Numerical solution of fractional differential equations with Caputo derivative by using numerical fractional predict–correct technique

Abstract

Fractional differential equations have recently demonstrated their importance in a variety of fields, including medicine, applied sciences, and engineering. The main objective of this study is to propose an Adams-type multistep method for solving differential equations of fractional order. The method is developed by implementing the Lagrange interpolation and taking into account the idea of the Adams–Moulton method for fractional case. The fractional derivative applied in this study is in the Caputo derivative operator. The analysis of the proposed method is presented in terms of order of the method, order of accuracy, and convergence analysis, with the proposed method being proved to converge. The stability of the method is also examined, where the stability regions appear to be symmetric to the real axis for various values of α. In order to validate the competency of the proposed method, several numerical examples for solving linear and nonlinear fractional differential equations are included. The method will be presented in the numerical predict–correct technique for the condition where $\alpha \in (0,1)$ α ∈ ( 0 , 1 ) , in which α represents the order of fractional derivatives of $D^{\alpha }y(t)$ D α y ( t ) .

Investigation of Egyptian Banks’ Competition through a Riesz–Caputo Fractional Model

“…Similarly, in the case of the proposed optimal controlled model in (33) to (37) with constraints from (37) to (41), it can be numerically solved after substituting Equation (52) using both the four banking categories and their controlling functions (P(t), A(t), F(t), I(t), φ P (t), φ A (t), φ F (t), φ I (t)). Assuming that the corresponding smoothing functions are, respectively, G 1 , G 2 , G 3 , G 4 , P , A , F , and I , then:…”

“…Many numerical techniques can be used to solve (43) [31,32]. In our case, we used the Adams-Bashford method [33] for partitioning the interval [0, T] into n equal subdivisions [t i , t i+1 ] with a width h = T/n, t i = ch for c = 0,1, . .…”

Numerical Simulations of One-Directional Fractional Pharmacokinetics燤odel

“…The advantage of implementing the Caputo derivative is that the initial conditions for fractional order differential equations with the Caputo derivative are the same for integer differential equations, hence avoiding solvability difficulties and its advantages in applied problems [9]. Besides, many experts choose to utilise the Caputo definition because one may typically have a well-intelligible physical meaning that can be measured when using Caputo's idea [20].…”

Efficient semianalytical investigation of a fractional model describing human cornea shape

“…23 While exact solutions to nonlinear fractional-order differential equations remain elusive, numerous numerical and analytical methods, originally designed for integer-derivative differential equations have been adapted to address their fractional-derivative counterparts. Noteworthy among these methods are the fractional Adams-Bashforth-Moulton method, 24 the fractional predict-correct method, 25 various spectral methods, 26,27 and artificial network methods. 28,29 While these meth-ods demonstrated efficiency and accuracy, they are not without challenges, such as issues of numerical schemes and the necessity to adjust parameters to align with numerical data.…”