CaputoD

The Caputo fractional derivative of order $\alpha > 0$ of $f$ with respect to $x$ , at base point 0.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageCaputoD!

CaputoD(f, {x, alpha})

D^\alpha_C f(x) = \frac{1}{\Gamma(n-\alpha)}\int_0^x f^{(n)}(t)\,(x-t)^{n-\alpha-1}\,dt

,

n=\lceil\alpha\rceil

.

Typed signatures

Type	Package	Applies
`(expression, tuple<symbol, real>) -> expression`	`analytic`	always

Domain: Analytic

Details

By linearity: any constant (x-free subexpression) is 0; sums and a constant factor pass through termwise.
The power function $x^\beta$ ( $\beta$ a concrete real number; bare $x$ is $\beta=1$ ): exactly 0 when $\beta$ is a nonnegative INTEGER less than $n$ — the classical vanishing case ( $f$ 's own $n$ -th ordinary derivative is already 0 there), confirmed numerically (mpmath.quad on the definition above evaluates to 0 for these). Otherwise, when $\beta > n-1$ : $\Gamma(\beta+1)/\Gamma(\beta-\alpha+1)\cdot x^{\beta-\alpha}$ , verified against wolframscript's own CaputoD AND independently against mpmath.quad on the definition above.
Declined: a non-numeric-literal $\alpha$ or $\alpha\le0$ ; a symbolic $\beta$ ; a $\beta$ real but not covered by either case above (a genuine base-point singularity — the same numerical integral diverges there); two or more $x$ -dependent factors in one product (no product rule); and any transcendental of $x$ (Exp(ax)'s Caputo derivative is a Mittag-Leffler function, and this repo has no MittagLefflerE head, so it declines rather than inventing one for this head alone).

Examples

Basic #

$D^{0.5}_C\{x^2\} = \frac{\Gamma(3)}{\Gamma(2.5)}x^{1.5}$

In

\mathrm{CaputoD}(x^2, \bigl\lbrack x, 0.5\bigr\rbrack)

Out

1.504\,505\,556\,127\,350\,098\,532x^{1.5}

a constant's Caputo derivative is 0

In

\mathrm{CaputoD}(5, \bigl\lbrack x, 0.5\bigr\rbrack)

Out

0

$D^{1.5}_C\{x\} = 0$ — $\beta=1 < n=2$ , the classical vanishing case (Wolfram's own CaputoD leaves this symbolic rather than reducing it, though it numerically evaluates to 0 — confirmed independently against mpmath.quad)

In

\mathrm{CaputoD}(x, \bigl\lbrack x, 1.5\bigr\rbrack)

Out

0

a constant's Caputo derivative is 0, exact order $\alpha=1/2$

In

\mathrm{CaputoD}(5, \bigl\lbrack x, \frac{1}{2}\bigr\rbrack)

Out

0

$D^{1/2}_C\{x\} = \frac{2\sqrt{x}}{\sqrt{\pi}}$ — exact rational order stays exact

In

\mathrm{CaputoD}(x, \bigl\lbrack x, \frac{1}{2}\bigr\rbrack)

Out

\frac{2\sqrt{x}}{\sqrt{π}}

the exact-order analogue of integer-edge-case-vanishes — $\beta=1 < n=2$ , still 0 (Wolfram leaves it symbolic)

In

\mathrm{CaputoD}(x, \bigl\lbrack x, \frac{3}{2}\bigr\rbrack)

Out

0

$D^{1/2}_C\{x^2\} = \frac{8x^{3/2}}{3\sqrt{\pi}}$ — Wolfram's own CaputoD[x^2, {x, 1/2}], exactly

In

\mathrm{CaputoD}(x^2, \bigl\lbrack x, \frac{1}{2}\bigr\rbrack)

Out

\frac{8\sqrt{x}^{3}}{3\sqrt{π}}

$D^{3/2}_C\{x^2\} = \frac{4\sqrt{x}}{\sqrt{\pi}}$ — past the vanishing boundary ( $\beta=2 \not< n=2$ )

In

\mathrm{CaputoD}(x^2, \bigl\lbrack x, \frac{3}{2}\bigr\rbrack)

Out

\frac{4\sqrt{x}}{\sqrt{π}}

$D^{1/3}_C\{x^2\} = \frac{2x^{5/3}}{\Gamma(8/3)}$ — an order that doesn't halve the exponent

In

\mathrm{CaputoD}(x^2, \bigl\lbrack x, \frac{1}{3}\bigr\rbrack)

Out

\frac{2x^{\frac{5}{3}}}{Γ(\frac{8}{3})}

Half-derivative of $x^2$ brings in a $\sqrt\pi$ and a power $3/2$

In

\mathrm{CaputoD}(x^2, \bigl\lbrack x, \frac{1}{2}\bigr\rbrack)

Out

\frac{8\sqrt{x}^{3}}{3\sqrt{π}}

The Caputo derivative of a constant is 0

In

\mathrm{CaputoD}(c, \bigl\lbrack x, \frac{2}{3}\bigr\rbrack)

Out

0

Scope #

linearity — the constant term (1) drops out entirely

In

\mathrm{CaputoD}(x^2+3x+1, \bigl\lbrack x, 0.5\bigr\rbrack)

Out

1.504\,505\,556\,127\,350\,098\,532x^{1.5}+3.385\,137\,501\,286\,537\,721\,7x^{0.5}

linearity holds under an exact order too

In

\mathrm{CaputoD}(x^2+3x+1, \bigl\lbrack x, \frac{1}{2}\bigr\rbrack)

Out

\frac{6\sqrt{x}}{\sqrt{π}}+\frac{8\sqrt{x}^{3}}{3\sqrt{π}}

In

\mathrm{FullSimplify}(\mathrm{LaplaceTransform}(\mathrm{CaputoD}(\sin(t), \bigl\lbrack t, 3/2\bigr\rbrack), t, s))

Out

\mathrm{LaplaceTransform}(\mathrm{CaputoD}(\sin(t), \bigl\lbrack t, \frac{3}{2}\bigr\rbrack), t, s)

Applications #

Half-derivative of a cubic, with exponent $5/2$

In

\mathrm{CaputoD}(x^3, \bigl\lbrack x, \frac{1}{2}\bigr\rbrack)

Out

\frac{16\sqrt{x}^{5}}{5\sqrt{π}}

Properties #

A fractional order $4/5$ gives a gamma-function coefficient

In

\mathrm{CaputoD}(x^2, \bigl\lbrack x, \frac{4}{5}\bigr\rbrack)

Out

\frac{2x^{\frac{6}{5}}}{Γ(\frac{11}{5})}

Constants are killed for any order

In

\mathrm{CaputoD}(c, \bigl\lbrack x, \frac{1}{2}\bigr\rbrack)

Out

0

Possible issues #

A power at order $2/3$ picks up a negative exponent

In

\mathrm{CaputoD}(\sqrt[3]{x}, \bigl\lbrack x, \frac{2}{3}\bigr\rbrack)

Out

\frac{Γ(\frac{4}{3})}{Γ(\frac{2}{3})\sqrt[3]{x}}

About

Examples	18
Checked against	Wolfram Language 13 agree3 differ2 unevaluated
Crosswalk	Wolfram Language

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/caputo-derivative.ts

See also: SeriesCoefficient