Shifted Chebyshev Polynomials Research Articles

Numerical solution method for multi-term variable order fractional differential equations by shifted Chebyshev polynomials of the third kind

Multi-term variable-order fractional differential equations (VO-FDEs) are considered to be one of the tools to illustrate the behavior of transient-regime real-life phenomena precisely. The analytical solutions of VO-FDEs are generally very hard to obtain. Therefore, in this study, an approximation method for the solution of multi-term VO-FDEs is proposed using shifted Chebyshev polynomials of the third kind (SCP3). The method employs the operational matrices of SCP3 to proximate the VO-FDEs with a system of algebraic equations. The solution of which also yields the approximate result for the multi-term VO-FDE. An error analysis is also included. The SCP3 method is examined on several examples demonstrating the precision and prowess of the method.

A Pseudo-spectral Method for Time Distributed Order Two-sided Space Fractional Differential Equations

Time distributed order two-sided space differential equations of arbitrary order offer a robust approach to modelling complex dynamical systems. In this study, we describe a scheme for obtaining the numerical solutions of time distributed order multidimensional two-sided space fractional differential equations. The numerical discretization scheme is a hybrid scheme, comprising a Newton–Cotes quadrature formula and a spectral collocation method. The time distributed order fractional differential operator is approximated using the composite Simpson's rule, and the solution of the resulting differential equation is expressed as a linear combination of shifted Chebyshev polynomials in all variables. Convergence analysis of the numerical scheme is presented. Some one- and two-dimensional time distributed order two-sided space fractional differential equations, such as the fractional advection-dispersion and diffusion equations, are presented to demonstrate the accuracy and computational efficiency of the numerical scheme, and numerical solutions are compared with the exact solutions, where these are available.

Numerical solutions of fractional optimal control with Caputo\u2013Katugampola derivative

In this paper, we present a numerical technique for solving fractional optimal control problems with a fractional derivative called Caputo–Katugampola derivative. This derivative is a generalization of the Caputo fractional derivative. The proposed technique is based on a spectral method using shifted Chebyshev polynomials of the first kind. The Clenshaw and Curtis scheme for the numerical integration and the Rayleigh–Ritz method are used to estimate the state and control variables. Moreover, the error bound of the fractional derivative operator approximation of Caputo–Katugampola is derived. Illustrative examples are provided to show the validity and applicability of the presented technique.

A collocation approach for multiterm variable-order fractional delay-differential equations using shifted Chebyshev polynomials

In this article, a general form of multiterm variable-order fractional delay differential equations (GVOFDDEs) is presented. The introduced GVOFDDEs have variable-order multi-terms and integer-order derivatives as well for all terms delayed or with normal argument. The variable order derivative is a generalization of fractional and integer orders, so it considered in this work. The collocation approach is applied with the aid of shifted Chebyshev polynomials to solve the presented GVOFDDEs as a matrix discretization technique. The presented technique transforms all terms of GVOFDDEs to a matrix equation with novel operational matrices. The qualification of the presented scheme is measured by many numerical test examples.

Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials

In this paper, the finite integration method and the operational matrix of fractional integration are implemented based on the shifted Chebyshev polynomial. They are utilized to devise two numerical procedures for solving the systems of fractional and classical integro-differential equations. The fractional derivatives are described in the Caputo sense. The devised procedure can be successfully applied to solve the stiff system of ODEs. To demonstrate the efficiency, accuracy and numerical convergence order of these procedures, several experimental examples are given. As a consequence, the numerical computations illustrate that our presented procedures achieve significant improvement in terms of accuracy with less computational cost.

Bivariate Chebyshev polynomials of the fifth kind for variable-order time-fractional partial integro-differential equations with weakly singular kernel

The shifted Chebyshev polynomials of the fifth kind (SCPFK) and the collocation method are employed to achieve approximate solutions of a category of the functional equations, namely variable-order time-fractional weakly singular partial integro-differential equations (VTFWSPIDEs). A pseudo-operational matrix (POM) approach is developed for the numerical solution of the problem under study. The suggested method changes solving the VTFWSPIDE into the solution of a system of linear algebraic equations. Error bounds of the approximate solutions are obtained, and the application of the proposed scheme is examined on five problems. The results confirm the applicability and high accuracy of the method for the numerical solution of fractional singular partial integro-differential equations.

Parameter Estimation of the Lotka–Volterra Model with Fractional Order Based on the Modulation Function and Its Application

The Lotka–Volterra model is widely applied in various fields, and parameter estimation is important in its application. In this study, the Lotka–Volterra model with universal applicability is established by introducing the fractional order. Modulation function is multiplied by both sides of the Lotka–Volterra model, and the model is converted into linear equations with parameters to be estimated by the fractional integration method. The parameters are obtained by solving the equations. The state of the system is estimated by shifted Chebyshev polynomial. Last, the implementation program of the model is compiled. The concrete implementation method of the improved model is proposed by an example in this study.

