Spectral Tau Research Articles

Solving Nonholonomic Systems with the Tau Method

A numerical procedure based on the spectral Tau method to solve nonholonomic systems is provided. Nonholonomic systems are characterized as systems with constraints imposed on the motion. The dynamics is described by a system of differential equations involving control functions and several problems that arise from nonholonomic systems can be formulated as optimal control problems. Applying the Pontryagins maximum principle, the necessary optimality conditions along with the transversality condition, a boundary value problem is obtained. Finally, a numerical approach to tackle the boundary value problem is required. Here we propose the Lanczos spectral Tau method to obtain an approximate solution of these problems exploiting the Tau toolbox software library, which allows for ease of use as well as accurate results.

Spectral Tau method for solving general fractional order differential equations with linear functional argument

In this paper, a numerical technique for solving new generalized fractional order differential equations with linear functional argument is presented. The spectral Tau method is extended to study this problem, where the derivatives are defined in the Caputo fractional sense. The proposed equation with its functional argument represents a general form of delay and advanced differential equations with fractional order derivatives. The obtained results show that the proposed method is very effective and convenient.

Multi-dimensional spectral tau methods for distributed-order fractional diffusion equations

The distributed-order fractional diffusion equation is a generalization of the standard fractional diffusion equation that can model processes lacking power-law scaling over the whole time-domain. An important application of distributed-order diffusions is to model ultraslow diffusion where a plume of particles spreads at a logarithmic rate. To broaden the range of applicability of distributed-order fractional diffusion models, efficient numerical methods are needed to solve the model equation. In this work, we develop spectral tau schemes to discretize the fractional diffusion equation with distributed-order fractional derivative in time and Dirichlet boundary conditions. The model solution is expanded in multi-dimensions in terms of Legendre polynomials and the discrete equations are obtained with the tau method. Numerical examples are provided to highlight the convergence rate and the flexibility of this approach. The proposed spectral tau methods yield an exponential rate of convergence when the solution is smooth. Our results confirm that nonlocal numerical methods are best suited to discretize distributed-order fractional differential equations as they naturally take the global behavior of the solution into account.

Approximating the solution of integro-differential problems via the spectral Tau method with filtering

Computing approximate solutions of integro-differential problems can be difficult and time consuming, requiring large expertize. Tau Toolbox is a numerical library that produces approximate polynomial solutions of differential, integral and integro-differential equations via the spectral Lanczos' Tau method. This approach, however, may fail to ensure the spectral rate of convergence, and even to reach convergence, whenever the exact solution shows singularities. This paper describes a post-processing phase based on a Frobenius-Padé approximation method to build rational approximations from the polynomial Tau approximation. This filtering extension improves the accuracy of the spectral approximation when working in the vicinity of solutions with singularities. Furthermore, some insights on how to perform such a filtering process on a piecewise version of the Tau method to tackle larger domains and/or stiff problems are offered.

Numerical solution of a class of fractional order integro-differential algebraic equations using Müntz–Jacobi Tau method

This paper presents a spectral Tau method based on Müntz–Jacobi basis functions for approximating the solutions of fractional order Volterra integro-differential algebraic equations. We examine the solvability of the considered equation, and we show that the exact solutions suffer from a singularity at the origin. Numerical solvability and convergence analysis of the proposed approximate approach are also studied, and the error bounds are obtained in a weighted L 2 -norm. Some illustrative examples are presented to validate the theoretical results.

Operational shifted hybrid Gegenbauer functions method d for solving multi-term time fractional differential equations

In this paper, we propose an efficient operational formulation of spectral tau method d for solving multi-term time fractional differential equations with initial-b boundary conditions. The shifted hybrid Gegenbauer functions (SHGFs) operational matrices of Riemann-Liouville fractional integral and Caputo fractional derivatives are presented. By using these operational matrices, the shifted hybrid Gegenbauer functions tau method for both temp oral and spatial discretization are presented, which allow us to introduce an efficient spectral method for solving such problems. Finally, numerical results show good deal with the theoretical analysis.

