Potentials for solenoidal fields using the three-dimensional $\varphi$-harmonic cyclic algebra

Homero G. Díaz-Marín; Elifalet López-González; Osvaldo Osuna

doi:10.3842/umzh.v76i11.7982

Homero G. Díaz-Marín Facultad de Ciencias Físico-Matemáticas, Universidad Michoacana, Ciudad Universitaria, Morelia, México
Elifalet López-González Extensión Multidisciplinaria Cuauhtémoc en Cuauhtémoc, Universidad Autónoma de Ciudad Juárez, Carretera Cuauhtémoc-Anáhuac, Chih, México
Osvaldo Osuna Instituto de Física y Matemáticas, Universidad Michoacana, Ciudad Universitaria, Morelia, México

DOI: https://doi.org/10.3842/umzh.v76i11.7982

Keywords: Functions of hypercomplex variables, Laplace operator, Solenoidal vector fields, Calculus over algebras, Differentiation theory.

Abstract

UDC 517.5

Given a PDE, in [E. López-González, E. A. Martínez-García, R.~Torres-Córdoba, Chaos, Solitons and Fractals, 73, Article 113757 (2023)], the authors proposed a method for the construction of solutions by considering an associative real algebra $\mathbb A$ and a suitable affine vector field $\varphi$ with respect to which the components of all functions $\mathcal L\circ\varphi$ are solutions, where $\mathcal L$ is differentiable in a sense of Lorch with respect to $\mathbb A.$ If we consider the 3D cyclic algebra and a suitable 3D affine map $\varphi,$ then we get families of solutions for the Laplace equation with three independent variables.

References

A. Alvarez-Parrilla, M. E. Frías-Armenta, E. López-González, C. Yee-Romero, On solving systems of autonomous ordinary differential equations by reduction to a variable of an algebra, Int. J. Math. and Math. Sci., 2012, Article ID 753916 (2012). DOI: https://doi.org/10.1155/2012/753916

E. Blum, Theory of analytic functions in banach algebras, Trans. Amer. Math. Soc., 78, № 2, 343–370 (1955). DOI: https://doi.org/10.1090/S0002-9947-1955-0069405-2

J. S. Cook, Introduction to $A$-calculus; Preprint arXiv: 1708.04135v1 (2017).

H. A. von Beckh-Widmanstetter, Lässt sich die Eigenshaft der analytischen Funktionen einer gemeinen komplexed Varanserlichen, Potentiale als Bestandteile zu lieferb, auf Zahlsysteme mir drei Einveiten verallgemeunern?, Monatsh. Math. und Phys., 23, 257–260 (1912) DOI: https://doi.org/10.1007/BF01707687

M. E. Frías-Armenta, E. López-González, On geodesibility of algebrizable planar vector fields, Bol. Soc. Mat. Mexicana (2017). DOI: https://doi.org/10.1007/s40590-017-0186-2

M. E. Frías-Armenta, E. López-González, On geodesibility of algebrizable three-dimensional vector fields; Preprint https://arxiv.org/abs/1912.00105.

W. Fedoroff, Sur la monogénéité, Rec. Math. (Mat. Sb.) N.S., 18(60), № 3, 353–378 (1946).

V. S. Fedorov, Basic properties of generalized monogenic functions, Izv. Vyssh. Uchebn. Zaved. Mat., № 6, 257–265 (1958).

P. W. Ketchum, Analytic functions of hypercomplex variables, Trans. Amer. Math. Soc., 30, 641–667 (1928). DOI: https://doi.org/10.1090/S0002-9947-1928-1501452-7

P. W. Ketchum, A complete solution of Laplace's equation by an infinite hypervariable, Amer. J. Math., 51, 179–188 (1929). DOI: https://doi.org/10.2307/2370704

E. López-González, E. A. Martínez-García, R. Torres-Córdoba, Pre-twisted calculus and differential equations, Chaos, Solitons and Fractals, 73, Article 113757 (2023). DOI: https://doi.org/10.1016/j.chaos.2023.113757

E. López-González, On solutions of PDEs by using algebras, Math. Methods Appl. Sci., 1–19 (2022); DOI:10.1002/mma.8073. DOI: https://doi.org/10.22541/au.163423888.80699888/v1

E. Lorch, The theory of analytic functions in normed Abelian vector rings, Trans. Amer. Math. Soc., 54, 414–425 (1943). DOI: https://doi.org/10.1090/S0002-9947-1943-0009090-0

