What is time fractional derivative?

Time fractional diffusion equations (TFDE) arise by replacing the standard time partial derivative in the diffusion equation with a time fractional partial derivative, attempting to generalize the classical Fick (or Fourier) law to describe phenomena with long memory where the rate of diffusion might be inconsistent …

What does fractional derivative represent?

fractional derivatives are non-local so the 1/2 derivative can not have a local meaning like tangent or curvature but would have to take into account the properties of the curve over a large extent (boundary conditions). Fractional differential equations can explain time delay, fractal properties etc…

What is fractal fractional derivative?

Fractal-fractional derivative is a new class of fractional derivative with power Law kernel which has many applications in real world problems. This operator is used for the first time in such kind of fluid flow.

What is Caputo derivative?

The Caputo derivative is the most appropriate fractional operator to be used in modeling real world problem. From: Fractional Operators with Constant and Variable Order with Application to Geo-Hydrology, 2018.

What is fractional order differential equation?

A fractional order differential equation (FODE) is a generalized form of an integer order differential equation. The FODE is useful in many areas, e.g., for the depiction of a physical model of various phenomena in pure and applied science (see [1–4] and the references therein).

What is fractional equation?

Definition of fractional equation : an equation containing the unknown in the denominator of one or more terms (as a/x + b/(x + 1) = c)

What is Caputo Fabrizio fractional derivative?

The Caputo–Fabrizio fractional-order derivative (CF) is defined as follows [43] (2) 0 C F D t α u t = 1 1 − α ∫ 0 t u ′ τ exp − α t − τ 1 − α d τ , 0 < α ≤ 1 , in this definition, the derivative of a constant is equal to zero, but unlike the usual Liouville–Caputo definition (1), the kernel does not have a singularity …

Is the fractional derivative linear?

For α ∈ [ n − 1 , n ) , the derivative of is. Now, all definitions including (i) and (ii) above satisfy the property that the fractional derivative is linear. This is the only property inherited from the first derivative by all of the definitions.

What is fractional partial differential equations?

Fractional order partial differential equations, as generalizations of classical integer order partial differential equations, are increasingly used to model problems in fluid flow, finance, physical and biological processes and systems [4, 10, 11, 18, 19, 28–30, 43–45].

What is Atangana Baleanu fractional derivative?

The Atangana–Baleanu derivative is a nonlocal fractional derivative with nonsingular kernel which is connected with variety of applications, see [5], [7], [9], [10], [15], [20]. Definition 2.1. Let p ∈ [1, ∞) and Ω be an open subset of the Sobolev space Hp(Ω) is defined by. Definition 2.2.

What is a numerical fraction?

Fractions are numerical quantities that represent values of less than one. Also known as fractional numbers, they are commonly used to measure parts of a whole, such as: One half (1/2) One fifth (1/5) Two thirds (2/3)

Is there an implicit numerical method for the time fractional derivative?

A novel implicit numerical method for the equation is proposed and the stability of the approximation is investigated. As for the convergence of the numerical method, we only consider a special case, i.e., the time fractional derivative is independent of the time variable .

Is the time fractional derivative converged with the time variable?

As for the convergence of the numerical method, we only consider a special case, i.e., the time fractional derivative is independent of the time variable . The case where the time fractional derivative depends on both the time variable and the space variable will be considered in a future work.

What are the benefits of using fractional order derivatives in physics?

Fractional order derivatives give a detailed presentation of the real-world problems in a more significant way as compared to integer order derivatives. One can broadly observe anomalous diffusion phenomena in a variety of physics and engineering fields.

Is there an integral method for solving time fractional differential equations?

New approach for exact solutions of time fractional Cahn–Allen equation and time fractional Phi-4 equation. Phys. A, Stat. Mech. Appl. 473, 352–362 (2017) Lu, B.: The first integral method for some time fractional differential equations. J.