Jacobi Spectral Collocation Method Research Articles

Jacobi spectral collocation method of space-fractional Navier-Stokes equations

In this paper, we study the Jacobi spectral collocation method for two-dimensional space-fractional Navier-Stokes equations with Laplacian and fractional Laplacian. We first derive modified fractional differentiation matrices to accommodate the singularity in two dimensions and verify the boundedness of its spectral radius. Next, we construct a fully discrete scheme for the space-fractional Navier-Stokes equations, combined with the first-order implicit-explicit Euler time-stepping scheme at the Jacobi-Gauss-Lobatto collocation points. Through some two-dimensional numerical examples, we present the influence of different parameters in the equations on numerical errors. Various numerical examples verify the effectiveness of our method and suggest the smoothness of the solution for further regularity analysis.

Implicitly linear Jacobi spectral-collocation methods for two-dimensional weakly singular Volterra-Hammerstein integral equations

Weakly singular Volterra integral equations of the second kind typically have nonsmooth solutions near the initial point of the interval of integration, which seriously affects the accuracy of spectral methods. We present Jacobi spectral-collocation method to solve two-dimensional weakly singular Volterra-Hammerstein integral equations based on smoothing transformation and implicitly linear method. The solution of the smoothed equation is much smoother than the original one after smoothing transformation and the spectral method can be used. For the nonlinear Hammerstein term, the implicitly linear method is applied to simplify the calculation and improve the accuracy. The weakly singular integral term is discretized by Jacobi Gauss quadrature formula which can absorb the weakly singular kernel function into the quadrature weight function and eliminate the influence of the weakly singular kernel on the method. Convergence analysis in the L∞-norm is carried out and the exponential convergence rate is obtained. Finally, we demonstrate the efficiency of the proposed method by numerical examples.

A reduced-order Jacobi spectral collocation method for solving the space-fractional FitzHugh–Nagumo models with application in myocardium

PurposeThe space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is the main aim of most researchers. This paper devotes to developing a novel spectral algorithm to solve the FitzHugh–Nagumo models with space fractional derivatives.Design/methodology/approachThe fractional derivative is defined based upon the Riesz derivative. First, a second-order finite difference formulation is used to approximate the time derivative. Then, the Jacobi spectral collocation method is employed to discrete the spatial variables. On the other hand, authors assume that the approximate solution is a linear combination of special polynomials which are obtained from the Jacobi polynomials, and also there exists Riesz fractional derivative based on the Jacobi polynomials. Also, a reduced order plan, such as proper orthogonal decomposition (POD) method, has been utilized.FindingsA fast high-order numerical method to decrease the elapsed CPU time has been constructed for solving systems of space fractional PDEs.Originality/valueThe spectral collocation method is combined with the POD idea to solve the system of space-fractional PDEs. The numerical results are acceptable and efficient for the main mathematical model.

Applications of Fractional Differentiation Matrices in Solving Caputo Fractional Differential Equations

This paper pursues obtaining Jacobi spectral collocation methods to solve Caputo fractional differential equations numerically. We used the shifted Jacobi–Gauss–Lobatto or Jacobi–Gauss–Radau quadrature nodes as the collocation points and derived the fractional differentiation matrices for Caputo fractional derivatives. With the fractional differentiation matrices, the fractional differential equations were transformed into linear systems, which are easier to solve. Two types of fractional differential equations were used for the numerical simulations, and the numerical results demonstrated the fast convergence and high accuracy of the proposed methods.

An Efficient Jacobi Spectral Collocation Method with Nonlocal Quadrature Rules for Multi-Dimensional Volume-Constrained Nonlocal Models

In this paper, an efficient Jacobi spectral collocation method is developed for multi-dimensional weakly singular volume-constrained nonlocal models including both nonlocal diffusion (ND) models and peridynamic (PD) models. The model equation contains a weakly singular integral operator with the singularity located at the center of the integral domain, and the numerical approximation of it becomes an essential difficulty in solving nonlocal models. To approximate such singular nonlocal integrals in an accurate way, a novel nonlocal quadrature rule is constructed to accurately compute these integrals for the numerical solutions produced by spectral methods. Numerical experiments are given to show that spectral accuracy can be obtained by using the proposed Jacobi spectral collocation methods for several different nonlocal models. Besides, we numerically verify that the numerical solution of our Jacobi spectral method can converge to its correct local limit as the nonlocal interactions vanish.

