 556–571 (2021).
S. G. Mikhlin, “Plane problem of the theory of elasticity,” Trud. Seism. Inst. Akad. Nauk SSSR, No. 65 (1934).
N. E. Tovmasyan, Non-Regular Differential Equations and Calculations of Electromagnetic Fields, World Scientific, Singapore (1998).
E. A. Buryachenko, “On the dimension of the kernel of Dirichlet problem for the fourth-order equations,” Differents. Uravn., 51, No. 4, 472–480 (2015).
I. P. Mel’nichencko, “Biharmonic bases in algebras of the second rank,” Ukr. Mat. Zh., 38, No. 2, 252–254 (1986); English translation: Ukr. Math. J., 38, No. 2, 224–226 (1986).
S. V. Hryshchuk, “Commutative complex algebras of the second rank with unity and some cases of plane orthotropy. I,” Ukr. Mat. Zh., 70, No. 8, 1058–1071 (2018); English translation: Ukr. Math. J., 70, No. 8, 1221–1236 (2019).
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 Scientific Publishers, Cambridge (2020), pp. 33–48.
S. V. Gryshchuk, “Monogenic functions in two-dimensional commutative algebras for the equations of plane orthotropy,” in: Proc. of the Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine [in Ukrainian], 32 (2018), pp. 18–29.
P. W. Ketchum, “Solution of partial differential equations by means of hypervariables,” Amer. J. Math., 54, No. 2, 253–264 (1932).
V. F. Kovalev and I. P. Mel’nichenko, “Biharmonic functions on the biharmonic plane,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 8, 25–27 (1981).
R. Z. Yeh, “Hyperholomorphic functions and higher order partial differential equations in the plane,” Pacif. J. Math., 142, No. 2, 379–399 (1990).
V. S. Shpakivs’kyi, “Hypercomplex method for the solution of linear partial differential equations,” in: Proc. of the Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine [in Ukrainian], 32 (2018), pp. 147–168.
V. S. Shpakivskyi, “Monogenic functions in finite-dimensional commutative associative algebras,” in: Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, 12, No. 3 (2015), pp. 251–268.
S. A. Plaksa and R. P. Pukhtaevych, “Constructive description of monogenic functions in a finite-dimensional semisimple commutative algebra,” Dop. Nats. Akad. Nauk Ukr., No. 1, 14–21 (2014).
S. A. Plaksa and R. P. Pukhtaievych, “Monogenic functions in a finite-dimensional semi-simple commutative algebra,” An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat., 22, No. 1, 221–235 (2014).
E. Study, “Über Systeme komplexer Zahlen und ihre Anwendungen in der Theorie der Transformationsgruppen,” Monatsh. Math., 1, No.1, 283–354 (1890).
N. G. Chebotarev, Introduction to the Theory of Algebras, 3rd edn. [in Russian], LKI, Moscow (2008).
W. E. Baylis (editor), Clifford (Geometric) Algebras: with Applications to Physics, Mathematics, and Engineering, Birkhäuser, Boston (1996).
S. V. Gryshchuk and S. A. Plaksa, “On the logarithmic residue of monogenic functions of biharmonic variable,” in: Proc. of the Institute of Mathematics, National Academy of Sciences of Ukraine, 7, No. 2 (2010), pp. 227–234.
S. V. Grishchuk and S. A. Plaksa, “Monogenic functions in a biharmonic algebra,” Ukr. Mat. Zh., 61, No. 12, 1587–1596 (2009); English translation: Ukr. Math. J., 61, No. 12, 1865–1876 (2009).
I. P. Mel’nichenko and S. A. Plaksa, Commutative Algebras and Space Potential Fields [in Russian], Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev (2008).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 1, pp. 14–23, January, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i1.6948.
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Gryshchuk, S.V. Monogenic Functions with Values in Commutative Algebras of the Second Rank with Unit and the Generalized Biharmonic Equation with Double Characteristic. Ukr Math J 74, 15–26 (2022). https://doi.org/10.1007/s11253-022-02045-x
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DOI: https://doi.org/10.1007/s11253-022-02045-x