Jacobi-weighted Sobolev Spaces Research Articles

A Jacobi spectral method for calculating fractional derivative based on mollification regularization

In this article, we construct a Jacobi spectral collocation scheme to approximate the Caputo fractional derivative based on Jacobi–Gauss quadrature. The convergence analysis is provided in anisotropic Jacobi-weighted Sobolev spaces. Furthermore, the convergence rate is presented for solving Caputo fractional derivative with noisy data by invoking the mollification regularization method. Lastly, numerical examples illustrate the effectiveness and stability of the proposed method.

Solution of weakly singular fractional integro-differential equations by using a new operational approach

In this study, an operational approach, based on the shifted Jacobi polynomials, is presented to solve a class of weakly singular fractional integro-differential equations. The fractional derivative operators are considered as the Caputo sense. In addition to finding the operational matrices of integration and product, a new operational matrix is derived to be approximated the singular kernels of this class of the functional equations. These operational matrices together with the collocation method are applied to convert the main equation into a linear or non-linear system of algebraic equations. By presenting some theorems, conditions of the existence and uniqueness of the solution of the main equation and its corresponding algebraic system are investigated. Also, the stability and convergence of the proposed method are studied and an error bound is obtained in a Jacobi-weighted Sobolev space. Finally, some illustrative examples are provided to demonstrate the efficiency and simplicity of the suggested approach.

Galerkin–Legendre spectral method for Neumann boundary value problems in three dimensions

ABSTRACTIn this paper, we investigate the Legendre spectral methods for problems with the essential imposition of Neumann boundary condition in three dimensions. A double diagonalization process has been employed, instead of the full stiffness matrices encountered in the classical variational formulation of the problem with a weak natural imposition of Neumann boundary condition. For analysing numerical errors, some results on Legendre orthogonal approximation in Jacobi weighted Sobolev space are established. As examples of applications, the spectral schemes are provided for two model problems. The convergences of the proposed schemes are proved, too. Numerical results demonstrate the spectral accuracy in space, and which confirm theoretical analysis well.

Low cost numerical solution for three-dimensional linear and nonlinear integral equations via three-dimensional Jacobi polynomials

In recent years, numerical methods have been introduced to solve two-dimensional Volterra and Fredholm integral equations. In this study, a numerical scheme is constructed to solve classes of linear and nonlinear three-dimensional integral equations (Volterra, Fredholm, and mixed Volterra–Fredholm). This operational approach is proposed to easily and directly solve these equations at low computational costs. The scheme is based on the Jacobi polynomials on the interval [0, 1] where three-variable Jacobi polynomials are introduced and their operational matrices of integration and product are derived. Compared to other existing methods for multidimensional problems, the Jacobi operational method eliminates the time-consuming computations and solely employs the one-dimensional operational matrix to construct corresponding multidimensional operational matrices. The absolute error of the proposed method is almost constant on the studied interval even at higher dimensions, confirming the stability of the proposed operational Jacobi method. Required theorems on the convergence of the method are proved in Jacobi-weighted Sobolev space. It is established that the error function vanishes as N increases. The method is evaluated using several illustrative examples which indicate the proposed method with lesser computational size compared to the Block–Pulse functions, differential transform, and degenerate kernel methods.

Spectral approximation methods and error estimates for Caputo fractional derivative with applications to initial-value problems

In this paper, we revisit two spectral approximations, including truncated approximation and interpolation for Caputo fractional derivative. The two approaches have been studied to approximate Riemann–Liouville (R–L) fractional derivative by Chen et al. and Zayernouri et al. respectively in their most recent work. For truncated approximation the reconsideration partly arises from the difference between fractional derivative in R–L sense and Caputo sense: Caputo fractional derivative requires higher regularity of the unknown than R–L version. Another reason for the reconsideration is that we distinguish the differential order of the unknown with the index of Jacobi polynomials, which is not presented in the previous work. Also we provide a way to choose the index when facing multi-order problems. By using generalized Hardy's inequality, the gap between the weighted Sobolev space involving Caputo fractional derivative and the classical weighted space is bridged, then the optimal projection error is derived in the non-uniformly Jacobi-weighted Sobolev space and the maximum absolute error is presented as well. For the interpolation, analysis of interpolation error was not given in their work. In this paper we build the interpolation error in non-uniformly Jacobi-weighted Sobolev space by constructing fractional inverse inequality. With combining collocation method, the approximation technique is applied to solve fractional initial-value problems (FIVPs). Numerical examples are also provided to illustrate the effectiveness of this algorithm.