A Spectral Penalty Method for Two-Sided Fractional Differential Equations with General Boundary Conditions

We consider spectral approximations to the conservative form of the two-sided Riemann-Liouville (R-L) and Caputo fractional differential equations (FDEs) with nonhomogeneous Dirichlet (fractional and classical, respectively) and Neumann (fractional) boundary conditions. In particular, we develop a spectral penalty method (SPM) by using the Jacobi poly-fractonomial approximation for the conservative R-L FDEs while using the polynomial approximation for the conservative Caputo FDEs. We establish the well-posedness of the corresponding weak problems and analyze sufficient conditions for the coercivity of the SPM for different types of fractional boundary value problems. This analysis allows us to estimate the proper values of the penalty parameters at boundary points. We present several numerical examples to verify the theory and demonstrate the high accuracy of SPM, both for stationary and time dependent FDEs. Moreover, we compare the results against a Petrov-Galerkin spectral tau method (PGS-$\tau$, an extension of [Z. Mao, G.E. Karniadakis, SIAM J. Numer. Anal., 2018]) and demonstrate the superior accuracy of SPM for all cases considered.

A Stefan problem with variable thermal coefficients and moving phase change material

This article describes a one-phase Stefan problem in a semi-infinite domain that involves temperature-dependent thermal coefficients and moving phase change material with a speed in the direction of the positive x-axis. The convective boundary condition at a fixed boundary is also considered in the problem. An approximate approach to the problem is discussed to solve the problem with the aid of spectral tau method. The existence and uniqueness of the analytical solution to the problem are also established for a particular case, and the obtained approximate solution is compared with this analytical solution which shows that the approximate results are sufficiently accurate. The impact of a few parameters on the moving interface is also analysed.

On solving fractional logistic population models with applications

The current manuscript focuses on solving fractional logistic population models (FLPMs). The spectral tau method is developed for solving FLPMs with shifted Jacobi polynomials as basis functions. We express the nonlinear terms of such FLPMs as linear expansions in shifted Jacobi polynomials. The proposed method is applied on two real-life applications and compared with other numerical schemes.

Spectral tau algorithm for solving a class of fractional optimal control problems via Jacobi polynomials

This paper is dedicated to analyzing and presenting an efficient numerical algorithm for solving a class of fractional optimal control problems (FOCPs). The basic idea behind the suggested algorithm is based on transforming the FOCP under investigation into a coupled system of fractional-order differential equations whose solutions can be expanded in terms of the Jacobi basis. With the aid of the spectral-tau method, the problem can be reduced into a system of algebraic equations which can be solved via any suitable solver. Some illustrative examples and comparisons are presented aiming to demonstrate the accuracy, applicability, and efficiency of the proposed algorithm.

A time–space spectral tau method for the time fractional cable equation and its inverse problem

This paper is devoted to the numerical solution of the time fractional cable equation and its inverse problem. The time–space spectral Legendre tau method based on the shifted Legendre polynomial and its operational matrices is used to solve the direct problem. Furthermore, we prove that the approximated solution of this method converges to the exact solution. In addition, the inverse problem is formulated by using the Tikhonov regularization, the stability and convergence for the inverse problem are provided, then we analyse the sensitivity coefficients and apply the nonlinear conjugate gradient method to solve the regularized problem. Finally, some numerical results are carried out to support the theoretical claims.

Dealing with Functional Coefficients Within Tau Method

The spectral tau method was originally proposed by Lanczos for the solution of linear differential problems with polynomial coefficients. In this contribution we present three approaches to tackle integro-differential equations with non-polynomial coefficients: (i) making use of functions of matrices, (ii) applying orthogonal interpolation and (iii) solving auxiliary differential problems by the tau method itself. These approaches are part of the Tau Toolbox efforts for deploying a numerical library for the solution of integro-differential problems. Numerical experiments illustrate the use of all these polynomial approximations in the context of the tau method.

Spectral Tau Algorithm for Certain Coupled System of Fractional Differential Equations via Generalized Fibonacci Polynomial Sequence

This paper concerns new numerical solutions for certain coupled system of fractional differential equations through the employment of the so-called generalized Fibonacci polynomials. These polynomials include two parameters and they generalize some important well-known polynomials such as Fibonacci, Pell, Fermat, second kind Chebyshev, and second kind Dickson polynomials. The proposed numerical algorithm is essentially built on applying the spectral tau method together with utilizing a Fejer quadrature formula. For the implementation of our algorithm, we introduce a new operational matrix of fractional-order differentiation of generalized Fibonacci polynomials. A careful investigation of convergence and error analysis of the proposed generalized Fibonacci expansion is performed. The robustness of the proposed algorithm is tested through presenting some numerical experiments.

A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations

In this study, a Legendre spectral tau method is revisited to handle the multi-term time-fractional diffusion equations (MTT-FDEs). An error estimate and rigorous convergence analysis are carried out using some critical theorems. The applicability and accuracy of the solution method are demonstrated by a numerical example to verify the theoretical analysis.

