This work is trying to introduce a fractional order floated pole controller as a fast and robust approach. We designed a robust Variable structure control that yields a continuous and constrained control signal, also a fast response in the presence of model uncertainties and external disturbances. In the proposed controller, we employed the pole placement algorithm, then by designing proper polynomials gave it robust property, then due to a simple optimization routine, we make it fast and faster within the stability region. Finally, to evaluate the proposed method, numerical examples in different situations of the presence of noise, disturbance, and model uncertainties, also comparative results are presented. This paper proposed an accurate, fast, and robust controller. This can improve the performance of the perturbed functional systems used in the industrial fields. It is proposed to spread the benefit of fractional calculus in the control of complex systems in practical situations.

A numerical method for the Variable-order fractional functional differential equations (VO-FFDEs) has been developed. This method is based on approximation with shifted Legendre polynomials. The properties of the latter were stated, first. These properties, together with the shifted Gauss-Legendre nodes were then utilized to reduce the VO-FFDEs into a solution of matrix equation. Sequentially, the error estimation of the proposed method was investigated. The validity and efficiency of our method were examined and verified via numerical examples.

This study is devoted to introducing a computational technique based on Bernstein polynomials to solve Variable order fractional optimal control problems (VO-FOCPs). This class of problems generated by dynamical systems describe with Variable order fractional derivatives in the Caputo sense. In the proposed method, the Bernstein operational matrix of the fractional Variable-order derivatives will be derived. Then, this matrix is used to obtain an approximate solution to mentioned problems. With the use of Gauss-Legendre quadrature rule and the mentioned operational matrix, the considered VO-FOCPs are reduced to a system of equations that are solved to get approximate solutions. The obtained results show the accuracy of the numerical technique.

In this work, a new mathematical model of thermoelasticity theory has been considered in the context of a new consideration of heat conduction with fractional order theory. A functionally graded isotropic unbounded medium is considered subjected to a periodically varying heat source in the context of space-time non-local generalization of three-phase-lag thermoelastic model and Green-Naghdi models, in which the thermophysical properties are temperature dependent. The governing equations are expressed in Laplace-Fourier double transform domain and solved in that domain. Then the inversion of the Fourier transform is carried out by using residual calculus, where poles of the integrand are obtained numerically in complex domain by using Laguerre’s method and the inversion of Laplace transform is done numerically using a method based on Fourier series expansion technique. The numerical estimates of the thermal displacement, temperature and thermal stress are obtained for a hypothetical material. Finally, the obtained results are presented graphically to show the effect of non-local fractional parameter on thermal displacement, temperature and thermal stress. A comparison of the results for different theories (three-phase-lag model, GN model II, GN model III) is presented and the effect of non-homogeneity is also shown. The results, corresponding to the cases, when the material properties are temperature independent, agree with the results of the existing literature.

In this paper, the radial basis functions (RBFs) method is used for solving a class of Variable-order time fractional telegraph equation (V-TFTE), which appears extensively in various fields of science and engineering. Fractional derivatives based on Caputo's fractional derivative as a function of the independent Variable are defined of order $1<\alpha(x, t)\leq2$. The proposed method combines the radial basis functions and finite difference scheme to produce a semi-discrete algorithm. In the first stage the Variable-order time-dependent derivative is discreticized, and then we approximate the solution by the radial basis functions. The aim of this paper is to show that the collocation method based on RBFs is suitable for the treatment of the Variable-order fractional partial differential equations. The efficiency and accuracy of the proposed method are shown for some concrete examples. The results reveal that the proposed method is very efficient and accurate.

The aim of this article is to find an effective method for solving Variable-order fractional integro-differential equations. This method transforms the problem into a system of algebraic equations. For this purpose, we first express Vieta-Lucas orthogonal polynomials, then, we express the operational matrices of these polynomials. At this stage, all components of the equation will be expressed in terms of the new shifted Vieta-Lucas operational matrices. After that, by placing these operational matrices in the main equation and using the spectral collocation method, the Variable-order fractional integro-differential equation will become an algebraic system. By solving this algebraic system, we will find an approximate solution to the original equation. In the following, an analysis of the error is also presented by preparing some theorems. In the end, in order to express the efficiency and capability of the method, some numerical examples are given. Additionally, for the numerical examples, the condition number, numerical convergence order, and the computed CPU time are evaluated. Based on the obtained results, it was concluded that the proposed method is relatively stable, highly accurate and efficient, and has an appropriate convergence rate.

In this article, we use a finite difference technique to solve Variable-order fractional integro-differential equations (VOFIDEs, for short). In these equations, the Variable-order fractional integration (VOFI) and Variable-order fractional derivative (VOFD) are described in the Riemann-Liouville's and Caputo's sense, respectively. Numerical experiments, consisting of two examples, are studied. The obtained numerical results reveal that the proposed finite difference technique is very effective and convenient for solving VOFIDEs.

In this paper, a high-order numerical method is designed and implemented to solve a boundary value problem governed by the Variable-order fractional diffusion equation. This equation contains a Variable-order fractional time-derivative and a second-order spatial-derivative. To develop this novel method, a compact finite difference formula and a weighted shifted Grunwald-Letnikov operator are used for spatial and temporal discretization, respectively. It is shown that this method is of fourth- and second-order of convergence accuracy in spatial and time directions, respectively. Also, the solvability, stability and convergence of the peresent method are investigated. To verify the efficiency and high accuracy of this method, some numerical examples and comparative results are presented.

In this paper, we introduce the Lebesgue-Sobolev spaces critical points theory then we consider the boundary value problem involving an ordinary differential equation with p(x)-Laplacian operator, and nonhomogeneous Neumann conditions. Existence results for ordinary differential equations with elliptic Neumann problems that depending on two real parameters are investigated. Precisely, by using the critical point theory, we show the existence of three weak solutions for p(x)-Laplacian problems. Using the critical point theorems we have proved, we give some conclusions