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This paper introduces a new set of the basis functions called the fractional Chebyshev cardinal wavelets and details their properties. These wavelets have a greater degree of freedom than the classical Chebyshev cardinal wavelets. Moreover, they retain the cardinality and the spectral accuracy of these wavelets. The fractional derivative and integral matrices of these fractional basis functions are obtained exactly in the explicit forms. In this study, we aim to devise an efficient and powerful approximation method using these new basis functions. Then, we employ them for a new category of nonlinear fractional quadratic integro-differential equations. By employing their fractional integral matrix and their cardinality, the problem under study is transformed into solving a nonlinear system of algebraic equations. The error analysis of the presented technique is first investigated theoretically and then computational efficiency is examined for two numerical examples. [ABSTRACT FROM AUTHOR]

In this study, a new family of cardinal wavelets called the extended Chebyshev cardinal wavelets is introduced to investigate problems on arbitrary finite domains. These wavelets possess many beneficial properties, such as spectral (exponential) accuracy, cardinality and orthogonality. The classical and fractional derivative matrices together with the fractional integral matrix of these wavelets are obtained. Two formulations of delay optimal control problems with a dynamical system involved with fractional derivatives are introduced. Two direct approaches based on these wavelets, together with their fractional derivative and integral matrices, are adopted for solving such problems. The presented methods transform solving the problems under study into solving constrained minimisation problems by approximating the state and control variables via the expressed wavelets. Some numerical examples are provided to show the efficiency of the expressed wavelet methods. [ABSTRACT FROM AUTHOR]

In this paper, we solve the Caputo's fractional parabolic partial integro-differential equations (FPPI-DEs) by Gaussian-radial basis functions (G-RBFs) method. The main idea for solving these equations is based on the radial basis functions (RBFs) which also provides approaches to higher dimensional spaces. In the suggested method, FPPI-DEs are reduced to nonlinear algebraic systems. We propose to apply the collocation scheme using G-RBFs to approximate the solutions of FPPI-DEs. Numerical examples are provided to show the convenience of the numerical scheme based on the G-RBFs. The results reveal that the presented method is very efficient and convenient for solving such problems. [ABSTRACT FROM AUTHOR]

The generalized Couette flow of Jeffrey nanofluid through porous medium, subjected to the oscillating pressure gradient and mixed convection, is numerically simulated using variable-order fractional calculus. The effect of several involving parameters such as chemical reactions, heat generation, thermophoresis, radiation, channel inclination, and Soret effect is also considered. To the best of the authors' knowledge, the described general form of the Couette flow problem is not tackled by the researchers yet. The non-dimensional form of heat, mass, and momentum equations is solved as a coupled set. The effect of several parameters such as Grashof, Hartmann, Prandtl, Soret, and Schmidt numbers in addition to oscillation frequency, retardation time, radiation, heat absorption, and reaction rate are determined and presented graphically. An operational matrix method based on the second kind shifted Chebyshev polynomials is proposed to investigate the behavior of the interested problem. In fact, regarding the established method, the unknown solutions are expanded by the mentioned basis polynomials. Then, the operational matrix of the variable-order fractional derivative is utilized to transfer the problem into solving an algebraic system of equations. According to the obtained results, the growth of fractional order from 0 to 1 changes the skin friction coefficient, Nu and flow rate by - 18.1 , 35.5, and 10 % , respectively. [ABSTRACT FROM AUTHOR]

In this paper, the radial basis functions (RBFs) method is used for solving a class of variable-order time fractional telegraph equa- tAiobnst(Vra-TctF.TE), 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 < a(x,t) < 2. 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. [ABSTRACT FROM AUTHOR]

In this paper, a new operational matrix of variable-order fractional derivative (OMV-FD) is derived for the second kind Chebyshev wavelets (SKCWs). Moreover, a new optimization wavelet method based on SKCWs is proposed to solve multi variable-order fractional differential equations (MV-FDEs). In the proposed method, the solution of the problem under consideration is expanded in terms of SKCWs. Then, the residual function and its errors in 2-norm are employed for converting the problem under study to an optimization one, which optimally chooses the unknown coefficients. Finally, the method of constrained extremum is applied, which consists of adjoining the constraint equations obtained from the given initial conditions to the object function obtained from residual function by a set of unknown Lagrange multipliers. The main advantage of this approach is that it reduces such problems to those optimization problems, which greatly simplifies them and also leads to obtain a good approximate solution for them. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the fractional biharmonic equation and an orthogonal system of basis functions for the space of continuous functions on the interval [0, L], generated by the shifted Chebyshev polynomials. Moreover, we propose a computational method based on the operational matrix of fractional derivatives of these basis functions for solving the fractional biharmonic equation. The main characteristic behind this approach is that it reduces the problem under consideration to solving a system of algebraic equations which greatly simplifies the problem. Convergence of the shifted Chebyshev polynomials expansion in two-dimensions is investigated. Also the power of this manageable method is illustrated. [ABSTRACT FROM AUTHOR]

An efficient Chebyshev wavelets method for solving a class of nonlinear fractional integrodifferential equations in a large interval is developed, and a new technique for computing nonlinear terms in such equations is proposed. Existence of a unique solution for such equations is proved. Convergence and error analysis of the proposed method are investigated. Moreover in order to show efficiency of the proposed method, the new approach is compared with some numerical methods. [ABSTRACT FROM AUTHOR]

The operational matrices of fractional-order integration for the Legendre and Chebyshev wavelets are derived. Block pulse functions and collocation method are employed to derive a general procedure for forming these matrices for both the Legendre and the Chebyshev wavelets. Then numerical methods based on wavelet expansion and these operational matrices are proposed. In this proposed method, by a change of variables, the multiorder fractional differential equations (MOFDEs) with nonhomogeneous initial conditions are transformed to the MOFDEs with homogeneous initial conditions to obtain suitable numerical solution of these problems. Numerical examples are provided to demonstrate the applicability and simplicity of the numerical scheme based on the Legendre and Chebyshev wavelets. [ABSTRACT FROM AUTHOR]