On solving systems of multi-pantograph equations via spectral tau method

The current manuscript focuses on solving systems of multi-pantograph equations. The spectral tau method is applied for solving systems of multi-pantograph equations with shifted Jacobi polynomials as basis functions. The convergence analysis of the proposed technique is also investigated. We introduced the numerical solutions of some test problems and compared the obtained numerical solutions of such problems with those given using different numerical methods.

Bivariate Legendre approximation

Spectral methods are among the numerical methods commonly used for approximating solutions of boundary value problems. In this paper we propose, a generalization of the spectral Tau method in dimension 2, this method is generalized by the use of a new two-dimensional polynomial basis constructed by a three terms recurrence relation. We also present an estimation of error committed by the proposed method.

Design and Stability Analysis of an Integral Time-Delay Feedback Control Combined With an Open-Loop Control for an Infinitely Variable Transmission System

The kinematic model of an infinitely variable transmission (IVT) is introduced, and the nonlinear differential equation for the dynamic model of the IVT system with a permanent magnetic direct current (DC) motor and a magnetic brake is derived. To make the average of the input speed converge to a desired constant for any input power and output load, an integral time-delay feedback control combined with an open-loop control is used to adjust the speed ratio of the IVT. The speed ratio for the open-loop control is obtained by a modified incremental harmonic balance (IHB) method. Existence and convergence of a periodic solution are proved under a condition for parameters of the IVT system, and uniqueness of the periodic solution is proved by converting the nonlinear differential equation to a new differential equation that is Lipchitz in the dependent variable and piecewise continuous in the independent variable. A time-delay variable that is an approximation of the average of the input speed is used as the feedback to control the changing rate of the speed ratio. The IVT system with the time-delay control variable can be converted to a distributed-parameter system. Thus, the spectral Tau method is used to design the time-delay feedback control so that the IVT system is locally exponentially stable. The static error from the open-loop control is eliminated; the feedback control variable with time-delay is smoother than that without time-delay, which yields a lower control effort and more robust control design, since the time-delay variable that acts as a low-pass filter reduces the effect of the instantaneous change of the IVT system.

Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations

The principal aim of the current paper is to present and analyze two new spectral algorithms for solving some types of linear and nonlinear fractional-order differential equations. The proposed algorithms are obtained by utilizing a certain kind of shifted Chebyshev polynomials called the shifted fifth-kind Chebyshev polynomials as basis functions along with the application of a modified spectral tau method. The class of fifth-kind Chebyshev polynomials is a special class of a basic class of symmetric orthogonal polynomials which are constructed with the aid of the extended Sturm–Liouville theorem for symmetric functions. An investigation for the convergence and error analysis of the proposed Chebyshev expansion is performed. For this purpose, a new connection formulae between Chebyshev polynomials of the first and fifth kinds are derived. The obtained numerical results ascertain that our two proposed algorithms are applicable, efficient and accurate.

B-spline wavelet operational method for numerical solution of time-space fractional partial differential equations

In this paper, the linear B-spline scaling functions and wavelets operational matrix of fractional integration are derived. A new approach implementing the linear B-spline scaling functions and wavelets operational matrices combining with the spectral tau method is introduced for approximating the numerical solutions of time-space fractional partial differential equations with initial-boundary conditions. They are utilized to reduce the main problem to a system of algebraic equations. The uniform convergence analysis for the linear B-spline scaling functions and wavelets expansion and an efficient error estimation of the presented method are also introduced. Illustrative examples are given and numerical results are presented to demonstrate the efficiency and accuracy of the proposed method. Special attention is given to a comparison between the numerical results obtained by our new technique and those found by other known methods.

Uncertain viscoelastic models with fractional order: A new spectral tau method to study the numerical simulations of the solution

The analysis of the behaviors of physical phenomena is important to discover significant features of the character and the structure of mathematical models. Frequently the unknown parameters involve in the models are assumed to be unvarying over time. In reality, some of them are uncertain and implicitly depend on several factors. In this study, to consider such uncertainty in variables of the models, they are characterized based on the fuzzy notion. We propose here a new model based on fractional calculus to deal with the Kelvin–Voigt (KV) equation and non-Newtonian fluid behavior model with fuzzy parameters. A new and accurate numerical algorithm using a spectral tau technique based on the generalized fractional Legendre polynomials (GFLPs) is developed to solve those problems under uncertainty. Numerical simulations are carried out and the analysis of the results highlights the significant features of the new technique in comparison with the previous findings. A detailed error analysis is also carried out and discussed.