En matemáticas , la derivada fraccionaria de Caputo , también llamada derivada fraccionaria de tipo Caputo, es una generalización de las derivadas para órdenes no enteros que recibe su nombre de Michele Caputo. Caputo definió por primera vez esta forma de derivada fraccionaria en 1967. [ 1 ]
Motivación La derivada fraccionaria de Caputo está motivada por la integral fraccionaria de Riemann-Liouville . SeaF {\textstyle f} ( 0 , ∞ ) {\displaystyle \left(0,\,\infty \right)} RL I {\textstyle {^{\text{RL}}\operatorname {I} }}
0 RL I incógnita α [ F ( incógnita ) ] = 1 Γ ( α ) ⋅ ∫ 0 incógnita F ( t ) ( incógnita − t ) 1 − α d t {\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(xt\right)^{1-\alpha }}}\,\operatorname {d} t}
dóndeΓ ( ⋅ ) {\textstyle \Gamma \left(\cdot \right)} función Gamma .
Vamos a definirD incógnita α := d α d incógnita α {\textstyle \operatorname {D} _{x}^{\alpha }:={\frac {\operatorname {d} ^{\alpha }}{\operatorname {d} x^{\alpha }}}} D incógnita α D incógnita β = D incógnita α + β {\textstyle \operatorname {D} _{x}^{\alpha }\operatorname {D} _{x}^{\beta }=\operatorname {D} _{x}^{\alpha +\beta }} D incógnita α = RL I incógnita − α {\textstyle \operatorname {D} _{x}^{\alpha }={^{\text{RL}}\operatorname {I} _{x}^{-\alpha }}} α = metro + z ∈ R ∧ metro ∈ norte 0 ∧ 0 < z < 1 {\textstyle \alpha =m+z\in \mathbb {R} \wedge m\in \mathbb {N} _{0}\wedge 0<z<1} D incógnita α = D incógnita metro + z = D incógnita z + metro = D incógnita z − 1 + 1 + metro = D incógnita z − 1 D incógnita 1 + metro = RL I incógnita 1 − z D incógnita 1 + metro {\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} do metro ( 0 , ∞ ) {\displaystyle C^{m}\left(0,\,\infty \right)}
D incógnita metro + z [ F ( incógnita ) ] = 1 Γ ( 1 − z ) ⋅ ∫ 0 incógnita F ( 1 + metro ) ( t ) ( incógnita − t ) z d t . {\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.}
Esto se conoce como la derivada fraccionaria de tipo Caputo, que a menudo se escribe comodo D incógnita α {\textstyle {^{\text{C}}\operatorname {D} }_{x}^{\alpha }}
Definición La primera definición de la derivada fraccionaria de tipo Caputo fue dada por Caputo como:
do D incógnita metro + z [ F ( incógnita ) ] = 1 Γ ( 1 − z ) ⋅ ∫ 0 incógnita F ( metro + 1 ) ( t ) ( incógnita − t ) z d t {\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}
dóndedo metro ( 0 , ∞ ) {\displaystyle C^{m}\left(0,\,\infty \right)} metro ∈ norte 0 ∧ 0 < z < 1 {\textstyle m\in \mathbb {N} _{0}\wedge 0<z<1} [ 2 ]
Una definición equivalente popular es:
do D incógnita α [ F ( incógnita ) ] = 1 Γ ( ⌈ α ⌉ − α ) ⋅ ∫ 0 incógnita F ( ⌈ α ⌉ ) ( t ) ( incógnita − t ) α + 1 − ⌈ α ⌉ d t {\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}
dóndeα ∈ R > 0 ∖ norte {\textstyle \alpha \in \mathbb {R} _{>0}\setminus \mathbb {N} } ⌈ ⋅ ⌉ {\textstyle \left\lceil \cdot \right\rceil } función techo . Esto se puede derivar sustituyendoα = metro + z {\textstyle \alpha =m+z} ⌈ α ⌉ = metro + 1 {\textstyle \left\lceil \alpha \right\rceil =m+1} ⌈ α ⌉ + z = α + 1 {\textstyle \left\lceil \alpha \right\rceil +z=\alpha +1} [ 3 ]
Otra definición equivalente popular la proporciona:
do D incógnita α [ F ( incógnita ) ] = 1 Γ ( norte − α ) ⋅ ∫ 0 incógnita F ( norte ) ( t ) ( incógnita − t ) α + 1 − norte d t {\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}
dóndenorte − 1 < α < norte ∈ norte . {\textstyle n-1<\alpha <n\in \mathbb {N} .}
El problema con estas definiciones es que solo permiten argumentos en( 0 , ∞ ) {\textstyle \left(0,\,\infty \right)} a {\textstyle a} a do D incógnita α [ F ( incógnita ) ] = 1 Γ ( ⌈ α ⌉ − α ) ⋅ ∫ a incógnita F ( ⌈ α ⌉ ) ( t ) ( incógnita − t ) α + 1 − ⌈ α ⌉ d t {\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} ( a , ∞ ) {\textstyle \left(a,\,\infty \right)} [ 4 ]
Propiedades y teoremas