Tanh Jacobi spectral collocation method for the numerical simulation of nonlinear Schrödinger equations on unbounded domain

We present a class of orthogonal functions on infinite domain based on Jacobi polynomials. These functions are generated by applying a tanh transformation to Jacobi polynomials. We construct interpolation and projection error estimates using weighted pseudo‐derivatives tailored to the involved mapping. Then, using the nodes of the newly introduced tanh Jacobi functions, we develop an efficient spectral tanh Jacobi collocation method for the numerical simulation of nonlinear Schrödinger equations on the infinite domain without using artificial boundary conditions. The applicability and accuracy of the solution method are demonstrated by two numerical examples for solving the nonlinear Schrödinger equation and the nonlinear Ginzburg–Landau equation.

Convergence analysis of Jacobi spectral collocation methods for weakly singular nonlocal diffusion equations with volume constraints

This paper considers efficient spectral solutions for weakly singular nonlocal diffusion equations with Dirichlet-type volume constraints. The equation we consider contains an integral operator that typically has a singularity at the midpoint of the integral domain, and the approximation of the integral operator is one of the essential difficulties in solving nonlocal equations. To overcome this problem, two-sided Jacobi spectral quadrature rules are proposed to develop a Jacobi spectral collocation method for nonlocal diffusion equations. A rigorous convergence analysis of the proposed method with the L∞ norm is presented, and we further prove that the Jacobi collocation solution converges to its corresponding local limit as nonlocal interactions vanish. Numerical examples are given to verify the theoretical results.

Spectral Collocation Methods for Fractional Integro‐Differential Equations with Weakly Singular Kernels

In this paper, we propose and analyze a spectral approximation for the numerical solutions of fractional integro‐differential equations with weakly kernels. First, the original equations are transformed into an equivalent weakly singular Volterra integral equation, which possesses nonsmooth solutions. To eliminate the singularity of the solution, we introduce some suitable smoothing transformations, and then use Jacobi spectral collocation method to approximate the resulting equation. Later, the spectral accuracy of the proposed method is investigated in the infinity norm and weighted L2 norm. Finally, some numerical examples are considered to verify the obtained theoretical results.

Solving Nonlinear Volterra-Fredholm Integral Equations using an Accurate Spectral Collocation Method

Abstract In this paper, we present a Jacobi spectral collocation method to solve nonlinear Volterra-Fredholm integral equations with smooth kernels. The main idea in this approach is to convert the original problem into an equivalent one through appropriate variable transformations so that the resulting equation can be accurately solved using spectral collocation at the Jacobi-Gauss points. The convergence and error analysis are discussed for both L ∞ and weighted L 2 norms. We confirm the theoretical prediction of the exponential rate of convergence by the numerical results which are compared with well-known methods.

Effect of vaccination on the transmission dynamics of COVID-19 in Ethiopia

Governments and health officials are eager to gain a thorough understanding of the dynamics of COVID-19 transmission in order to devise strategies to mitigate the pandemic’s negative effects. As a result, we created a new fractional order mathematical model to investigate the dynamics of Covid-19 vaccine transmission in Ethiopia. The nonlinear system of differential equations for the model is represented using Atangana–Baleanu fractional derivative in Caputo sense and the Jacobi spectral collocation method is used to convert this system into an algebraic system of equations, which is then solved using inexact Newton’s method. The fundamental reproduction number, R0 for the proposed model is determined using the next generation matrix approach.

Solving a class of local and nonlocal elliptic boundary value problems arising in heat transfer

AbstractThis paper deals with a class of Bratu's type, Troesch's, and nonlocal elliptic boundary value problems arising in the heat transfer process. Due to the strong nonlinearity and presence of parameter , it is very difficult to solve these problems analytically as well as numerically. By using the Jacobi spectral collocation method, these problems are solved fruitfully. We have shown the numerical as well as theoretical convergence of the suggested scheme. Numerical results are presented through figures and tables, demonstrating the accuracy of the scheme. Results are compared with some known methods to highlight its neglectable error.

Non-polynomial B-spline and shifted Jacobi spectral collocation techniques to solve time-fractional nonlinear coupled Burgers\u2019 equations numerically

This paper proposes two numerical approaches for solving the coupled nonlinear time-fractional Burgers’ equations with initial or boundary conditions on the interval [0, L]. The first method is the non-polynomial B-spline method based on L1-approximation and the finite difference approximations for spatial derivatives. The method has been shown to be unconditionally stable by using the Von-Neumann technique. The second method is the shifted Jacobi spectral collocation method based on an operational matrix of fractional derivatives. The proposed algorithms’ main feature is that when solving the original problem it is converted into a nonlinear system of algebraic equations. The efficiency of these methods is demonstrated by applying several examples in time-fractional coupled Burgers equations. The error norms and figures show the effectiveness and reasonable accuracy of the proposed methods.