Fractional modelling and numerical simulations of variable-section viscoelastic arches

In this paper, two fractional viscoelastic constitutive models are used to establish nonlinear fractional integro-differential governing equations of variable-section viscoelastic arches. Shifted Chebyshev polynomial algorithm is introduced to numerically solve the governing equations directly in time domain. The feasibility and accuracy of the proposed algorithm are verified by convergence analysis and error estimation of a mathematical example. In addition, the dynamic responses of variable-section viscoelastic arches with three materials under two fractional models are also studied to verify the effectiveness of shifted Chebyshev polynomial algorithm.

Shifted fifth-kind Chebyshev Galerkin treatment for linear hyperbolic first-order partial differential equations

Through the current article, a numerical technique to obtain an approximate solution of one-dimensional linear hyperbolic partial differential equations is implemented. A certain combination of the shifted Chebyshev polynomials of the fifth-kind is used as basis functions. The main idea behind the proposed technique is established on converting the governed boundary-value problem into a system of linear algebraic equations via the application of the spectral Galerkin method. The resulting linear system can be solved by expedients of the Gaussian elimination procedure. The convergence and error analysis of the shifted Chebyshev expansion are carefully investigated. Various numerical examples are given to demonstrate the power and accuracy of the given method.

An efficient algorithm for solving the conformable time-space fractional telegraph equations

Abstract In this paper, an efficient algorithm is proposed for solving one dimensional time-space-fractional telegraph equations. The fractional derivatives are described in the conformable sense. This algorithm is based on shifted Chebyshev polynomials of the fourth kind. The time-space fractional telegraph equations is reduced to a linear system of second order differential equations and the Newmark’s method is applied to solve this system. Finally, some numerical examples are presented to confirm the reliability and effectiveness of this algorithm.

Buckling of Two-Directional Functionally Graded Cylindrical Beams Based on a High-Order Cylindrical Beam Model

In this paper, a high-order cylindrical beam model, where the shear deformation is taken into account, will be used to analyze the buckling behaviors of the functionally graded cylindrical beams with radially and axially varying material inhomogeneities. The coupled governing equations for buckling of a cylindrical beam under axial compression are derived, which can be translated into a single differential equation by introducing an auxiliary function. The shifted Chebyshev polynomials are used to compute the critical buckling loads for kinds of boundary conditions. By comparing with the three-dimensional solutions for buckling of homogeneous circular beams, the validity of the introduced model is confirmed. Two typical material property distributions defined by the exponential- and power-law are considered. A parametric study is carried out to investigate the effects of material gradient indexes on the critical buckling loads.

A robust spectral treatment of a class of initial value problems using modified Chebyshev polynomials

New modified shifted Chebyshev polynomials (MSCPs) have been constructed over the interval [α, β]. These polynomials are utilized as basis functions with the application of the spectral collocation method. The operational matrix of integer order derivatives of these polynomials is introduced. The elements of this matrix are explicitly given. The introduced operational matrix along with the collocation method is used to find a direct solver of linear/nonlinear class of IVPs. Furthermore, the convergence and error analysis of the modified Chebyshev expansion are discussed. Some specific numerical examples are given to ascertain the wide applicability and the good efficiency of the suggested algorithm. The obtained results from the tested numerical examples are convincing. Also, the introduced approximate solutions are very close to the analytical ones.

An efficient 4-step block method for solution of first order initial value problems via shifted chebyshev polynomial

In this paper, we develop a four-step block method for solution of first order initial value problems of ordinary differential equations. The collocation and interpolation approach is adopted to obtain a continuous scheme for the derived method via Shifted Chebyshev Polynomials, truncated after sufficient terms. The properties of the proposed scheme such as order, zero-stability, consistency and convergence are also investigated. The derived scheme is implemented to obtain numerical solutions of some test problems, the result shows that the new scheme competes favorably with exact solution and some existing methods.

Hypergeometric fractional derivatives formula of shifted Chebyshev polynomials: tau algorithm for a type of fractional delay differential equations

Abstract This paper presents an explicit formula that approximates the fractional derivatives of Chebyshev polynomials of the first-kind in the Caputo sense. The new expression is given in terms of a terminating hypergeometric function of the type 4 F 3(1). The integer derivatives of Chebyshev polynomials of the first-kind are deduced as a special case of the fractional ones. The formula will be applied for obtaining a spectral solution of a certain type of fractional delay differential equations with the aid of an explicit Chebyshev tau method. The shifted Chebyshev polynomials of the first-kind are selected as basis functions and the spectral tau method is employed for obtaining the desired approximate solutions. The convergence and error analysis are discussed. Numerical results are presented illustrating the efficiency and accuracy of the proposed algorithm.

