Shifted Jacobi Polynomials Research Articles

A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations

This article adapts an operational matrix formulation of the collocation method for the one- and two-dimensional nonlinear fractional sub-diffusion equations (FSDEs). In the proposed collocation approach, the double and triple shifted Jacobi polynomials are used as base functions for approximate solutions of the one- and two-dimensional cases. The space and time fractional derivatives given in the underline problems are expressed by means of Jacobi operational matrices. This investigates spectral collocation schemes for both temporal and spatial discretizations. Thereby, the expansion coefficients are then determined by reducing the FSDEs, with their initial and boundary conditions, into systems of nonlinear algebraic equations which are far easier to be solved. Furthermore, the error of the approximate solution is estimated theoretically along with graphical analysis to confirm the exponential convergence rate of the proposed method in both spatial and temporal discretizations. In order to show the high accuracy of our algorithms, we report the numerical results of some numerical examples and compare our numerical results with those reported in the literature.

Numerical solution of fractional delay differential equation by shifted Jacobi polynomials

ABSTRACTIn this paper, the fractional delay differential equation (FDDE) is considered for the purpose to develop an approximate scheme for its numerical solutions. The shifted Jacobi polynomial scheme is used to solve the results by deriving operational matrix for the fractional differentiation and integration in the Caputo and Riemann–Liouville sense, respectively. In addition to it, the Jacobi delay coefficient matrix is developed to solve the linear and nonlinear FDDE numerically. The error of the approximate solution of proposed method is discussed by applying the piecewise orthogonal technique. The applicability of this technique is shown by several examples like a mathematical model of houseflies and a model based on the effect of noise on light that reflected from laser to mirror. The obtained numerical results are tabulated and displayed graphically.

A Jacobi Gauss–Lobatto and Gauss–Radau collocation algorithm for solving fractional Fokker–Planck equations

In this article, we construct a new numerical approach for solving the time-fractional Fokker–Planck equation. The shifted Jacobi polynomials are used as basis functions, and the fractional derivative is described in the sense of Caputo. The proposed approach is a combination of shifted Jacobi Gauss–Lobatto scheme for the spatial discretization and the shifted Jacobi Gauss–Radau scheme for temporal approximation. The problem is then reduced to a problem consisting of a system of algebraic equations that greatly simplifies the problem. In addition, our numerical algorithm is also applied for solving the space-fractional Fokker–Planck equation and the time–space-fractional Fokker–Planck equation. Numerical results are consistent with the theoretical analysis, indicating the high accuracy and effectiveness of the proposed algorithm.

Numerical Computation of Travelling Wave Solutions for the Nonlinear Ito System

The Ito equation (a coupled nonlinear wave equation which generalizes the KdV equation) has previously been shown to admit a reduction to a single nonlinear Casimir equation governing the wave solutions. Some analytical properties of the solutions to this equation in certain parameter regimes have been studied recently. However, for general parameter regimes where the analytical approach is not so useful, a numerical method would be desirable. Therefore, in this paper, we proceed to show that the Casimir equation for the Ito system can be solved numerically by use of the shifted Jacobi-Gauss collocation (SJC) spectral method. First, we present the general solution method, which is follows by implementation of the method for specific parameter values. The presented r esults in this article demonstrate the accuracy and efficiency of the metho d. In particular, we demonstrate that relatively few notes p ermit very low residual errors in the approximate numerical solutions. We are also able to show that the coefficients of the higher order terms in the shifted Jacobi polynomials decrease exponentially, meaning that accurate solutions can be obtained after relatively few terms are used. With this, we have a numerical method which can accurately and efficiently capture the behavior of nonlinear waves in the I to equation.

A new operational approach for numerical solution of generalized functional integro-differential equations

In this paper, a class of linear and nonlinear functional integro-differential equations are considered that can be found in the various fields of sciences such as: stress–strain states of materials, motion of rigid bodies and models of polymer crystallization. The operational collocation method with shifted Jacobi polynomial bases is applied to approximate the solution of these equations. In addition, some theoretical results are given to simplify and reduce the computational costs. Finally, some numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.

The use of Jacobi polynomials in the numerical solution of coupled system of fractional differential equations

In this paper, we study shifted Jacobi polynomials and develop a simple but highly accurate scheme for the numerical solution of coupled system of fractional differential equations. We derive some operational matrices of integration and differentiation of fractional order. By the application of these matrices we provide a theoretical treatment to approximate the solutions of the corresponding system. We use Matlab to perform necessary operations. The applicability of the technique is shown with some examples and the results are displayed graphically.

