On solutions of time‐fractional advection–diffusion equation

Abstract

In this paper, we present an attractive reliable numerical approach to find an approximate solution of the time‐fractional advection–diffusion equation (FADE) under the Atangana–Baleanu derivative in Caputo sense (ABC) with Mittag–Leffler kernel. The analytic and approximate solutions of FADE have been determined by using reproducing kernel Hilbert space method (RKHSM). The most valuable advantage of the RKHSM is its ease of use and its quick calculation to obtain the numerical solution of the FADE. Our main tools are reproducing kernel theory, some important Hilbert spaces, and a normal basis. The convergence analysis of the RKHSM is studied. The computational results are compared with other results of an appropriate iterative scheme and also by using specific examples, these results clearly show: On the one hand, the effect of the ABC‐fractional derivative with the Mittag–Leffler kernel in the obtained outcomes, and on the other hand, the superior performance of the RKHSM. From a numerical viewpoint, the RKHSM provides the solution's representation in a convergent series. Furthermore, the obtained results elucidate that the proposed approach gives highly accurate outcomes. It is worthy to observe that the numerical results of the specific examples show the efficiency and convenience of the RKHSM for dealing with various fractional problems emerging in the physical environment.

A reproducing kernel Hilbert space method for nonlinear partial differential equations: applications to physical equations

“…First, we express some RKHSs mostly needed through this study and can be found in many papers (see, e.g. [13,14]). Then we derive the expression of the reproducing kernel function (RKF) for each defined RKHS.…”

“…Here, we don't present the expression of the reproducing kernel function Z x m ( ) of the space 3 because it is so long. The interested reader is referred to[14] for more details about this point.2Theorem 2.11. We obtain the RK function R t h ( ) of the space…”

Solving Fractional Riccati Differential Equation with Caputo-Fabrizio Fractional Derivative

“…The Riccati differential equation is employed across diverse disciplines like physics, engineering, biology, control theory, signal processing, and finance [25], [6], [20]. The fractional Riccati equation holds significance in numerous physics and engineering contexts [36], [33], [43], [40], [8], [11], [23], [35], [45]. Many investigators have examined the numerical solution of this problem [24], [22], [21], [30], [42], [5], [7].More convenientreferences for this equation can be found in [18], [37], [1], [30], [38], [27].…”

An efficient technique based on cubic B-spline functions for solving time-fractional advection diffusion equation involving Atangana–Baleanu derivative

“…Akgül and Modanli [36] proposed the Crank-Nicholson difference scheme method for the third order fractional partial differential equation with ABFD. Attia et al [37] developed the computational approach to find an approximate solution of the TFADE.…”