Eigenfunctions and Fundamental Solutions of the Fractional Laplace and Dirac Operators: The Riemann-Liouville Case |
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Authors: | M Ferreira N Vieira |
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Institution: | 1.School of Technology and Management,Polytechnic Institute of Leiria,Leiria,Portugal;2.Department of Mathematics, CIDMA-Center for Research and Development in Mathematics and Applications,University of Aveiro,Aveiro,Portugal |
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Abstract: | In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator \(\Delta _+^{(\alpha , \beta , \gamma )}:= D_{x_0^+}^{1+\alpha } +D_{y_0^+}^{1+\beta } +D_{z_0^+}^{1+\gamma },\) where \((\alpha , \beta , \gamma ) \in \,]0,1]^3\), and the fractional derivatives \(D_{x_0^+}^{1+\alpha }, D_{y_0^+}^{1+\beta }, D_{z_0^+}^{1+\gamma }\) are in the Riemann–Liouville sense. Applying operational techniques via two-dimensional Laplace transform we describe a complete family of eigenfunctions and fundamental solutions of the operator \(\Delta _+^{(\alpha ,\beta ,\gamma )}\) in classes of functions admitting a summable fractional derivative. Making use of the Mittag–Leffler function, a symbolic operational form of the solutions is presented. From the obtained family of fundamental solutions we deduce a family of fundamental solutions of the fractional Dirac operator, which factorizes the fractional Laplace operator. We apply also the method of separation of variables to obtain eigenfunctions and fundamental solutions. |
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