Shifted Jacobi Collocation Research Articles

Numerical and computational analysis of fractional order mathematical models for chemical kinetics and carbon dioxide absorbed into phenyl glycidyl ether

The major objective of this effort is to use the collocation technique (CT) to examine fractional chemical kinetics(CK) and other problem that correlates the condensations of carbon dioxide (CO2) and phenyl glycidyl ether (PGE) with two varieties of Dirichlet and a mixed set of Neumann boundary and Dirichlet-type constraints respectively. The suggested approach depends on the shifted Jacobi collocation techniques along with the shifted Jacobi operational matrix for fractional derivatives of any order, defined in the Caputo sense. Examining a global approximation for temporal and spatial discretizations is the main benefit of the suggested methodology. Additionally, the mathematical method simplifies the fractional differential equations by reducing them to a straightforward problem that just needs the solution of a set of algebraic equations. The arithmetical results and figures show that the suggested approach is an effective algorithm with excellent accuracy for resolving arbitrary order differential equations. A few theorems regarding error analysis are presented and explained. We also compare the numerical results produced using the suggested technique with those results obtained through the existing techniques.

Compatibility of the Paraskevopoulos’s algorithm with operational matrices of Vieta–Lucas polynomials and applications

In this study, the numerically stable operational matrices are proposed to approximate the Caputo fractional-order derivatives by introducing an algorithm. The proposed operational matrices are named fractional Vieta–Lucas differentiation matrices that are constructed by using the basis of shifted Vieta–Lucas polynomials (VLPs) and Caputo-fractional derivatives. In addition, we numerically solve the multi-order linear and nonlinear fractional-order differential equations by introducing a new numerical algorithm that is based on Paraskevopoulos’s algorithm together with the newly proposed operational matrices of shifted VLPs. The applicability of Paraskevopoulos’s algorithm was previously studied with the Adomian decomposition technique to solve fractional-order ordinary differential equations. We extend its applicability to the operational matrices technique. However, to the best of our knowledge, no previous study has reported that discusses the applicability of Paraskevopoulos’s algorithm with the operational matrices of shifted VLPs. To demonstrate the advantages of the newly proposed numerical algorithm, the multi-order linear and nonlinear Caputo fractional-order differential equations are solved numerically. The solutions of the first, third, fourth, and fifth examples obtained by using the proposed algorithm are compared with the solutions obtained otherwise by using various numerical approaches including stochastic approach, Taylor matrix method, Bessel collocation method, shifted Jacobi collocation method, spectral Tau method, and Chelyshkov collocation method. It is shown that the proposed numerical algorithm and the fractional Vieta–Lucas differentiation matrices are highly efficient in solving all the aforementioned examples.

Shifted Jacobi collocation method for Volterra-Fredholm integral equation

In this paper, we evaluate the approximate numerical solution for the Volterra-Fredholm integral equation (V-FIE) using the shifted Jacobi collocation (SJC) method. This method depends on the operational matrices. We present some properties of the shifted Jacobi polynomials. These properties together with the shifted Jacobi polynomials transform the Volterra-Fredholm integral equation into a system of algebraic equations in the expansion coefficients of the solution. We discuss the convergence and error analysis of the shifted Jacobi polynomials in detail. The efficiency of this method is verified through numerical examples and compared with others.

Numerical solutions of two-dimensional nonlinear fractional Volterra and Fredholm integral equations using shifted Jacobi operational matrices via collocation method

In this paper, an efficient numerical method is presented to approximate solutions of two-dimensional nonlinear fractional Volterra and Fredholm integral equations. We derive new operational matrices of fractional-order integration and product based on two-variable shifted Jacobi polynomials. These operational matrices via shifted Jacobi collocation method are utilized to reduce the understudy equations to systems of linear or nonlinear algebraic equations. Then, the arising systems can be solved by the Newton method. Discussion on the error bound and convergence analysis of the proposed method is presented. The efficiency, accuracy, and validity of the presented method are demonstrated by its application to three test examples and by comparing our results with the results obtained by existing methods in the literature.

Jacobi spectral discretization for nonlinear fractional generalized seventh-order KdV equations with convergence analysis

We developed a numerical scheme to solve the nonlinear fractional (NF) generalized seventh order KdV (sKdV) equation with time-space-fractional using the shifted Jacobi collocation method. Basically, a time-space collocation approximation for temporal and spatial discretizations is employed efficiently to tackle these equations. The convergence and stability analyses of the suggested basis functions are presented in-depth. The validity and efficiency of the proposed method are investigated and verified through numerical examples.

