Die fraktionale Caputo-Ableitung ist in der Mathematik eine Verallgemeinerung von Ableitungen für nicht-ganzzahlige Ordnungen benannt nach Michele Caputo. 1967 definierte Caputo diese Form der fraktionalen Ableitung das erste Mal.[ 1]
Die fraktionale Caputo-Ableitung ist motiviert aus den fraktionalen Riemann–Liouville-Intergal . Sei
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{\textstyle f}
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{\displaystyle \left(0,\,\infty \right)}
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{\textstyle {^{\text{RL}}\operatorname {I} }}
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{\displaystyle {_{0}^{\text{RL}}\operatorname {I} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(-\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f\left(t\right)}{\left(x-t\right)^{1-\alpha }}}\,\operatorname {d} t}
wobei
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{\textstyle \Gamma \left(\cdot \right)}
Gammafunktion ist.
Man definiert
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{\textstyle \operatorname {D} _{x}^{\alpha }:={\frac {\operatorname {d} ^{\alpha }}{\operatorname {d} x^{\alpha }}}}
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{\textstyle \operatorname {D} _{x}^{\alpha }\operatorname {D} _{x}^{\beta }=\operatorname {D} _{x}^{\alpha +\beta }}
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{\textstyle \operatorname {D} _{x}^{\alpha }={^{\text{RL}}\operatorname {I} _{x}^{-\alpha }}}
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{\textstyle \alpha =m+z\in \mathbb {R} \wedge m\in \mathbb {N} _{0}\wedge 0<z<1}
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{\textstyle \operatorname {D} _{x}^{\alpha }=\operatorname {D} _{x}^{m+z}=\operatorname {D} _{x}^{z+m}=\operatorname {D} _{x}^{z-1+1+m}=\operatorname {D} _{x}^{z-1}\operatorname {D} _{x}^{1+m}={^{\text{RL}}\operatorname {I} }_{x}^{1-z}\operatorname {D} _{x}^{1+m}}
f
{\displaystyle f}
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{\displaystyle C^{m}\left(0,\,\infty \right)}
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{\displaystyle {\operatorname {D} _{x}^{m+z}}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(1-z\right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(1+m\right)}\left(t\right)}{\left(x-t\right)^{z}}}\,\operatorname {d} t.}
Der Zusammenhang wird als fraktionale Caputo-Ableitung bezeichnet, mit der häufig genutzten Notation
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{\textstyle {^{\text{C}}\operatorname {D} }_{x}^{\alpha }}
Die erste Definition der fraktionale Caputo-Ableitung wurde von Caputo gegeben durch:
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{\displaystyle {^{\text{C}}\operatorname {D} _{x}^{m+z}}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(1-z\right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(m+1\right)}\left(t\right)}{\left(x-t\right)^{z}}}\,\operatorname {d} t}
mit
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{\displaystyle C^{m}\left(0,\,\infty \right)}
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{\textstyle m\in \mathbb {N} _{0}\wedge 0<z<1}
[ 2]
Eine andere beliebte äquivalente Definition ist gegeben durch:
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{\displaystyle {^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(\left\lceil \alpha \right\rceil \right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t}
wobei
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{\textstyle \alpha \in \mathbb {R} _{>0}\setminus \mathbb {N} }
⌈
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{\textstyle \left\lceil \cdot \right\rceil }
Aufrundungsfunktion . Das kann aus der vorherigen Formel durch die Substitution
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{\textstyle \left\lceil \alpha \right\rceil =m+1}
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{\textstyle \left\lceil \alpha \right\rceil +z=\alpha +1}
[ 3]
Eine weitere beliebte äquivalente Definition ist gegeben durch:
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{\displaystyle {^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(n-\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(n\right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-n}}}\,\operatorname {d} t}
mit
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{\textstyle n-1<\alpha <n\in \mathbb {N} .}
Das Problem mit diesen Definitionen ist, dass sie nur Argumente in
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{\textstyle \left(0,\,\infty \right)}
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{\textstyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {f^{\left(\left\lceil \alpha \right\rceil \right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t}
