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Variable‐order fractional calculus operators can be used as a valuable tool to simulate many nonlinear models with a memory property in fractional calculus applications. In this paper, we developed a novel predictor–corrector method for solving numerically nonlinear differential equations with variable‐order Liouville–Caputo‐type fractional derivatives. First, we found an inversion integral formula that is equivalent to the studied problem, and then we used it to formulate our numerical algorithm. Numerical approximate solutions of some variable‐order fractional models have been presented to display the efficiency and accuracy of the proposed algorithm. The influence of the variable‐order on the dynamic behavior of the considered problems is described. [ABSTRACT FROM AUTHOR]

In this study, two predictor–corrector schemes are developed for the numerical solution of variable exponent fractional‐type integral equations. The proposed schemes considered a Volterra‐type integral equation and then employed Trapezoidal and Simpson's 1/3 rules to obtain numerically approximate solutions. Some test problems are examined to demonstrate the performance, validity, and effectiveness of the proposed schemes. Comparisons are made between the proposed schemes to highlight the accuracy and convergence of these techniques. As an application, the effect of the variable‐order exponent on the dynamics of the studied systems is discussed. [ABSTRACT FROM AUTHOR]

In this article, a powerful modification of the optimized decomposition method is presented for dealing with nonlinear fractional differential equations, where the fractional derivatives are described in the Caputo sense. The new modification, the Legendre‐based approach of the optimized decomposition method, is suggested by employing the Legendre orthogonal polynomials to construct an optimal decomposition of the series solutions. An illustrative comparison between the Legendre‐based approaches of the Adomian decomposition method and the optimized decomposition method is performed to highlight the features and the computational efficiency of the suggested approach. The resulting numerical approximations demonstrate that the approximate solutions of the proposed approach have better accuracy and convergence. [ABSTRACT FROM AUTHOR]

In this research study, the generalized differential transform scheme has been applied to simulate impulsive differential equations with the noninteger order. One specific tool of the implemented scheme is that it converts the problems into a recurrence equation that finally leads easily to the solution of the considered problem. The validity and reliability of this method have successfully been accomplished by applying it to simulate the solution of some equations. It is shown that the considered method is very suitable and efficient for solving classes of fractional-order initial value problems for impulsive differential equations and might find wide applications. [ABSTRACT FROM AUTHOR]