Propiedades y teoremas básicos Algunas propiedades básicas son: [ 5 ]
No conmutación La ley de índices no siempre cumple la propiedad de conmutación:
a do D incógnita α a do D incógnita β = a do D incógnita α + β ≠ a do D incógnita β a do D incógnita α {\displaystyle \operatorname {_{a}^{\text{C}}D} _{x}^{\alpha }\operatorname {_{a}^{\text{C}}D} _{x}^{\beta }=\operatorname {_{a}^{\text{C}}D} _{x}^{\alpha +\beta }\neq \operatorname {_{a}^{\text{C}}D} _{x}^{\beta }\operatorname {_{a}^{\text{C}}D} _{x}^{\alpha }}
dóndeα ∈ R > 0 ∖ norte ∧ β ∈ norte {\displaystyle \alpha \in \mathbb {R} _{>0}\setminus \mathbb {N} \wedge \beta \in \mathbb {N} }
Regla fraccionaria de Leibniz La regla de Leibniz para la derivada fraccionaria de Caputo viene dada por:
a do D incógnita α [ gramo ( incógnita ) ⋅ h ( incógnita ) ] = ∑ k = 0 ∞ [ ( a k ) ⋅ gramo ( k ) ( incógnita ) ⋅ a RL D incógnita α − k [ h ( incógnita ) ] ] − ( incógnita − a ) − α Γ ( 1 − α ) ⋅ gramo ( a ) ⋅ h ( a ) {\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)}
dónde( a b ) = Γ ( a + 1 ) Γ ( b + 1 ) ⋅ Γ ( a − b + 1 ) {\textstyle {\binom {a}{b}}={\frac {\Gamma \left(a+1\right)}{\Gamma \left(b+1\right)\cdot \Gamma \left(a-b+1\right)}}} coeficiente binomial . [ 6 ] [ 7 ]
Relación con otros operadores diferenciales fraccionarios La derivada fraccionaria de tipo Caputo está estrechamente relacionada con la integral fraccionaria de Riemann-Liouville a través de su definición:
a do D incógnita α [ F ( incógnita ) ] = a RL I incógnita ⌈ α ⌉ − α [ D incógnita ⌈ α ⌉ [ F ( incógnita ) ] ] {\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]}
Además, se aplica la siguiente relación:
a do D incógnita α [ F ( incógnita ) ] = a RL D incógnita α [ F ( incógnita ) ] − ∑ k = 0 ⌊ α ⌋ [ incógnita k − α Γ ( k − α + 1 ) ⋅ F ( k ) ( 0 ) ] {\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\lfloor \alpha \right\rfloor }\left[{\frac {x^{k-\alpha }}{\Gamma \left(k-\alpha +1\right)}}\cdot f^{\left(k\right)}\left(0\right)\right]}
dóndea RL D incógnita α {\displaystyle {_{a}^{\text{RL}}\operatorname {D} _{x}^{\alpha }}}
La transformada de Laplace de la derivada fraccionaria de tipo Caputo viene dada por:
L incógnita { a do D incógnita α [ F ( incógnita ) ] } ( s ) = s α ⋅ F ( s ) − ∑ k = 0 ⌈ α ⌉ − 1 [ s α − k − 1 ⋅ F ( k ) ( 0 ) ] {\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 -1}\left[s^{\alpha -k-1}\cdot f^{\left(k\right)}\left(0\right)\right]}
dóndeL incógnita { F ( incógnita ) } ( s ) = F ( s ) {\textstyle {\mathcal {L}}_{x}\left\{f\left(x\right)\right\}\left(s\right)=F\left(s\right)} [ 8 ]
Derivada fraccionaria de Caputo de algunas funciones La derivada fraccionaria de Caputo de una constantedo {\displaystyle c}
a do D incógnita α [ do ] = 1 Γ ( ⌈ α ⌉ − α ) ⋅ ∫ a incógnita D t ⌈ α ⌉ [ do ] ( incógnita − t ) α + 1 − ⌈ α ⌉ d t = 1 Γ ( ⌈ α ⌉ − α ) ⋅ ∫ a incógnita 0 ( incógnita − t ) α + 1 − ⌈ α ⌉ d t a do D incógnita α [ do ] = 0 {\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}}}
La derivada fraccionaria de Caputo de una función potencialincógnita b {\displaystyle x^{b}} [ 9 ]
a do D incógnita α [ incógnita b ] = a RL I incógnita ⌈ α ⌉ − α [ D incógnita ⌈ α ⌉ [ incógnita b ] ] = Γ ( b + 1 ) Γ ( b − ⌈ α ⌉ + 1 ) ⋅ a RL I incógnita ⌈ α ⌉ − α [ incógnita b − ⌈ α ⌉ ] a do D incógnita α [ incógnita b ] = { Γ ( b + 1 ) Γ ( b − α + 1 ) ( incógnita b − α − a b − α ) , para ⌈ α ⌉ − 1 < b ∧ b ∈ R 0 , para ⌈ α ⌉ − 1 ≥ b ∧ b ∈ norte {\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}}}
La derivada fraccionaria de Caputo de una función exponencial mi a ⋅ incógnita {\displaystyle e^{a\cdot x}}
a do D incógnita α [ mi b ⋅ incógnita ] = a RL I incógnita ⌈ α ⌉ − α [ D incógnita ⌈ α ⌉ [ mi b ⋅ incógnita ] ] = b ⌈ α ⌉ ⋅ a RL I incógnita ⌈ α ⌉ − α [ mi b ⋅ incógnita ] a do D incógnita α [ mi b ⋅ incógnita ] = b α ⋅ ( mi incógnita ( ⌈ α ⌉ − α , b ) − mi a ( ⌈ α ⌉ − α , b ) ) {\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}}}