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{\textstyle \left(a,\,\infty \right)}
[ 4]
Fraktionale Caputo-Ableitung einiger Funktionen
Bearbeiten
Die fraktionale Caputo-Ableitung einer Konstante
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{\displaystyle {\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[c\right]&={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {\operatorname {D} _{t}^{\left\lceil \alpha \right\rceil }\left[c\right]}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {0}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[c\right]&=0\end{aligned}}}
Die fraktionale Caputo-Ableitung einer Potenzfunktion
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[ 9]
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{\displaystyle {\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[x^{b}\right]&={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[x^{b}\right]\right]={\frac {\Gamma \left(b+1\right)}{\Gamma \left(b-\left\lceil \alpha \right\rceil +1\right)}}\cdot {_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[x^{b-\left\lceil \alpha \right\rceil }\right]\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[x^{b}\right]&={\begin{cases}{\frac {\Gamma \left(b+1\right)}{\Gamma \left(b-\alpha +1\right)}}\left(x^{b-\alpha }-a^{b-\alpha }\right),\,&{\text{for }}\left\lceil \alpha \right\rceil -1<b\wedge b\in \mathbb {R} \\0,\,&{\text{for }}\left\lceil \alpha \right\rceil -1\geq b\wedge b\in \mathbb {N} \\\end{cases}}\end{aligned}}}
Die fraktionale Caputo-Ableitung einer Exponentialfunktion
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{\displaystyle e^{a\cdot x}}
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{\displaystyle {\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[e^{b\cdot x}\right]&={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[e^{b\cdot x}\right]\right]=b^{\left\lceil \alpha \right\rceil }\cdot {_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[e^{b\cdot x}\right]\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[e^{b\cdot x}\right]&=b^{\alpha }\cdot \left(E_{x}\left(\left\lceil \alpha \right\rceil -\alpha ,\,b\right)-E_{a}\left(\left\lceil \alpha \right\rceil -\alpha ,\,b\right)\right)\\\end{aligned}}}
wobei
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{\textstyle E_{x}\left(\nu ,\,a\right):={\frac {a^{-\nu }\cdot e^{a\cdot x}\cdot \gamma \left(\nu ,\,a\cdot x\right)}{\Gamma \left(\nu \right)}}}
E
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{\textstyle \operatorname {E} _{t}}
γ
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{\textstyle \gamma \left(a,\,b\right)}
unvollständige Gammafunktion ist.[ 10]
↑ Kai Diethelm: Fractional Differential Equations . 2019, ISBN 978-3-11-057166-0 , General theory of Caputo-type fractional differential equations, S. 1–20 , doi :10.1515/9783110571660-001 kobv.de [abgerufen am 10. August 2023]). ↑ Michele Caputo: Linear Models of Dissipation whose Q is almost Frequency Independent-II . In: ResearchGate . 13. Jahrgang, Nr. 5 , 1967, S. 530 , doi :10.1111/j.1365-246X.1967.tb02303.x bibcode :1967GeoJ...13..529C (englisch, researchgate.net [abgerufen am 27. September 2024]).
↑ Mihailo Lazarević, Milan Rade Rapaić, Tomislav Šekara: Introduction to Fractional Calculus with Brief Historical Background . In: ResearchGate . 2014, S. 8 (englisch, researchgate.net [abgerufen am 27. September 2024]).
↑ Yuri Dimitrov, Slavi Georgiev, Venelin Todorov: Approximation of Caputo Fractional Derivative and Numerical Solutions of Fractional Differential Equations . In: Fractal and Fractional . 7. Jahrgang, Nr. 10 , 2023, S. 750 , doi :10.3390/fractalfract7100750
↑ Beata Sikora: Remarks on the Caputo fractional derivative . In: Matematyka I Informatyka Na Uczelniach Technicznych . Nr. 5 , 2023, S. 78–79 (englisch, polsl.pl [PDF]).
↑ Ismail Huseynov, Arzu Ahmadova, Nazim Mahmudov: Fractional Leibniz integral rules for Riemann-Liouville and Caputo fractional derivatives and their applications . In: ResearchGate . 2020, S. 1 , arxiv :2012.11360 (englisch, researchgate.net ).
↑ Eric W. Weisstein: Binomial Coefficient. mathworld.wolfram.com. 2024, abgerufen am 20. Mai 2024 (englisch). ↑ Bhausaheb Rajba Sontakke, Amjad Shaikh: Properties of Caputo Operator and Its Applications to Linear Fractional Differential Equations . In: Journal of Engineering Research and Applications . 5. Jahrgang, Nr. 5 , 2015, ISSN 2248-9622 S. 23–24 (englisch, ijera.com [PDF]).
↑ Eric W. Weisstein: Fractional Derivative. mathworld.wolfram.com. Abgerufen am 20. Mai 2024 (englisch). ↑ Eric W. Weisstein: E_t-Function. mathworld.wolfram.com. 2024, abgerufen am 20. Mai 2024 (englisch). 
