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In this article, an effective numerical technique for solving the variable order Fredholm–Volterra integro-differential equations (VO-FV-IDEs), systems of VO-FV-IDEs and variable order Volterra partial integro-differential equations (VO-V-PIDEs) is given. The suggested technique is built on the combination of the spectral collocation method with some types of operational matrices of the variable order fractional differentiation and integration of the shifted fractional Gegenbauer polynomials (SFGPs). The proposed technique reduces the considered problems to systems of algebraic equations that are straightforward to solve. The error bound estimation of using SFGPs is discussed. Finally, the suggested technique’s authenticity and efficacy are tested via presenting several numerical applications. Comparisons between the outcomes achieved by implementing the proposed method with other numerical methods in the existing literature are held, the obtained numerical results of these applications reveal the high precision and performance of the proposed method.

In this article, a novel shifted Gegenbauer operational matrix (SGOM) of fractional derivative in the Caputo sense is derived. Based on this operational matrix, an accurate and effective numerical algorithm is proposed. The SGOM of fractional derivative in conjunction with the tau method are used for solving the constant and variable coefficients space–time fractional telegraph equations (FTE) with various types of boundary conditions, namely, Neumann, Dirichlet and Robin conditions. The convergence analysis of the proposed method is established in $$\mathcal {L}^{2}_{\omega _{\alpha }}$$ . Finally, miscellaneous test examples are given and compared with other methods to clarify the accuracy and efficiency of the presented algorithm.

This paper declares a new spectral collocation technique to provide accurate approximate solutions of the one- and two-dimensional time and space fractional coupled Burgers’ equations (TSFCBEs) in which the fractional derivatives are defined according to Caputo’s definition. The suggested method is based on the shifted Gegenbauer polynomials (SGPs) for approximating the solution of TSFCBEs. The suggested technique reduces the considered problems to the solution of nonlinear algebraic equations (NLAEs). Moreover, the accuracy and reliability of the proposed method are confirmed through numerical examples. Finally, the obtained numerical results are compared with those previously reported in the literature.

In this paper, by introducing the fractional derivative in sense of Caputo, the Laplace- variational iteration method (LVIM) and the Laplace-Adomian decomposition method (LADM) are directly extended to study the linear and nonlinear systems of fractional partial differential equations, as a result the approximated numerical solutions are acquired in the form of rapidly convergent series with easily computable components. Numerical results show that the two approaches are easy to implement and accurate when are applied. Compassions are made between the two methods and exact solutions. Figures are used to show the efficiency as well as the accuracy of the achieved approximated results.

We solve the one- and two-dimensional fractional coupled Burgers’ equations (FCBEs) by three different methods. The proposed methods are the Laplace–Adomian decomposition method (LADM), the Laplace-variational iteration method (LVIM) and the reduced differential transform method (RDTM). The solutions are obtained as rapidly convergent series with simply calculable terms. Numerical studies of the application of these approaches for a number of sample problems are given and are illustrated graphically. With these methods, it is possible to investigate the nature of solutions when the fractional derivative parameters are changed. The numerical results reveal the effectiveness and the correctness of the proposed methods.

This paper presents a method to solve numerically the linear quadratic optimal control problems (LQOCPs) of the fractional singular control systems with Caputo fractional derivatives. Under certain conditions on the singular fractional control systems, the problems under consideration are transformed into LQOCPs of the standard normal fractional control systems. Based on the Grunwald–Letnikov approximation (GLA) of the fractional derivatives (FDs), a numerical technique is used to solve the LQOCPs of the standard normal fractional control systems. An illustrative example is introduced to demonstrate the applicability of the proposed technique.

The response of Scenedesmus spp. to three phenylureas, namely Maloran, Dicuran, and Patoran was investigated. Chloropyll (a) content, ratio of chlorophyll (a/b), growth rate and dry weight of algae tended to decrease as the concentration of herbicide in the bioassay media was increased. The inhibitory effect of the studied herbicides followed the order; Maloran > Dicuran > Patoran. At low concentration levels, patoran led to an increase in Chl. (a) content, Chl. (a/b) ratio and maximum growth rate. Uptake of the studied compounds by algal cells was at its maximum value in the presence of low herbicide concentrations and tended to decrease as the herbicide concentration was increased. Scenedesmus algal Cells were able to recover and attained their normal growth rate and morphological characteristics when the cells were transferred to a herbicide-free medium. The pH value of the algal media gradually increased as algal growth proceeded. In media containing inhibitory levels of phenylureas, the pH value was unchanged. Statistical analysis indicated that changes in Chl. (a) induced by the herbicides were significant.

The effect of three triazine herbicides namely, Gardoprim, Gesapax and Igran on the freshwater alga, Scenedesmus sp., was studied. Growth rate, chlorophyll (a) content, and ratio of chlorophylls (a/b) tend to decrease as the concentration of herbicides in the bioassay media was increased. The inhibitory effect of the studied herbicides was in the following the order: Igran > Gesapax > Gardoprim. Obtainable results reflect the relation between chemical structure and toxicity even within a chemically related group of herbicides. Uptake of the studied compounds by Scenedesmus algal cells decreased as the concentration of herbicides in the media was increased. The pH of the bioassay media showed gradual increase as algal growth proceeded, whereas in media containing inhibitory levels of the triazines, the pH value was unaffected. Microscopical examinations of Scenedesmus in all tested cultures showed marked morphological changes. In the case of inhibitory concentrations, the unicellular forms increased in numbers compared to Scenedesmus in the normal four-cell state. Statistical analysis revealed that significant relation existed between herbicide concentrations and chlorophyll content.