Numerical treatment of non-integer order partial differential equations by omitting discretization of data

In this manuscript, we derived the shifted Jacobi operational matrices of fractional derivatives and integration which are applied for numerical solution of general linear multi-term fractional partial differential equations (FPDEs). A new approach implementing shifted Jacobi operational matrix without using the shifted Jacobi collocation technique is introduced for the numerical solution of multi-term FPDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. Further the proposed method needs no discretization of data. The proposed method is applied for solving linear multi-term FPDEs subject to initial conditions and the approximate solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the considered method with those found by other known methods like homotopy analysis (HAM) method. For computation purposes, we use Matlab 2016.

Jacobi collocation scheme for variable-order fractional reaction-subdiffusion equation

We developed a numerical scheme to solve the variable-order fractional linear subdiffusion and nonlinear reaction-subdiffusion equations using the shifted Jacobi collocation method. Basically, a time-space collocation approximation for temporal and spatial discretizations is employed efficiently to tackle these equations. The convergence and stability analyses of the suggested basis functions are presented in-depth. The validity and efficiency of the proposed method are investigated and verified through numerical examples.

Numerical solution of linear and nonlinear hyperbolic telegraph type equations with variable coefficients using shifted Jacobi collocation method

The telegraph equation is one of the important equations of mathematical physics. In this work, a spectral collocation scheme is proposed for the numerical solutions of one- and two-dimensional linear telegraph equations and telegraph equations with nonlinear forcing term. The homogeneous initial and boundary conditions are satisfied exactly by expanding the unknown variable using polynomial bases of functions which are built upon the Jacobi polynomials. The suggested scheme is successfully developed for the aforementioned problem with nonhomogeneous data. Extensive numerical experiments are presented to verify the efficiency of the proposed scheme.

Shifted Jacobi collocation method for solving multi-dimensional fractional Stokes’ first problem for a heated generalized second grade fluid

This paper reports a new spectral collocation technique for solving fractional Stokes’ first problem for a heated generalized second grade fluid (FSFP-HGSGF). We develop a collocation scheme to approximate FSFP-HGSGF by means of the shifted Jacobi-Gauss-Lobatto collocation (SJ-GL-C) and shifted Jacobi-Gauss-Radau collocation (SJ-GR-C) methods. The discussed numerical tests illustrate the capability and high accuracy of the proposed methodologies.

New spectral collocation algorithms for one- and two-dimensional Schrödinger equations with a Kerr law nonlinearity

A shifted Jacobi collocation method in two stages is constructed and used to numerically solve nonlinear Schrodinger equations (NLSEs) with a Kerr law nonlinearity, subject to initial-boundary conditions. An expansion in a series of spatial shifted Jacobi polynomials with temporal coefficients for the approximate solution is considered. The first stage, collocation at the shifted Jacobi Gauss-Lobatto (SJ-GL) nodes, is applied for a spatial discretization; its spatial derivatives occur in the NLSE with a treatment of the boundary conditions. This in all will produce a system of ordinary differential equations (SODEs) for the coefficients. The second stage is to collocate at the shifted Jacobi Gauss-Radau (SJ-GR-C) nodes in the temporal discretization to reduce the SODEs to a system of algebraic equations which is solved by an iterative method. Both stages can be extended to solve the two-dimensional NLSEs. Numerical examples are carried out to confirm the spectral accuracy and the efficiency of the proposed algorithms.

Shifted Jacobi Collocation Method Based on Operational Matrix for Solving the Systems of Fredholm and Volterra Integral Equations

This paper aims to construct a general formulation for the shifted Jacobi operational matrices of integration and product. The main aim is to generalize the Jacobi integral and product operational matrices to the solving system of Fredholm and Volterra equations. These matrices together with the collocation method are applied to reduce the solution of these problems to the solution of a system of algebraic equations. The method is applied to solve system of linear and nonlinear Fredholm and Volterra equations. Illustrative examples are included to demonstrate the validity and efficiency of the presented method. Also, several theorems, which are related to the convergence of the proposed method, will be presented.

