Abstract
This paper reviews and summarizes the relevant literature on Dirichlet problems for monogenic functions on classic Clifford Algebras and the Clifford algebras depending on parameters on. Furthermore, our aim is to explore the properties when extending the problem to and, illustrating it using the concept of fibres. To do so, we explore ways in which the Dirichlet problem can be written in matrix form, using the elements of a Clifford's base. We introduce an algorithm for finding explicit expressions for monogenic functions for Dirichlet problems using matrices in Finally, we illustrate how to solve an initial value problem related to a fibre.
Highlights
1.0.1 A higher-dimensional commutative structure and its disadvantagesTo define a product of vectors in Rn+1 with n ≥ 2, we may consider ordinary polynomials in X1, ..., Xn, where Xj represents the xj -axis, j = 1, ..., n
By identifying squares Xj2 with real numbers −1, one obtains a finite-dimensional extension of Rn+1 whose elements are finite linear combinations of the 2n basis elements 1,X1, ..., X1X2, ..., X1X2, · · · Xn
We can ask ourselves which structure is more useful for factorizing the Laplace operator. This is our motivation for the introduction of Clifford algebras
Summary
We can ask ourselves which structure is more useful for factorizing the Laplace operator This is our motivation for the introduction of Clifford algebras. The usual Clifford Algebras over Rn+1 can be constructed as equivalence classes of the n free independent variables X1, ..., Xn, where one has to distinguish the two terms Xμ1 Xμ2 · · · Xμm with factors that are in different order. The dimension for the Clifford Algebra An(p | 2, αj (p), γij (p)) for n ≥ 2 is 2n It has the usual basis 1, e1, ..., en, e12, . Any Clifford-algebra-valued function u = A uAeA, which is monogenic with respect to the usual Cauchy-Riemann operator D and, the Clifford Algebra. Remark 2 Hereinafter, we will use An,2 =An(2, αj , γij ) to refer to the case αj , γij constants and A∗n,2 =An(2, αj , γij ) to refer to no constant case
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