Jacobi Spectral Collocation Technique for Time-Fractional Inverse Heat Equations

In this paper, we introduce a numerical solution for the time-fractional inverse heat equations. We focus on obtaining the unknown source term along with the unknown temperature function based on an additional condition given in an integral form. The proposed scheme is based on a spectral collocation approach to obtain the two independent variables. Our approach is accurate, efficient, and feasible for the model problem under consideration. The proposed Jacobi spectral collocation method yields an exponential rate of convergence with a relatively small number of degrees of freedom. Finally, a series of numerical examples are provided to demonstrate the efficiency and flexibility of the numerical scheme.

Method of lines for multi-dimensional coupled viscous Burgers’ equations via nodal Jacobi spectral collocation method

This paper deals with the multi-dimensional coupled viscous Burgers’ equations, using the method of lines (MOL). Indeed, the solutions are approximated by their Lagrange interpolation on a set of Jacobi-type nodes. Then, an appropriate differentiation matrix is used to approximate the first and the second partial derivatives with respect to spatial variables. This procedure approximates the original PDE problem with an ODE system for the time variable. Then the resulting ODE is solved with an existing ODE-solver. This MOL approach is implemented for 1D, 2D and 3D coupled viscous Burgers’ equations. The numerical test shows efficient numerical accuracy with fewer nodes in comparison with some previous methods.

A Jacobi Spectral Collocation Method for Solving Fractional Integro-Differential Equations

A Jacobi Spectral Collocation Method for Solving Fractional Integro-Differential Equations

An hp-Version Jacobi Spectral Collocation Method for the Third-Kind VIEs

In this paper, we present an hp-version Jacobi spectral collocation method for the third-kind VIEs. We establish several new approximation results of the Jacobi polynomial interpolations. In addition, we also derive the convergence of the hp-version of Jacobi spectral collocation method under the $$L^{\infty }$$ -norm. Numerical experiments confirm the theoretical expectations.

Jacobi spectral collocation method for solving fractional pantograph delay differential equations

In this study, we mainly discuss the application of the Jacobi collocation method to a class of fractional-order pantograph delay differential equations. We first convert the problem to a nonlinear Volterra integral equation with a weakly singular kernel. Under reasonable assumptions of nonlinearity, the existence and uniqueness of the obtained integral equation are derived. Then, we apply a numerical scheme based on the Jacobi collocation approximation to solve the equivalent integral equation. Furthermore, an error analysis for the numerical scheme is performed. Finally, numerical examples are presented to validate our theoretical analysis.

Numerical Investigation of the Fractional-Order Liénard and Duffing Equations Arising in Oscillating Circuit Theory

In this paper, we present the Jacobi spectral collocation method to solve the fractional model of Lienard and Duffing equations with the Liouville-Caputo fractional derivative. These equations are the generalisation of the spring-mass system equation and describe the oscillating circuit. The main reason for using this technique is high accuracy and low computational cost compared to some other methods. The main solution behaviours of these equations are due to fractional orders which are explained graphically. The convergence analysis of the proposed method is also provided. A comparison is made between the exact and approximate solutions.

Efficient Method for Fractional Le ́vy-Feller Advection- Dispersion Equation Using Jacobi Polynomials

In this paper, a novel formula expressing explicitly the fractional-order derivatives, in the sense of Riesz-Feller operator, of Jacobi polynomials is presented. Jacobi spectral collocation method together with trapezoidal rule are used to reduce the fractional L\'{e}vy-Feller advection-dispersion equation (LFADE) to a system of algebraic equations which greatly simplifies solving like this fractional differential equation. Numerical simulations with some comparisons are introduced to confirm the effectiveness and reliability of the proposed technique for the L\'{e}vy-Feller fractional partial differential equations.

The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutions

<p style='text-indent:20px;'>In recent years, many numerical methods have been extended to fractional integro-differential equations. But most of them ignore an important problem. Even if the input function is smooth, the solutions of these equations would exhibit some weak singularity, which leads to non-smooth solutions, and a deteriorate order of convergence. To overcome this problem, we first study in detail the singularity of the fractional integro-differential equation, and then eliminate the singularity by introducing some smoothing transformation. We can maximize the convergence rate by adjusting the parameters in the auxiliary transformation. We use the Jacobi spectral-collocation method with global and high precision characteristics to solve the transformed equation. A comprehensive and rigorous error estimation under the <inline-formula><tex-math id="M2">$ L^{\infty} $</tex-math></inline-formula>- and <inline-formula><tex-math id="M3">$ L^{2}_{\omega^{\alpha, \beta}} $</tex-math></inline-formula>-norms is derived. Finally, we give specific numerical examples to show the accuracy of the theoretical estimation and the feasibility and effectiveness of the proposed method.</p>