dóndemi incógnita ( ν , a ) = a − ν ⋅ mi a ⋅ incógnita ⋅ γ ( ν , a ⋅ incógnita ) Γ ( ν ) {\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)}}} mi t {\textstyle \operatorname {E} _{t}} γ ( a , b ) {\textstyle \gamma \left(a,\,b\right)} función gamma incompleta inferior . [ 10 ]
Referencias ↑ Diethelm, Kai (2019). "Teoría general de las ecuaciones diferenciales fraccionarias de tipo Caputo" . Ecuaciones diferenciales fraccionarias . págs. 1–20 . doi : 10.1515/9783110571660-001 . ISBN 978-3-11-057166-0 . Consultado el 10 de agosto de 2023 . ↑ Caputo, Michele (1967). "Modelos lineales de disipación cuyo Q es casi independiente de la frecuencia-II" . ResearchGate . 13 (5): 530. Bibcode : 1967GeoJ...13..529C . doi : 10.1111/j.1365-246X.1967.tb02303.x . ↑ Lazarević, Mihailo; Rapaić, Milan Rade; Šekara, Tomislav (2014). "Introducción al cálculo fraccionario con breves antecedentes históricos" . Puerta de investigación : 8. ↑ Dimitrov, Yuri; Georgiev, Slavi; Todorov, Venelin (2023). "Aproximación de la derivada fraccionaria de Caputo y soluciones numéricas de ecuaciones diferenciales fraccionarias" . Fractal y Fraccional . 7 (10): 750. doi : 10.3390/fractalfract7100750 . ↑ Sikora, Beata (2023). "Observaciones sobre la derivada fraccionaria de Caputo" (PDF) . Matematyka I Informatyka Na Uczelniach Technicznych (5): 78– 79. ↑ Huseynov, Ismail; Ahmadova, Arzu; Mahmudov, Nazim (2020). "Reglas integrales fraccionarias de Leibniz para derivadas fraccionarias de Riemann-Liouville y Caputo y sus aplicaciones" . ResearchGate : 1. arXiv : 2012.11360 . ↑ Weisstein, Eric W. (2024). "Coeficiente binomial" . mathworld.wolfram.com . Consultado el 20 de mayo de 2024 . ↑ Sontakke, Bhausaheb Rajba; Shaikh, Amjad (2015). "Propiedades del operador de Caputo y sus aplicaciones a ecuaciones diferenciales fraccionarias lineales" (PDF) . Journal of Engineering Research and Applications . 5 (5): 23– 24. ISSN 2248-9622 . ↑ Weisstein, Eric W. "Derivada fraccionaria" . mathworld.wolfram.com . Consultado el 20 de mayo de 2024 . ↑ Weisstein, Eric W. (2024). "Función E_t" . mathworld.wolfram.com . Consultado el 20 de mayo de 2024 .
Lecturas adicionales Ricardo Almeida, Una derivada fraccionaria de Caputo de una función con respecto a otra función 
Hispanopedia Wiki
![{\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(xt\right)^{1-\alpha }}}\,\operatorname {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32b2c4d2a0458a53deedd31fd6e42ddd163c8674)
![{\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(xt\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(xt\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(xt\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(xt\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(xt\right)^{\alpha +1-\left\lceil \alpha \right\rceil }}}\,\operatorname {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5535c35870b70feb48a7ac931d6c4e88b7be5c6)
![{\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(xa\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\lfloor \alpha \right\rfloor }\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/135959b3ffc6a242b4d95b8cf970f756adf0af9f)
![{\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 -1}\left[s^{\alpha -k-1}\cdot f^{\left(k\right)}\left(0\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e9932da6197a1383c7e866a9cc52cdf6116ac67)
![{\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(xt\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(xt\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{para }}\left\lceil \alpha \right\rceil -1<b\wedge b\in \mathbb {R} \\0,\,&{\text{para }}\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)
