Differential Equations Over Octonions

Lüdkovsky, S. V.

doi:10.1007/s00006-011-0286-4

Cited by 9 publications

(4 citation statements)

References 6 publications

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“…The presented above results of this paper and from works [16,[20][21][22][23][24][25][26][27][28][29][30] can be used for further developments of the operator theory over the Cayley-Dickson algebras including that of PDO.…”

Section:  mentioning

confidence: 83%

Meta-Invariant Operators over Cayley-Dickson Algebras and Spectra

Lüdkovsky¹

2013

APM

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Section:  mentioning

confidence: 83%

Meta-Invariant Operators over Cayley-Dickson Algebras and Spectra

Lüdkovsky¹

2013

APM

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“…, m} denotes the standard orthonormal base in the Euclidean space R m , where m = max(n, h); R n is embedded into A n r,C as i 0 R n . Therefore, we deduce using Formulas (28), (29), and (33) that…”

Section: Definitionmentioning

confidence: 99%

Noncommutative Integration of Generalized Diffusion PDE

Lüdkovsky

2022

Symmetry

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The article is devoted to the noncommutative integration of a diffusion partial differential equation (PDE). Its generalizations are also studied. This is motivated by the fact that many existing approaches for solutions of PDEs are based on evolutionary operators obtained as solutions of the corresponding stochastic PDEs. However, this is restricted to PDEs of an order not higher than 2 over the real or complex field. This article is aimed at extending such approaches to PDEs of an order higher than 2. For this purpose, measures and random functions having values in modules over complexified Cayley–Dickson algebras are investigated. Noncommutative integrals of hypercomplex random functions are investigated. By using them, the noncommutative integration of the generalized diffusion PDE is scrutinized. Possibilities are indicated for a subsequent solution of higher-order PDEs using their decompositions and noncommutative integration.

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“…Moreover, an application of the noncommutative integral transforms for solutions of partial differential equations is described. It can serve as an effective means (tool) to solve partial differential equations with real or complex coefficients with or without boundary conditions and their systems of different types (see also [16]). An algorithm is described which permits to write fundamental solutions and functions of Green's type.…”

Section: Introductionmentioning

confidence: 99%

Multidimensional Laplace Transforms over Quaternions, Octonions and Cayley-Dickson Algebras, Their Applications to PDE

Lüdkovsky¹

2012

APM

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Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial differential equations of higher order with real and complex coefficients and with variable coefficients with or without boundary conditions are considered.

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Differential Equations Over Octonions

Cited by 9 publications

References 6 publications

Meta-Invariant Operators over Cayley-Dickson Algebras and Spectra

Meta-Invariant Operators over Cayley-Dickson Algebras and Spectra

Noncommutative Integration of Generalized Diffusion PDE

Multidimensional Laplace Transforms over Quaternions, Octonions and Cayley-Dickson Algebras, Their Applications to PDE

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