Shifted Jacobi Collocation Method Based on Operational Matrix for Solving the Systems of Fredholm and Volterra Integral Equations

Shifted Jacobi Collocation Method Based on Operational Matrix for Solving the Systems of Fredholm and Volterra Integral Equations

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

The cable equation plays a central role in many areas of electrophysiology and in modeling neuronal dynamics. This paper reports an accurate spectral collocation method for solving one- and two-dimensional variable-order fractional nonlinear cable equations. The proposed method is based on shifted Jacobi collocation procedure in conjunction with the shifted Jacobi operational matrix for variable-order fractional derivatives, described in the sense of Caputo. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations. In addition, the method reduces the variable-order fractional nonlinear cable equation to a simpler problem that consists of solving a system of algebraic equations. The validity and effectiveness of the method are demonstrated by solving three numerical examples. The convergence of the method is graphically analyzed. The results demonstrate that the proposed method is a powerful algorithm with high accuracy for solving the variable-order nonlinear partial differential equations.

A shifted Jacobi collocation algorithm for wave type equations with non-local conservation conditions

Abstract In this paper, we propose an efficient spectral collocation algorithm to solve numerically wave type equations subject to initial, boundary and non-local conservation conditions. The shifted Jacobi pseudospectral approximation is investigated for the discretization of the spatial variable of such equations. It possesses spectral accuracy in the spatial variable. The shifted Jacobi-Gauss-Lobatto (SJ-GL) quadrature rule is established for treating the non-local conservation conditions, and then the problem with its initial and non-local boundary conditions are reduced to a system of second-order ordinary differential equations in temporal variable. This system is solved by two-stage forth-order A-stable implicit RK scheme. Five numerical examples with comparisons are given. The computational results demonstrate that the proposed algorithm is more accurate than finite difference method, method of lines and spline collocation approach

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.

Numerical Solution of a Class of Functional-Differential Equations Using Jacobi Pseudospectral Method

The shifted Jacobi-Gauss-Lobatto pseudospectral (SJGLP) method is applied to neutral functional-differential equations (NFDEs) with proportional delays. The proposed approximation is based on shifted Jacobi collocation approximation with the nodes of Gauss-Lobatto quadrature. The shifted Legendre-Gauss-Lobatto Pseudo-spectral and Chebyshev-Gauss-Lobatto Pseudo-spectral methods can be obtained as special cases of the underlying method. Moreover, the SJGLP method is extended to numerically approximate the nonlinear high-order NFDE with proportional delay. Some examples are displayed for implicit and explicit forms of NFDEs to demonstrate the computation accuracy of the proposed method. We also compare the performance of the method with variational iteration method, one-legθ-method, continuous Runge-Kutta method, and reproducing kernel Hilbert space method.

On shifted Jacobi spectral method for high-order multi-point boundary value problems

This paper reports a spectral tau method for numerically solving multi-point boundary value problems (BVPs) of linear high-order ordinary differential equations. The construction of the shifted Jacobi tau approximation is based on conventional differentiation. This use of differentiation allows the imposition of the governing equation at the whole set of grid points and the straight forward implementation of multiple boundary conditions. Extension of the tau method for high-order multi-point BVPs with variable coefficients is treated using the shifted Jacobi Gauss–Lobatto quadrature. Shifted Jacobi collocation method is developed for solving nonlinear high-order multi-point BVPs. The performance of the proposed methods is investigated by considering several examples. Accurate results and high convergence rates are achieved.

Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations

A shifted Jacobi Galerkin method is introduced to get a direct solution technique for solving the third‐ and fifth‐order differential equations with constant coefficients subject to initial conditions. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. A quadrature Galerkin method is introduced for the numerical solution of these problems with variable coefficients. A new shifted Jacobi collocation method based on basis functions satisfying the initial conditions is presented for solving nonlinear initial value problems. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms. The algorithms are easy to implement and yield very accurate results.

A new Jacobi operational matrix: An application for solving fractional differential equations

In this paper, we derived the shifted Jacobi operational matrix (JOM) of fractional derivatives which is applied together with spectral tau method for numerical solution of general linear multi-term fractional differential equations (FDEs). A new approach implementing shifted Jacobi operational matrix in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of nonlinear multi-term FDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. The proposed methods are applied for solving linear and nonlinear multi-term FDEs subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.

Spectral Shifted Jacobi Tau and Collocation Methods for Solving Fifth‐Order Boundary Value Problems

We have presented an efficient spectral algorithm based on shifted Jacobi tau method of linear fifth‐order two‐point boundary value problems (BVPs). An approach that is implementing the shifted Jacobi tau method in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of fifth‐order differential equations with variable coefficients. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplify the problem. Shifted Jacobi collocation method is developed for solving nonlinear fifth‐order BVPs. Numerical examples are performed to show the validity and applicability of the techniques. A comparison has been made with the existing results. The method is easy to implement and gives very accurate results.