![{\displaystyle {_{0}^{\text{RL}}\operatorname {I} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(-\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f\left(t\right)}{\left(x-t\right)^{1-\alpha }}}\,\operatorname {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd70814fda4cbee4288ca757850b88057e3dddc4)
![{\displaystyle {\operatorname {D} _{x}^{m+z}}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(1-z\right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(1+m\right)}\left(t\right)}{\left(x-t\right)^{z}}}\,\operatorname {d} t.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a081af9954ae43d6839b0cc4da29bb9c3d2a3f4f)
![{\displaystyle {^{\text{C}}\operatorname {D} _{x}^{m+z}}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(1-z\right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(m+1\right)}\left(t\right)}{\left(x-t\right)^{z}}}\,\operatorname {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/138f94697ffd5b346e2e43660224e77b9de1cae9)
![{\displaystyle {^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(\left\lceil \alpha \right\rceil \right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71777eade0c4722f4bacae90571d8da69bc512f7)
![{\displaystyle {^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(n-\alpha \right)}}\cdot \int \limits _{0}^{x}{\frac {f^{\left(n\right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-n}}}\,\operatorname {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1ec7714bac06791e632ef971e979fc84c02bc2f)
![{\textstyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {f^{\left(\left\lceil \alpha \right\rceil \right)}\left(t\right)}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5535c35870b70feb48a7ac931d6c4e88b7be5c6)
![{\displaystyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd8cacbd2e730ad68dbad7868e4f7e5a11f5d19c)
![{\displaystyle b\cdot {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[g\left(x\right)\right]+c\cdot {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[h\left(x\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c067167108b7a82c81284037102d8791b575b06f)
![{\displaystyle \operatorname {_{a}^{\text{C}}D} _{x}^{\alpha }\left[g\left(x\right)\cdot h\left(x\right)\right]=\sum \limits _{k=0}^{\infty }\left[{\binom {a}{k}}\cdot g^{\left(k\right)}\left(x\right)\cdot \operatorname {_{a}^{\text{RL}}D} _{x}^{\alpha -k}\left[h\left(x\right)\right]\right]-{\frac {\left(x-a\right)^{-\alpha }}{\Gamma \left(1-\alpha \right)}}\cdot g\left(a\right)\cdot h\left(a\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12bf52380a489942d650c30cd48f0fa02cb0f60d)
![{\displaystyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[f\left(x\right)\right]\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31b0212b2f9973ffc903eb596d075cd1fdb2397c)
![{\displaystyle {_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]={_{a}^{\text{RL}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]-\sum \limits _{k=0}^{\left\lceil \alpha \right\rceil }\left[{\frac {x^{k-\alpha }}{\Gamma \left(k-\alpha +1\right)}}\cdot f^{\left(k\right)}\left(0\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cac1591dd4516ee8f603666f9828d91158842ca4)
![{\displaystyle {\mathcal {L}}_{x}\left\{{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[f\left(x\right)\right]\right\}\left(s\right)=s^{\alpha }\cdot F\left(s\right)-\sum \limits _{k=0}^{\left\lceil \alpha \right\rceil }\left[s^{\alpha -k-1}\cdot f^{\left(k\right)}\left(0\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12b929ca9b9401e6152a475f87fb04a30bfec5fb)
![{\displaystyle {\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[c\right]&={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {\operatorname {D} _{t}^{\left\lceil \alpha \right\rceil }\left[c\right]}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t={\frac {1}{\Gamma \left(\left\lceil \alpha \right\rceil -\alpha \right)}}\cdot \int \limits _{a}^{x}{\frac {0}{\left(x-t\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[c\right]&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01dd0bb475bc01fa68ee24b5a24b635b2580b0a6)
![{\displaystyle {\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[x^{b}\right]&={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[x^{b}\right]\right]={\frac {\Gamma \left(b+1\right)}{\Gamma \left(b-\left\lceil \alpha \right\rceil +1\right)}}\cdot {_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[x^{b-\left\lceil \alpha \right\rceil }\right]\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[x^{b}\right]&={\begin{cases}{\frac {\Gamma \left(b+1\right)}{\Gamma \left(b-\alpha +1\right)}}\left(x^{b-\alpha }-a^{b-\alpha }\right),\,&{\text{for }}\left\lceil \alpha \right\rceil -1<b\wedge b\in \mathbb {R} \\0,\,&{\text{for }}\left\lceil \alpha \right\rceil -1\geq b\wedge b\in \mathbb {N} \\\end{cases}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f9f9d9bfc07939992110d7c4a38b85ff0f408ea)
![{\displaystyle {\begin{aligned}{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[e^{b\cdot x}\right]&={_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[\operatorname {D} _{x}^{\left\lceil \alpha \right\rceil }\left[e^{b\cdot x}\right]\right]=b^{\left\lceil \alpha \right\rceil }\cdot {_{a}^{\text{RL}}\operatorname {I} _{x}^{\left\lceil \alpha \right\rceil -\alpha }}\left[e^{b\cdot x}\right]\\{_{a}^{\text{C}}\operatorname {D} _{x}^{\alpha }}\left[e^{b\cdot x}\right]&=b^{\alpha }\cdot \left(E_{x}\left(\left\lceil \alpha \right\rceil -\alpha ,\,b\right)-E_{a}\left(\left\lceil \alpha \right\rceil -\alpha ,\,b\right)\right)\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b7d8409f8544e0b3f07907a4973db9ec168071e)
