Subdiffusion Equation with Fractional Caputo Time Derivative with Respect to Another Function in Modeling Superdiffusion
Superdiffusion is usually defined as a random walk process of a molecule, in which the time evolution of the mean-squared displacement, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>σ</mi>...
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2025-01-01
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| author | Tadeusz Kosztołowicz |
| author_facet | Tadeusz Kosztołowicz |
| author_sort | Tadeusz Kosztołowicz |
| collection | DOAJ |
| description | Superdiffusion is usually defined as a random walk process of a molecule, in which the time evolution of the mean-squared displacement, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>σ</mi><mn>2</mn></msup></semantics></math></inline-formula>, of the molecule is a power function of time, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>∼</mo><msup><mi>t</mi><mrow><mn>2</mn><mo>/</mo><mi>γ</mi></mrow></msup></mrow></semantics></math></inline-formula>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>. An equation with a Riesz-type fractional derivative of the order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> with respect to a spatial variable (a fractional superdiffusion equation) is often used to describe superdiffusion. However, this equation leads to the formula <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>κ</mi><msup><mi>t</mi><mrow><mn>2</mn><mo>/</mo><mi>γ</mi></mrow></msup></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>κ</mi><mo>=</mo><mo>∞</mo></mrow></semantics></math></inline-formula>, which, in practice, makes it impossible to define the parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>. Moreover, due to the nonlocal nature of this derivative, it is generally not possible to impose boundary conditions at a thin partially permeable membrane. We show a model of superdiffusion based on an equation in which there is a fractional Caputo time derivative with respect to another function, <i>g</i>; the spatial derivative is of the second order. By choosing the function in an appropriate way, we obtain the <i>g</i>-superdiffusion equation, in which Green’s function (GF) in the long time limit approaches GF for the fractional superdiffusion equation. GF for the <i>g</i>-superdiffusion equation generates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>σ</mi><mn>2</mn></msup></semantics></math></inline-formula> with finite <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>κ</mi></semantics></math></inline-formula>. In addition, the boundary conditions at a thin membrane can be given in a similar way as for normal diffusion or subdiffusion. As an example, the filtration process generated by a partially permeable membrane in a superdiffusive medium is considered. |
| format | Article |
| id | doaj-art-2970ed1af9a443598935c4afda4421ba |
| institution | Kabale University |
| issn | 1099-4300 |
| language | English |
| publishDate | 2025-01-01 |
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| series | Entropy |
| spelling | doaj-art-2970ed1af9a443598935c4afda4421ba2025-01-24T13:31:48ZengMDPI AGEntropy1099-43002025-01-012714810.3390/e27010048Subdiffusion Equation with Fractional Caputo Time Derivative with Respect to Another Function in Modeling SuperdiffusionTadeusz Kosztołowicz0Institute of Physics, Jan Kochanowski University, Uniwersytecka 7, 25-406 Kielce, PolandSuperdiffusion is usually defined as a random walk process of a molecule, in which the time evolution of the mean-squared displacement, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>σ</mi><mn>2</mn></msup></semantics></math></inline-formula>, of the molecule is a power function of time, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>∼</mo><msup><mi>t</mi><mrow><mn>2</mn><mo>/</mo><mi>γ</mi></mrow></msup></mrow></semantics></math></inline-formula>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>. An equation with a Riesz-type fractional derivative of the order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> with respect to a spatial variable (a fractional superdiffusion equation) is often used to describe superdiffusion. However, this equation leads to the formula <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>κ</mi><msup><mi>t</mi><mrow><mn>2</mn><mo>/</mo><mi>γ</mi></mrow></msup></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>κ</mi><mo>=</mo><mo>∞</mo></mrow></semantics></math></inline-formula>, which, in practice, makes it impossible to define the parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>. Moreover, due to the nonlocal nature of this derivative, it is generally not possible to impose boundary conditions at a thin partially permeable membrane. We show a model of superdiffusion based on an equation in which there is a fractional Caputo time derivative with respect to another function, <i>g</i>; the spatial derivative is of the second order. By choosing the function in an appropriate way, we obtain the <i>g</i>-superdiffusion equation, in which Green’s function (GF) in the long time limit approaches GF for the fractional superdiffusion equation. GF for the <i>g</i>-superdiffusion equation generates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>σ</mi><mn>2</mn></msup></semantics></math></inline-formula> with finite <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>κ</mi></semantics></math></inline-formula>. In addition, the boundary conditions at a thin membrane can be given in a similar way as for normal diffusion or subdiffusion. As an example, the filtration process generated by a partially permeable membrane in a superdiffusive medium is considered.https://www.mdpi.com/1099-4300/27/1/48anomalous diffusion<i>g</i>-superdiffusion<i>g</i>-subdiffusionfractional calculusfractional Caputo derivative with respect to another function |
| spellingShingle | Tadeusz Kosztołowicz Subdiffusion Equation with Fractional Caputo Time Derivative with Respect to Another Function in Modeling Superdiffusion Entropy anomalous diffusion <i>g</i>-superdiffusion <i>g</i>-subdiffusion fractional calculus fractional Caputo derivative with respect to another function |
| title | Subdiffusion Equation with Fractional Caputo Time Derivative with Respect to Another Function in Modeling Superdiffusion |
| title_full | Subdiffusion Equation with Fractional Caputo Time Derivative with Respect to Another Function in Modeling Superdiffusion |
| title_fullStr | Subdiffusion Equation with Fractional Caputo Time Derivative with Respect to Another Function in Modeling Superdiffusion |
| title_full_unstemmed | Subdiffusion Equation with Fractional Caputo Time Derivative with Respect to Another Function in Modeling Superdiffusion |
| title_short | Subdiffusion Equation with Fractional Caputo Time Derivative with Respect to Another Function in Modeling Superdiffusion |
| title_sort | subdiffusion equation with fractional caputo time derivative with respect to another function in modeling superdiffusion |
| topic | anomalous diffusion <i>g</i>-superdiffusion <i>g</i>-subdiffusion fractional calculus fractional Caputo derivative with respect to another function |
| url | https://www.mdpi.com/1099-4300/27/1/48 |
| work_keys_str_mv | AT tadeuszkosztołowicz subdiffusionequationwithfractionalcaputotimederivativewithrespecttoanotherfunctioninmodelingsuperdiffusion |