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<p>The fractional-diffraction-optics theory has been elaborated using the Green function technique. The fractional differential equation describing the diffraction X-ray scattering by imperfect crystals has been derived as the matrix integral Fredholm-Volterra equation of the second kind. In the paper, to solve the Cauchy problem, the Liouville-Neumann-type series formalism has been used to build up the matrix resolvent-function solution. In the case when the crystal-lattice elastic displacement field is the linear function <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7605_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="127" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{f(} {\textbf {R}}\varvec{) = a x+b}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">f</mi> <mo mathvariant="bold" stretchy="false">(</mo> </mrow> <mi mathvariant="bold">R</mi> <mrow> <mo mathvariant="bold" stretchy="false">)</mo> <mo mathvariant="bold">=</mo> <mi mathvariant="bold-italic">a</mi> <mi mathvariant="bold-italic">x</mi> <mo mathvariant="bold">+</mo> <mi mathvariant="bold-italic">b</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7605_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="110" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{a, b = const,}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">a</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold-italic">b</mi> <mo mathvariant="bold">=</mo> <mi mathvariant="bold-italic">c</mi> <mi mathvariant="bold-italic">o</mi> <mi mathvariant="bold-italic">n</mi> <mi mathvariant="bold-italic">s</mi> <mi mathvariant="bold-italic">t</mi> <mo mathvariant="bold">,</mo> </mrow> </math></EquationSource> </InlineEquation> the analytical solution of the fractional-diffraction-optics Cauchy problem has been obtained and analyzed for arbitrary fractional-order-parameter <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7605_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{\alpha }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">α</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7605_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{\alpha \in (0, 1].}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">α</mi> <mo mathvariant="bold">∈</mo> <mo mathvariant="bold" stretchy="false">(</mo> <mn mathvariant="bold">0</mn> <mo mathvariant="bold">,</mo> <mn mathvariant="bold">1</mn> <mo mathvariant="bold" stretchy="false">]</mo> <mo mathvariant="bold">.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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ON THEORY OF FRACTIONAL DIFFRACTION OPTICS: THE CAUCHY PROBLEM SOLUTION

  • 期刊论文

  • 发表日期：

    2025年3月3日

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  • Murat O. Mamchuev,
  • Felix N. Chukhovskii

摘要

The fractional-diffraction-optics theory has been elaborated using the Green function technique. The fractional differential equation describing the diffraction X-ray scattering by imperfect crystals has been derived as the matrix integral Fredholm-Volterra equation of the second kind. In the paper, to solve the Cauchy problem, the Liouville-Neumann-type series formalism has been used to build up the matrix resolvent-function solution. In the case when the crystal-lattice elastic displacement field is the linear function \(\varvec{f(} {\textbf {R}}\varvec{) = a x+b}\) f ( R ) = a x + b , \(\varvec{a, b = const,}\) a , b = c o n s t , the analytical solution of the fractional-diffraction-optics Cauchy problem has been obtained and analyzed for arbitrary fractional-order-parameter \(\varvec{\alpha }\) α , \(\varvec{\alpha \in (0, 1].}\) α ∈ ( 0 , 1 ] .