I. P. Mel'nichenko, The representation of harmonic mappings by monogenic functions, Ukr. Math. J., 27, № 5, 499–505 (1975). DOI: https://doi.org/10.1007/BF01089142

I. P. Mel'nichenko, Algebras of functionally invariant solutions of the three-dimensional Laplace equation}. Ukr. Math. J., 55, № 9, 1551–1557 (2003). DOI: https://doi.org/10.1023/B:UKMA.0000018016.99061.d7

E. P. Miles, Three dimensional harmonic functions generated by analytic functions of a hypervariable, Amer. Math. Monthly, 61, № 10, 694–697 (1954); https://www.jstor.org/stable/2307325. DOI: https://doi.org/10.2307/2307325

A. Morev, On a generalization of the concept of monogenic functions, Mat. Sb. (N.S.), 42(84), № 2, 197–206 (1957).

S. Olariu, Complex numbers in three dimensions}; arXiv:math.CV/0008120.

R. Pierce, Associative algebras, Springer-Verlag, New York etc. (1982). DOI: https://doi.org/10.1007/978-1-4757-0163-0

S. A. Plaksa, V. S. Shpakivskyi, Monogenic functions in spaces with commutative multiplication and applications, Front. Math., Birkhäuser, Cham (2023). DOI: https://doi.org/10.1007/978-3-031-32254-9

S. A. Plaksa, Monogenic functions and harmonic vectors, Proc. Int. Geom. Center, 16, № 1, 59–76 (2023). DOI: https://doi.org/10.15673/tmgc.v16i1.2385

S. A. Plaksa, Monogenic functions in commutative algebras associated with classical equations of mathematical physics, J. Math. Sci., 242, 432–456 (2019); https://doi.org/10.1007/s10958-019-04488-3. DOI: https://doi.org/10.1007/s10958-019-04488-3

A. Pogorui, R. M. Rodriguez-Dagnino, Solutions of some partial differential equations with variable coefficients by properties of monogenic functions, J. Math. Sci., 220, № 5, 624–632 (2017). DOI: https://doi.org/10.1007/s10958-016-3205-3

A. Pogorui, R. M. Rodriguez-Dagnino, M. Shapiro, Solutions for PDEs with constant coefficients and derivability of functions ranged in commutative algebras, Math. Methods Appl. Sci., 37, № 17, 2799–2810 (2014). DOI: https://doi.org/10.1002/mma.3019

M. N. Roşculeƫ, O teorie a funcƫiilor de o variabilǎ hipercomplexǎ în spaƫiul cu trei dimensiuni, Stud. Cerc. Mat., 5, № 3-4, 361–401 (1954).

M. N. Roşculeƫ, Functii monogene pe algebre comutative, Bucuresti, Acad. Rep. Soc. Romania {(1975)}.

V. S. Shpakivskyi, Hypercomplex method for solving linear PDEs with constant coefficients, Proc. Inst. Appl. Math. Mech. NAS Ukr., 32, 147–168 (2018) (in Ukrainian). DOI: https://doi.org/10.37069/1683-4720-2018-32-16

V. S. Shpakivskyi, σ-Monogenic functions in commutative algebras, Proc. Int. Geom. Center, 16, № 1, 17–41 (2023). DOI: https://doi.org/10.15673/tmgc.v16i1.2421

R. D. Wagner, The generalized Laplace equations in a function theory for commutative algebras, Duke Math. J., 15, № 2, 455–461 (1948). DOI: https://doi.org/10.1215/S0012-7094-48-01544-0

J. Ward, A theory of analytic functions in linear associative algebras, Duke Math. J., 7, 233–248 (1940). DOI: https://doi.org/10.1215/S0012-7094-40-00714-1

J. A. Ward, From generalized Cauchy–Riemann equations to linear algebra, Proc. Amer. Math. Soc., 4, № 3, 456–461 (1953). DOI: https://doi.org/10.1090/S0002-9939-1953-0055981-6

E. T. Whittaker, G. N. Watson, A course of modern analysis, 4th ed., Cambridge Univ. Press (1927).

G. Sheffers, Verallgemeinerung der Grundlagen der gew"ohnlich komplexen Funktionen, Leipziger Berichte, 45, 838–848 (1893); 46, 120–134 (1894).