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In this paper, we improved a mathematical model of monkeypox disease with a time delay to a crossover model by incorporating variable-order and fractional differential equations, along with stochastic fractional derivatives, in three different time intervals. The stability and positivity of the solutions for the proposed model are discussed. Two numerical methods are constructed to study the behavior of the proposed models. These methods are the nonstandard modified Euler Maruyama technique and the nonstandard Caputo proportional constant Adams-Bashfourth fifth step method. Many numerical experiments were conducted to verify the efficiency of the methods and support the theoretical results. This study's originality is the use of fresh data simulation techniques and different solution methodologies. [ABSTRACT FROM AUTHOR]

In this work, a higher-order nonstandard implicit compact finite difference technique is used to study a two-dimensional nonlinear distributed order Cable problem. The distributed fractional order is defined in the Atangana-Baleanu sense. The key advantage of this strategy is the large stability areas it implicitly has. A particular focus is on examining the stability analysis of the proposed scheme through the application of the Jon Von Neumann approach. We show the effectiveness of the numerical scheme using two numerical examples, and we compare our results with the published literature to check the accuracy of the approach we have presented. The technique is a helpful tool for modeling this model, as demonstrated by the results. [ABSTRACT FROM AUTHOR]

In this work, we generalize the multi‐vaccination for COVID‐19 mathematical model to the variable‐order fractional derivative case by replacing the first‐order derivative with the variable‐order fractional Atangana‐Baleanu‐Caputo operator. Moreover, the parameters of the proposed model are dependent on the variable‐order fractional derivative. We use an amended numerical technique to build an asymptotically stable difference scheme. The proposed scheme accurately simulates the behavior of the solution of the presented model. Some properties, boundedness, and positivity of the studied system are proved analytically. The introduced scheme preserves these properties of the analytic solutions. Numerical treatment is run out for testing the reliability, applicability, and simplicity of this scheme. Comparitive study bettween the obtained results using the discretization of Atangana‐Baleanu‐Caputo derivative and nonstandard finite differeance method with the results which obtained using the discretization of the operator Yang‐Abdel‐Cattani is given. [ABSTRACT FROM AUTHOR]

The purpose of this article is to present the Laplace variational iteration method, which combines the VIM with the Laplace transform approach (LVIM). This combination will result in a better and more quickly convergent sequence since nonlinear fractional differential equations (FDEs) cannot be solved using the Laplace transform. With the use of the fixed point theory, the stability analysis is specifically discussed and examined. The blood ethanol concentration system is solved numerically by using the suggested scheme. This model can be represented by a system of FDEs. The investigation will employ the Caputo-Fabrizio fractional derivative. To provide a more in-depth study of this model, we have taken it in its fractional form so that we can more accurately follow the behavior of the solution in the future and history based on the memory effect of fractional derivatives. We determine the accuracy and efficiency of the provided process by evaluating the absolute errors, and a comparison with the existing published work. The results show that the approach is a useful tool for simulating this model. [ABSTRACT FROM AUTHOR]

The article's goal is to implement a semi-analytical technique named, the Laplace variational iteration method (LVIM), which is the combination of VIM and Laplace transform method. Although both the Laplace transform method and VIM cannot be applied to some nonlinear fractional differential equations (FDEs) individually, this combination will give a fast-convergent solution to the problem under study. The proposed scheme is used to numerically solve a biodynamic system called the Lotka-Volterra system, i.e. Predator-Prey Equations (PPEs). The system of FDEs can be used to represent this scenario, as well as the Caputo-Fabrizio fractional derivative will be used throughout the study. By assessing the residual error function, we can confirm that the given procedure is effective and accurate. The outcomes demonstrate that the technique used is an effective tool for simulating such models. [ABSTRACT FROM AUTHOR]

In the present article, an approach to find the exact solution of the fractional Fokker–Planck equation (FFPE) is presented. It is based on transforming it to a system of first‐order partial differential equation via Hopf transformation, together with implementing the extended unified method. On the other hand, a theorem provides the reduction of the fractional derivatives to non‐autonomous ordinary derivative is given. Thus, the FFPE is reduced to non‐autonomous classical ones. Some explicit solutions of the classical, fractional time‐derivative Fokker–Planck equation are obtained. It is shown that the solution of the Fokker–Planck equation is bi‐Gaussian's, which was not found up to date. It is found that high friction coefficient plays a significant role in lowering the standard deviation. Further, it is found that the effect of the presence of the fractional derivative prevails that of the fractal derivative. Here, the most interesting result found is that mixed‐Gaussian solution is obtained. It is worthy to mention that the mixture of Gaussian's is a powerful tool in machine learning and also in the distribution of loads in networks. Further, varying the order of the fractional time derivatives results to slight effects in the probability distribution function. Also, it is shown that the mean and mean square of the velocity vary slowly. [ABSTRACT FROM AUTHOR]

In this article, numerical study for both nonlinear space-fractional Schrödinger equation and the coupled nonlinear space-fractional Schrödinger system is presented. We offer here the weighted average nonstandard finite difference method (WANSFDM) as a novel numerical technique to study such kinds of partial differential equations. The space fractional derivative is described in the sense of the quantum Riesz-Feller definition. Stability analysis of the proposed method is studied. To show that this method is reliable and computationally efficient different numerical examples are provided. We expect that the proposed schemes can be applicable to different systems of fractional partial differential equations. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1399-1419, 2017 [ABSTRACT FROM AUTHOR]

In this paper, the optimal control problem of HIV infection is presented. We introduce fractional order into a model of HIV infection where the derivatives are taken in the sense of Caputo. The necessary optimality conditions are characterized using Pontryagin’s maximum principle. The optimality system is approximated by shifted Legendre polynomials which transform the system of differential equations into a nonlinear system algebraic equations with unknown coefficients. Newton’s iteration method is used to solve this system of nonlinear algebraic equations. The value of the objective function which is obtained by using shifted Legendre polynomials is compared with the value of the objective function which id obtained by using numerical methods such as the iterative optimal control method and forward–backward sweep method. Numerical results are also given to demonstrate the validity and applicability of the presented technique. [ABSTRACT FROM AUTHOR]