Fractional-Diffraction-Optics Cauchy Problem: Resolvent-Function Solution of the Matrix Integral Equation
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arXiv
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| Format: | Preprint |
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2024
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| author | Mamchuev, Murat O. Chukhovskii, Felix N. |
| author_facet | Mamchuev, Murat O. Chukhovskii, Felix N. |
| contents | The fractional diffraction optics theory has been elaborated using the Green function technique. The optics-fractional equation describing the diffraction X-ray scattering by imperfect crystals has been derived as the fractional matrix integral Fredholm--Volterra equation of the second kind. In the paper, to solve the Cauchy problems, the Liouville--Neumann-type series formalism has been used to build up the matrix Resolvent-function solution. In the case when the imperfect crystal-lattice elastic displacement field is the linear function $f({\bf R}) = a x+b$, $a, b = const,$ the explicit solution of the diffraction-optics Cauchy problem has been obtained and analyzed for arbitrary fractional-order-parameter $α$, $α\in (0, 1].$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_11618 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fractional-Diffraction-Optics Cauchy Problem: Resolvent-Function Solution of the Matrix Integral Equation Mamchuev, Murat O. Chukhovskii, Felix N. General Mathematics Materials Science 35F35, 35F40, 35A08, 35C15, 35L40, 45F05, 35Q70, 35Q92 The fractional diffraction optics theory has been elaborated using the Green function technique. The optics-fractional equation describing the diffraction X-ray scattering by imperfect crystals has been derived as the fractional matrix integral Fredholm--Volterra equation of the second kind. In the paper, to solve the Cauchy problems, the Liouville--Neumann-type series formalism has been used to build up the matrix Resolvent-function solution. In the case when the imperfect crystal-lattice elastic displacement field is the linear function $f({\bf R}) = a x+b$, $a, b = const,$ the explicit solution of the diffraction-optics Cauchy problem has been obtained and analyzed for arbitrary fractional-order-parameter $α$, $α\in (0, 1].$ |
| title | Fractional-Diffraction-Optics Cauchy Problem: Resolvent-Function Solution of the Matrix Integral Equation |
| topic | General Mathematics Materials Science 35F35, 35F40, 35A08, 35C15, 35L40, 45F05, 35Q70, 35Q92 |
| url | https://arxiv.org/abs/2404.11618 |