A Shifted Jacobi-Gauss Collocation Scheme for Solving Fractional Neutral Functional-Differential Equations

The shifted Jacobi-Gauss collocation (SJGC) scheme is proposed and implemented to solve the fractional neutral functional-differential equations with proportional delays. The technique we have proposed is based upon shifted Jacobi polynomials with the Gauss quadrature integration technique. The main advantage of the shifted Jacobi-Gauss scheme is to reduce solving the generalized fractional neutral functional-differential equations to a system of algebraic equations in the unknown expansion. Reasonable numerical results are achieved by choosing few shifted Jacobi-Gauss collocation nodes. Numerical results demonstrate the accuracy, and versatility of the proposed algorithm.

Shifted Jacobi tau method for solving the space fractional diffusion equation

In this paper, approximation techniques based on the shifted Jacobi together with spectral tau technique are presented to solve a class of initial-boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain. The fractional derivatives are described in the Caputo sense. The technique is derived by expanding the required approximate solution as the elements of shifted Jacobi polynomials. Using the operational matrix of the fractional derivative, the problem can be reduced to a set of linear algebraic equations. Numerical examples are included to demonstrate the validity and applicability of the technique and a comparison is made with the existing results to show that the proposed method is easy to implement and produce accurate results.

Efficient Jacobi-Gauss collocation method for solving initial value problems of Bratu type

In this paper, we propose the shifted Jacobi-Gauss collocation spectral method for solving initial value problems of Bratu type, which is widely applicable in fuel ignition of the combustion the� ory and heat transfer. The spatial approximation is based on shifted Jacobi polynomials with α, β ∈ (-1, ∞), x ∈ (0, 1) and n the polynomial degree. The shifted Jacobi-Gauss points are used as collocation nodes. Illustrative examples have been discussed to demonstrate the validity and applica� bility of the proposed technique. Comparing the numerical results of the proposed method with some wellknown results show that the method is ef ficient and gives excellent numerical results.

Errata and comments on “Generic orthogonal moments: Jacobi–Fourier moments for invariant image description”

Ping et al. [Z. Ping, H. Ren, J. Zou, Y. Sheng, and W. Bo, Generic orthogonal moments: Jacobi–Fourier moments for invariant image description, Pattern Recognition 40 (4) (2007) 1245–1254] made a landmark contribution to the theory of two-dimensional orthogonal moments confined to the unit disk by unifying the radial kernels of existing polynomial-based circular orthogonal moments under the roof of shifted Jacobi polynomials. However, the work contains some errata that result mainly from the confusion between the two slightly different definitions of shifted Jacobi polynomials in the literature. Taking into account the great importance and the high impact of the work in the pattern recognition community, this paper points out the confusing points, corrects the errors, and gives some other relevant comments. The corrections developed in this paper are illustrated by some experimental evidence.

On shifted Jacobi spectral approximations for solving fractional differential equations

In this paper, a new formula of Caputo fractional-order derivatives of shifted Jacobi polynomials of any degree in terms of shifted Jacobi polynomials themselves is proved. We discuss a direct solution technique for linear multi-order fractional differential equations (FDEs) subject to nonhomogeneous initial conditions using a shifted Jacobi tau approximation. A quadrature shifted Jacobi tau (Q-SJT) approximation is introduced for the solution of linear multi-order FDEs with variable coefficients. We also propose a shifted Jacobi collocation technique for solving nonlinear multi-order fractional initial value problems. The advantages of using the proposed techniques are discussed and we compare them with other existing methods. We investigate some illustrative examples of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.

A Confirmed Model to Polymer Core-Shell Structured Nanofibers Deposited via Coaxial Electrospinning

A model to core-shell structured polymer nanofibers deposited via coaxial electrospinning is presented. Investigations are based on a modified Jacobi-Gauss collocation spectral method, proposed along with the Boubaker Polynomials Expansion Scheme (BPES), for providing solution to a nonlinear Lane-Emden-type equation. The spatial approximation has been based on shifted Jacobi polynomials with was n the polynomial degree. The Boubaker Polynomials Expansion Scheme (BPES) main features, concerning the embedded boundary conditions, have been outlined. The modified Jacobi-Gauss points are used as collocation nodes. Numerical examples are included to demonstrate the validity and applicability of the technique, and a comparison is made with existing results. It has been revealed that both methods are easy to implement and yield very accurate results.

A Jacobi–Gauss collocation method for solving nonlinear Lane–Emden type equations

In this paper, a shifted Jacobi–Gauss collocation spectral method is proposed for solving the nonlinear Lane–Emden type equation. The spatial approximation is based on shifted Jacobi polynomials P T , n ( α , β ) ( x ) with α, β ∈ (−1, ∞), T > 0, and n is the polynomial degree. The shifted Jacobi–Gauss points are used as collocation nodes. Numerical examples are included to demonstrate the validity and applicability of the technique and a comparison is made with existing results. The method is easy to implement and yields very accurate results.

