A numerical method for solving the fractional-order logistic differential equation with two different delays (FOLE) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce FOLE to a system of algebraic equations. Special attention is given to study the convergence and the error estimate of the presented method. Numerical illustrations are presented to demonstrate utility of the proposed method. Chaotic behavior is observed and the smallest fractional order for the chaotic behavior is obtained. Also, FOLE is studied using variational iteration method (VIM) and the fractional complex transform is introduced to convert fractional Logistic equation to its differential partner, so that its variational iteration algorithm can be simply constructed. Numerical experiment is presented to illustrate the validity and the great potential of both proposed techniques.

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We study the estimated investigative answers for one of the popular models in biomathematics, in particular, the nonlinear Anopheles mosquito model numerically. The optimal control (OC) for nonlinear Anopheles mosquito model is examined. Important and adequate conditions to ensure the presence and singularity of the arrangements of the control issue are assumed. Two control factors are suggested to limit the normal measure of eggs laid per treated female every day. The signal stream chart and Simulink[Formula: see text]Matlab of this model are constructed. The framework is designed utilizing the MULTISIM simulation program. We utilize the homotopy disruption strategy (HPM) to examine the logical surmised answer for the nonlinear control issue. We utilize the mathematical programming bundles, for example, Maple, to emphasize while ascertaining the rough arrangement. Results are displayed graphically and introduced to delineate the conduct of obtained inexact arrangements.

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This paper is devoted with numerical solution of the system fractional differential equations (FDEs) which are generated by optimization problem using the Chebyshev collocation method. The fractional derivatives are presented in terms of Caputo sense. The application of the proposed method to the generated system of FDEs leads to algebraic system which can be solved by the Newton iteration method. The method introduces a promising tool for solving many systems of non-linear FDEs. Two numerical examples are provided to confirm the accuracy and the effectiveness of the proposed methods. Comparisons with the fractional finite difference method (FDM) and the fourth order Runge-Kutta (RK4) are given.

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