# Questions tagged [fractional-calculus]

Questions on the differentiation/integration of functions to arbitrary order. Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation and integration operators.

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### Definition of Riemann-Liouville fractional integral

A real function $f(x),\;x>0$ is said to be in the space $C_{\mu},\;\mu\in\mathbb{R}$ if there exists a real number $p(>\mu)$ such that $f(x)=x^pf_1(x)$, where $f_1(x)\in C[0,\infty)$ and is said ...
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### Proof of laplace tansform

How to prove Laplace transform of Jumarie fractional derivative $$L\{f^{(\alpha)}(t)\} =s^\alpha F(s) -s^{\alpha-1} f(0),$$ in which $f^{(\alpha)}$ is a fractional derivative. Using the paper with ...
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### Inverse Laplace with fractional power

What is inverse laplace for $y=\{-16s^{v-1}+2s^{2v-1}/s^{2v}-4s^{v}+13\}$ where $v$ is fraction . The answer that I need to get after applying invers laplace is $y=[E(2t^v)][\cos(3t^v)-5\sin(3t^v)]$ ...
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### Inverse Laplace transform for fractional power

enter image description here My question in the picture ; because im new i found that difficult to write the question here .can any one help me please -
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### Inverse laplace with fractional order

enter image description here $$L^{-1}\left[ \frac{s^{\alpha}}{s^2\alpha+a^2}\right]\\ L^{-1}\left[ \frac{a}{s^2\alpha+a^2}\right]$$ Can any one help me with those where alpha is fraction/rational ...
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### Time-fractional Navier-Stokes Equation

In the paper Analytical solution of a time-fractional Navier–Stokes equation by Adomian decomposition method by Shaher Momani and Zaid Odibat ($2006$), the Navier-Stokes equation was written in time-...
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### Product of Mittag-Leffler functions $E_\alpha(x) \cdot E_\alpha(y)$

For the exponential function, the following relation holds: \begin{equation} e^x\cdot e^y = e^{x+y} \end{equation} The Mittag-Leffler function \begin{equation} E_\alpha(x)=\sum_{k=0}^\infty \frac{x^...
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### Add Mittag-Leffler functions with same parameters

I have two functions in the form \begin{equation} f_1(t) = a_1 + b_1 E_\nu \left( c_1 \frac{t^{\nu}}{\tau^{\nu}} \right)\\ f_2(t) = a_2 + b_2 E_\nu \left( c_2 \frac{t^{\nu}}{\tau^{\nu}} \right) \end{...
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### Theory for general fractional differintegral equations?

I am aware there exist ways to construct fractional calculus, fractional differential operators and integral operators, for example by using Cauchy integral theorem in complex analysis or by Fourier ...
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### What's Fractional Partial Differential Equation and its application.

I read about fractional partial differential equation on wiki and I now know its an expansion of the usual integral power partial differential equation. I also read that it had some applications in ...
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### On the origin of fractional Fokker Planck equation

I have recently encountered fractional Fokker Planck equation for the first time in literature to explain subdiffusive behaviour. However, I find it extremely uncomfortable since the classical ...
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### How do discrete factional order functions look like?

A fractional order transfer function looks like this $$\frac{Y}{U}=\frac{b_ns^{m\alpha}+\dots+b_1s^{\alpha}+b_0}{a_ns^{n\alpha}+\dots+a_1s^{\alpha}+a_0}$$ where $\alpha \in (0,1)$ and it is often ...
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### Compute the solution of fractional differential equations?

Recently, there is the definition of the following the generalized fractional differential operator (GFDEO): \begin{equation} {}^{C}_{0}D^{\gamma,\lambda(t)}_{t}u(t) = \frac{1}{\Gamma(1 -\gamma)}\int^{...
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