APROKSIMASI DISTRIBUSI WAKTU HIDUP YANG AKAN DATANG

This paper give an analitical technique to approximate future lifetime distributions. Approximations of the future lifetime distribution based on the shifted Jacobi polynomials, andit yielded the sequences of a exponentials combination. The results of approximations of the future lifetime distribution in this cases study based on Makeham’s Law. It is very accurate inthe case study.

On the use of shifted Jacobi polynomials in accurate evaluation of roots and weights of Rys polynomials

In this paper it is shown that shifted Jacobi polynomials G(n)(p,q,x) can be used in connection with the Gaussian quadrature modified moment technique to greatly enhance the accuracy of evaluation of Rys roots and weights used in Gaussian integral evaluation in quantum chemistry. A general four-term inhomogeneous recurrence relation is derived for the shifted Jacobi polynomial modified moments over the Rys weight function e(-Tx)/square root x. It is shown that for q=1/2 this general four-term inhomogeneous recurrence relation reduces to a three-term p-dependent inhomogeneous recurrence relation. Adjusting p to proper values depending on the Rys exponential parameter T, the method is capable of delivering highly accurate results for large number of roots and weights in the most difficult to treat intermediate T range. Examples are shown, and detailed formulas together with practical suggestions for their efficient implementation are also provided.

General methods for determining the droplet size distribution in emulsion systems

We present a general method that allows us to figure out the size distribution of an isolated collection of droplets of dilute emulsion system using nuclear magnetic resonance pulsed gradient spin echo measurements. We show that the expression to obtain the volume fraction distribution function is equivalent to a Fredholm integral equation of the first kind. We prove, using the Dirac notation, that a solution of this equation can be easily found if its kernel has a complete biorthogonal system of eigenvectors. Two numerical procedures are discussed. The first, termed indirect, is based on the expansion of the unknown distribution function in the eigenfunctions of the kernel. The second one, called direct, uses the properties of shifted Legendre polynomials to integrate numerically the integral equation and evaluates the unknown distribution by means of a constrained least square procedure. The computational limits are analyzed. To extract the distribution’s form directly by experimental data we have constructed a generating function using the shifted Jacobi polynomials. The procedures have been tested on simulated and experimental data and appear to be a powerful and flexible method to obtain the size distribution function directly by the experimental data.

Orthogonal collocation in the nonconforming boundary element method

This paper outlines the use of non-conforming (discontinuous) elements in the collocation boundary element method for solving two-dimensional potential and Poisson type problems. The roots of an orthogonal polynomial (shifted Jacobi polynomial) are used as the collocation points. This results in increased accuracy due to the least square minimization property of the orthogonal polynomials. The advantage of using non-conforming elements is realized when the method is applied (i) to problems with singularities (both due to geometry and boundary conditions) and (ii) in conjunction with domain decomposition techniques. Also, the collocation points can be relocated within an element by changing two user-defined parameters in the shifted Jacobi polynomial, thus providing an error indicator which can be used for mesh refinement purposes. This technique, called the rh method, is discussed and illustrated. The results obtained by using non-conforming boundary elements for standard test problems are shown to be accurate.

Evaluation of Bessel function integrals with algebraic singularities

A new method for the numerical evaluation of ∫ 1 0(1-x) αx βJ v(ax)dx is proposed, where, α, β, ν, a are given constants, and Jν is the Bessel function of the first kind and of order ν. It is assumed that f is a smooth function. The method is based on approximation of the function f by a polynomial given in the form of a linear combination of the shifted Jacobi polynomials R( α,β+ν) k , and on the use of Chebyshev series expansion for the function J ν ( ax).

Shifted-Jacobi series direct method for variational problems

The operational matrix or integration of a Jacobi vector whose elements are Jacobi polynomials is introduced and applied to solve variational problems by a direct method. Illustrative examples are given to demonstrate that only a small number of shifted-Jacobi polynomials are needed to obtain an accurate solution.

On Kutt's Gaussian quadrature rule for finite-part integrals

The simple remark that Kutt's Gaussian quadrature rule for the computation of divergent integrals of the form ▪ (with λ ⩽ - 1), defined in the sense of finite-part integrals, coincides with the Gauss-Jacobi quadrature rule and the corresponding orthogonal polynomials with shifted Jacobi polynomials is made. This remark is based on the well-known orthogonality properties of Jacobi polynomials and no analysis is required. Yet, it seems that the Gauss-Jacobi quadrature rule for the above class (or more general classes) of finite-part integrals was used for the first time by Kutt.