Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation
Firstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">...
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MDPI AG
2023-11-01
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| Online Access: | https://www.mdpi.com/2075-1680/12/11/1046 |
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| author | Asifa Tassaddiq Rekha Srivastava Ruhaila Md Kasmani Rabab Alharbi |
| author_facet | Asifa Tassaddiq Rekha Srivastava Ruhaila Md Kasmani Rabab Alharbi |
| author_sort | Asifa Tassaddiq |
| collection | DOAJ |
| description | Firstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="sans-serif">Γ</mi><mfenced separators="|"><mrow><mi mathvariant="normal">z</mi></mrow></mfenced><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>−</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">z</mi></mrow></msup></mrow></semantics></math></inline-formula> cannot be integrated over positive real numbers. Secondly, Dirac delta function is a linear functional under which every function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">f</mi></mrow></semantics></math></inline-formula> is mapped to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. This article combines both functions to solve the problems that have remained unsolved for many years. For instance, it has been demonstrated that the power law feature is ubiquitous in theory but challenging to observe in practice. Since the fractional derivatives of the delta function are proportional to the power law, we express the gamma function as a complex series of fractional derivatives of the delta function. Therefore, a unified approach is used to obtain a large class of ordinary, fractional derivatives and integral transforms. All kinds of <i>q</i>-derivatives of these transforms are also computed. The most general form of the fractional kinetic integrodifferential equation available in the literature is solved using this particular representation. We extend the models that were valid only for a class of locally integrable functions to a class of singular (generalized) functions. Furthermore, we solve a singular fractional integral equation whose coefficients have infinite number of singularities, being the poles of gamma function. It is interesting to note that new solutions were obtained using generalized functions with complex coefficients. |
| first_indexed | 2024-03-09T17:01:52Z |
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| spelling | doaj.art-bac6b70d81be4e8baa447ff6802d2c482023-11-24T14:28:59ZengMDPI AGAxioms2075-16802023-11-011211104610.3390/axioms12111046Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral EquationAsifa Tassaddiq0Rekha Srivastava1Ruhaila Md Kasmani2Rabab Alharbi3Department of Basic Sciences and Humanities, College of Computer and Information Sciences Majmaah University, Al Majmaah 11952, Saudi ArabiaDepartment of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, CanadaInstitute of Mathematical Sciences, Universiti Malaya, Kuala Lumpur 50603, MalaysiaDepartment of Mathematics, College of Sciences and Arts, Qassim University, ArRass 51921, Saudi ArabiaFirstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="sans-serif">Γ</mi><mfenced separators="|"><mrow><mi mathvariant="normal">z</mi></mrow></mfenced><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>−</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">z</mi></mrow></msup></mrow></semantics></math></inline-formula> cannot be integrated over positive real numbers. Secondly, Dirac delta function is a linear functional under which every function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">f</mi></mrow></semantics></math></inline-formula> is mapped to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. This article combines both functions to solve the problems that have remained unsolved for many years. For instance, it has been demonstrated that the power law feature is ubiquitous in theory but challenging to observe in practice. Since the fractional derivatives of the delta function are proportional to the power law, we express the gamma function as a complex series of fractional derivatives of the delta function. Therefore, a unified approach is used to obtain a large class of ordinary, fractional derivatives and integral transforms. All kinds of <i>q</i>-derivatives of these transforms are also computed. The most general form of the fractional kinetic integrodifferential equation available in the literature is solved using this particular representation. We extend the models that were valid only for a class of locally integrable functions to a class of singular (generalized) functions. Furthermore, we solve a singular fractional integral equation whose coefficients have infinite number of singularities, being the poles of gamma function. It is interesting to note that new solutions were obtained using generalized functions with complex coefficients.https://www.mdpi.com/2075-1680/12/11/1046fractional Taylor seriesH-functionsingular integral equationq-fractional derivatives |
| spellingShingle | Asifa Tassaddiq Rekha Srivastava Ruhaila Md Kasmani Rabab Alharbi Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation Axioms fractional Taylor series H-function singular integral equation q-fractional derivatives |
| title | Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation |
| title_full | Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation |
| title_fullStr | Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation |
| title_full_unstemmed | Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation |
| title_short | Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation |
| title_sort | complex generalized representation of gamma function leading to the distributional solution of a singular fractional integral equation |
| topic | fractional Taylor series H-function singular integral equation q-fractional derivatives |
| url | https://www.mdpi.com/2075-1680/12/11/1046 |
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