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<p>Using the Green function technique, foundations of the fractional 3D X-ray diffraction crystal-optics-theory has been elaborated. The fractional crystal-optics equations in particular derivatives that describe the X-ray diffractive scattering by distorted crystals has been derived in the form of the matrix Fredholm–Volterra integral equation of the second kind. To solve the mathematical Cauchy problems, the Liouville–Neumann-type series formalism has been used to build up the resolvent-function-solution. In the case of the distorted crystal-lattice with the elastic displacement field function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(f(\textbf{R})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi mathvariant="bold">R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> being equal to the linear function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(a x+b\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(a, b = const,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> the analytical solution of the 3D X-ray-diffraction optics Cauchy problem has been obtained and analyzed for arbitrary fractional-order-parameter <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\alpha \in (0, 1].\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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Fractional 3D Diffraction Optics: Resolvent Function Solution of Cauchy Problem

  • 期刊论文

  • 发表日期：

    2025年11月24日

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  • Murat O. Mamchuev,
  • Felix N. Chukhovskii

摘要

Using the Green function technique, foundations of the fractional 3D X-ray diffraction crystal-optics-theory has been elaborated. The fractional crystal-optics equations in particular derivatives that describe the X-ray diffractive scattering by distorted crystals has been derived in the form of the matrix Fredholm–Volterra integral equation of the second kind. To solve the mathematical Cauchy problems, the Liouville–Neumann-type series formalism has been used to build up the resolvent-function-solution. In the case of the distorted crystal-lattice with the elastic displacement field function \(f(\textbf{R})\) f ( R ) being equal to the linear function \(a x+b\) a x + b , \(a, b = const,\) a , b = c o n s t , the analytical solution of the 3D X-ray-diffraction optics Cauchy problem has been obtained and analyzed for arbitrary fractional-order-parameter \(\alpha \) α , \(\alpha \in (0, 1].\) α ∈ ( 0 , 1 ] .