Search

In this paper, we combine the fractional ψ−$$ \psi - $$hyperholomorphic function theory with the fractional calculus with respect to another function. As a main result, a fractional Borel–Pompeiu type formula related to a fractional ψ−$$ \psi - $$Fueter operator with respect to a vector‐valued function is proved. [ABSTRACT FROM AUTHOR]

This paper is a continuation of our work [J. O. González Cervantes and J. Bory Reyes, A quaternionic fractional Borel–Pompeiu type formula, Fractal30(1) (2022) 2250013], where we introduced a fractional operator calculus related to a fractional ψ -Fueter operator in the one-dimensional Riemann–Liouville derivative sense in each direction of the quaternionic structure, that depends on an additional vector of complex parameters with fractional real parts. This allowed us also to study a pair of lower order fractional operators and prove the associated analogues of both Stokes and Borel–Pompieu formulas for holomorphic functions in two complex variables. [ABSTRACT FROM AUTHOR]

In theoretical setting, associated with a fractional ψ -Fueter operator that depends on an additional vector of complex parameters with fractional real parts, this paper establishes a fractional analog of Borel–Pompeiu formula as a first step to develop a fractional ψ -hyperholomorphic function theory and the related operator calculus. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to combine the (q,q′)$$ \left(q,{q}^{\prime}\right) $$‐calculus in the quaternionic context, which is proposed via two kinds of (q,q′)$$ \left(q,{q}^{\prime}\right) $$‐operators, with the theory of slice regular functions. Specifically, we shall work in suitable subclasses of slice regular functions in which the (q,q′)$$ \left(q,{q}^{\prime}\right) $$‐operators can be related with the slice derivative. The paper presents some results such as an integral formula and series expansion. [ABSTRACT FROM AUTHOR]

One way to deal with physical problems on nowhere differentiable fractals is the mapping of these problems into the corresponding problems for continuum with a proper fractal metric. On this way different definitions of the fractal metric were suggested to account for the essential fractal features. In this work we develop the metric differential vector calculus in a three-dimensional continuum with a non-Euclidean metric. The metric differential forms and Laplacian are introduced, fundamental identities for metric differential operators are established and integral theorems are proved by employing the metric version of the quaternionic analysis for the Moisil–Teodoresco operator, which has been introduced and partially developed in this paper. The relations between the metric and conventional operators are revealed. It should be emphasized that the metric vector calculus developed in this work provides a comprehensive mathematical formalism for the continuum with any suitable definition of fractal metric. This offers a novel tool to study physics on fractals. [ABSTRACT FROM AUTHOR]

Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy-Riemann equations to the quaternion skew field [ABSTRACT FROM AUTHOR]

A theory of quaternion-valued functions, called hyperholomorphic, of two real variables has long been established. This theory is in the same relation to the two dimensional Helmholtz equation as the usual one-dimensional complex analysis is to the Laplace equation in R 2 . In this work we define a new Cauchy integral for domains with fractal boundary illustrating its applications and usage to study the jump and Dirichlet type boundary value problems in a fractal domain. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study a class of boundary value problems for a first order linear partial differential equation (all of whose solutions are harmonic functions), which is called the Cimmino system. With the help of the one-to-one correspondence between the theory of quaternion valued hyperholomorphic functions and that of Cimmino system’s solutions, necessary and sufficient conditions for the solvability of the non-homogeneous Cimmino system coupled by the boundary conditions are derived and its general solution is explicitly described. [Copyright &y& Elsevier]

Abstract: Let Ω be a simply connected bounded domain in with boundary an Ahlfors David regular surface Γ and f be a continuous function on Γ. Ahlfors David regular surfaces include a broad range of those from smooth to piece-wise Liapunov and Lipschitz surfaces. Using intimate relation between holomorphic function theory of two complex variables and some version of quaternionic analysis we prove that the Bochner–Martinelli integralhas continuous limit values on Γ if the truncated integrals. converge uniformly with respect to z on Γ as ϵ →0. This allows us to discuss, in the last part of the note, a formula for the square of the singular Bochner–Martinelli integral on Ahlfors David regular surfaces. Our formula is in agreement with that of obtained for the context of piece-wise Liapunov surface of integration. [Copyright &y& Elsevier]