Fractional Diffusion Equation under Singular and Non-Singular Kernel and Its Stability

Abstract

The fractional reaction–diffusion equation has been used in many real-world applications in fields such as physics, biology, and chemistry. Motivated by the huge application of fractional reaction–diffusion, we propose a numerical scheme to solve the fractional reaction–diffusion equation under different kernels. Our method can be particularly employed for singular and non-singular kernels, such as the Riemann–Liouville, Caputo, Fabrizio–Caputo, and Atangana–Baleanu operators. Moreover, we obtained general inequalities that guarantee that the stability condition depends explicitly on the kernel. As an implementation of the method, we numerically solved the diffusion equation under the power-law and exponential kernels. For the power-law kernel, we solved by considering fractional time, space, and both operators. In another example, we considered the exponential kernel acting on the time derivative and compared the numerical results with the analytical ones. Our results showed that the numerical procedure developed in this work can be employed to solve fractional differential equations considering different kernels.

Transient Dynamics of a Fractional Fisher Equation

“…The complete discussion about the method for the fractional reaction-diffusion equation under general kernels can be found in ref. [43]. To do that, we construct a grid defined by [0, X] × [0, T], where the space and time are discretized according to x i = i∆x and t j = j∆t, respectively, where i = 0, 1, .…”

“…A detailed derivation of Equation ( 8) is shown in ref. [43]. It is worth mentioning that this choice for the spatial operator allows us to obtain an equivalent to the Riesz differential operator for the interval [43,44], i.e., −∞ < x < ∞, which, for practical proposes is considered finite −X ≤ x ≤ X (with |X| → ∞) to perform the numerical calculations.…”

“…[43]. It is worth mentioning that this choice for the spatial operator allows us to obtain an equivalent to the Riesz differential operator for the interval [43,44], i.e., −∞ < x < ∞, which, for practical proposes is considered finite −X ≤ x ≤ X (with |X| → ∞) to perform the numerical calculations. In this manner, the Caputo differential operator applied for the spatial variable behaves like the Riesz differential operator.…”

Modeling and Analysis of Caputo–Fabrizio Definition-Based Fractional-Order Boost Converter with Inductive Loads

“…When studying practical systems, the Caputo definition is often used [17]. However, there is a bias in describing the full memory effect because of the singular kernel of the Caputo definition [18,19]. To overcome this issue, Caputo and Fabrizio proposed the C-F definition [20].…”