A Numerical Scheme Based on the Chebyshev Functions to Find Approximate Solutions of the Coupled Nonlinear Sine-Gordon Equations with Fractional Variable Orders

In this article, a numerical method based on the shifted Chebyshev functions for the numerical approximation of the coupled nonlinear variable-order fractional sine-Gordon equations is shown. The variable-order fractional derivative is considered in the sense of Caputo-Prabhakar. To solve the problem, first, we obtain the operational matrix of the Caputo-Prabhakar fractional derivative of shifted Chebyshev polynomials. Then, this matrix and collocation method are used to reduce the solution of the nonlinear coupled variable-order fractional sine-Gordon equations to a system of algebraic equations which is technically simpler for handling. Convergence and error analysis are examined. Finally, some examples are given to test the proposed numerical method to illustrate the accuracy and efficiency of the proposed method.

The construction of a new operational matrix of the distributed-order fractional derivative using Chebyshev polynomials and its applications

In this paper, the properties of Chebyshev polynomials and the Gauss–Legendre quadrature rule are employed to construct a new operational matrix of distributed-order fractional derivative. This operational matrix is applied for solving some problems such as distributed-order fractional differential equations, distributed-order time-fractional diffusion equations and distributed-order time-fractional wave equations. Our approach easily reduces the solution of all these problems to the solution of some set of algebraic equations. We also discuss the error analysis of approximation distributed-order fractional derivative by using this operational matrix. Finally, to illustrate the efficiency and validity of the presented technique five examples are given. Abbreviations: DFDEs: distributed-order fractional differential equations; DTFDEs: distributed-order time-fractional diffusion equations; DTFWEs: distributed-order time-fractional wave equations; OMFD: operational matrix of fractional derivative; SCP: shifted Chebyshev polynomial

Learning nonlinear dynamics with behavior ordinary/partial/system of the differential equations: looking through the lens of orthogonal neural networks

Applications of neural network algorithms have been grown in recent years and various architectures have been introduced by researchers for the purpose of solving different types of differential equations. Physics informed neural networks, functional link neural networks, and feed-forward differential equation neural networks are some of these architectures. In this paper, we introduce a new neural network for simulating the behavior of Emden–Fowler-type dynamic modeled as an ordinary/partial/system of differential equation, i.e. ODE/PDE/SDE which is based on the development of two introduced functional link neural network and feed-forward differential equation neural network for the partial/system of differential equations. This algorithm uses roots of shifted Chebyshev polynomials as a training data set and the The Levenberg–Marquardt algorithm is taken as an optimizer. To show the applicability of the proposed network, it is applied to some test problems and the obtained results are compared with some other neural network approaches and also, some other numerical algorithms. The reported results showed that the algorithm proposed in this paper is a powerful method for simulating the behavior of partial and system of differential equations and is more accurate than other methods in the literature.

Solving Linear Fractional Differential Equations with Time Delay by Steps Chebyshev-Tau Scheme

In this article, we obtain a new formula for obtaining an exact expression for the fractional derivatives of the shifted Chebyshev polynomials in the desired interval $$[\alpha , \beta ]$$ based on shifted Chebyshev polynomials themselves. We propose a novel scheme to solve linear fractional differential equations with the order $$\nu $$ and a single time delay. The proposed scheme consists of two stages. The first stage converts the linear fractional differential equations with a time delay to linear fractional differential equations using the method of steps. It employs the shifted Chebyshev polynomials to construct an approximate solution for the system of equations on the second stage. The proposed scheme’s convergence analysis has been performed, suggesting that the proposed scheme has a spectral rate of convergence for sufficiently smooth solutions. Also, the stability analysis is provided. Numerical benchmark problems are examined to show the proficiency and accuracy of the proposed scheme.

Numerical treatment of the space fractional advection–dispersion model arising in groundwater hydrology

This paper studies a new computational method for the approximate solution of the space fractional advection–dispersion equation in sense of Caputo derivatives. In the first method, a time discretization is accomplished via the compact finite difference, while the fourth kind shifted Chebyshev polynomials are used to discretize the spatial derivative. The unconditional stability and convergence order of the method are studied via the energy method. Three examples are given for illustrating the effectiveness and accuracy of the new scheme when compared with existing numerical methods reported in the literature.

Power and Chebyshev Series Transformation Formulas with Applications to Solving Ordinary Differential Equations via the Fröbenius and Taylor’s Methods

In this paper, we present formulas that turn finite power series into series of shifted Chebyshev polynomials of the first kind. Thereafter, we derive formulas for coefficients of economized power series obtained by truncating the resulting Chebyshev series. To illustrate the utility of our formulas, we apply them to the solution of first order ordinary differential equations via Taylor methods and to solving the Schr?dinger equation (SE) for a spherically symmetric hyperbolic potential via the Fr?benius method. In each of the two applications, we show that the use of our formulas makes it possible to reduce the computing time, while preserving the accuracy of the results.