A Spectral Method for Fourth-Order Mixed Inhomogeneous Boundary Value Problem in Three Dimensions

In this paper, we investigate spectral method for fourth- order mixed inhomogeneous boundary value problems in three dimensions. Some results on the three-dimensional Legendre approximation for fourth- order problem in Jacobi weighted Sobolev space are established, which improve and generalize the existing results, and play an important role in numerical solutions of partial differential equations. We also develop a lifting technique for fourth-order problems, with which we could handle mixed inhomogeneous boundary conditions easily. As examples of applications, spectral schemes are provided for three model problems with mixed inhomogeneous boundary conditions. The spectral accuracy in space of proposed algorithms are proved. Efficient implementations are presented. Numerical results demonstrate their high accuracy, and confirm the theoretical analysis well.

The Convergence of the H-P Version of the Finite Element Method with Quasi-Uniform Meshes for Three Dimensional Poisson Problems with Edge Singularity

This paper is concerned with the convergence of the h-p version of the finite element method for three dimensional Poisson problems with edge singularity on quasi-uniform meshes. First, we present the theoretical results for the convergence of the h-p version of the finite element method with quasi-uniform meshes for elliptic problems on polyhedral domains on smooth functions in the framework of Jacobi-weighted Sobolev spaces. Second, we investigate and analyze numerical results for three dimensional Poission problems with edge singularity. Finally, we verified the theoretical predictions by the numerical computation.

Spectral Method for Mixed Inhomogeneous Boundary Value Problems in Three Dimensions

In this paper, we investigate spectral method for mixed inhomogeneous boundary value problems in three dimensions. Some results on the three-dimensional Legendre approximation in Jacobi weighted Sobolev space are established, which improve and generalize the existing results, and play an important role in numerical solutions of partial differential equations. We also develop a lifting technique, with which we could handle mixed inhomogeneous boundary conditions easily. As examples of applications, spectral schemes are provided for three model problems with mixed inhomogeneous boundary conditions. The spectral accuracy in space of proposed algorithms is proved. Efficient implementations are presented. Numerical results demonstrate their high accuracy, and confirm the theoretical analysis well.

Petrov-Galerkin spectral element method for mixed inhomogeneous boundary value problems on polygons

The authors investigate Petrov-Galerkin spectral element method. Some results on Legendre irrational quasi-orthogonal approximations are established, which play important roles in Petrov-Galerkin spectral element method for mixed inhomogeneous boundary value problems of partial differential equations defined on polygons. As examples of applications, spectral element methods for two model problems, with the spectral accuracy in certain Jacobi weighted Sobolev spaces, are proposed. The techniques developed in this paper are also applicable to other higher order methods.

Spectral method on quadrilaterals

In this paper, we investigate the spectral method on quadrilaterals. We introduce an orthogonal family of functions induced by Legendre polynomials, and establish some results on the corresponding orthogonal approximation. These results play important roles in the spectral method for partial differential equations defined on quadrilaterals. As examples of applications, we provide spectral schemes for two model problems and prove their spectral accuracy in Jacobi weighted Sobolev space. Numerical results coincide well with the analysis. We also investigate the spectral method on convex polygons whose solutions possess spectral accuracy. The approximation results of this paper are also applicable to other problems.

Interpolation approximations based on Gauss–Lobatto–Legendre–Birkhoff quadrature

We derive in this paper the asymptotic estimates of the nodes and weights of the Gauss–Lobatto– Legendre–Birkhoff (GLLB) quadrature formula, and obtain optimal error estimates for the associated GLLB interpolation in Jacobi weighted Sobolev spaces. We also present a user-oriented implementation of the pseudospectral methods based on the GLLB quadrature nodes for Neumann problems. This approach allows an exact imposition of Neumann boundary conditions, and is as efficient as the pseudospectral methods based on Gauss–Lobatto quadrature for PDEs with Dirichlet boundary conditions.

Error analysis of spectral method on a triangle

In this paper, the orthogonal polynomial approximation on triangle, proposed by Dubiner, is studied. Some approximation results are established in certain non-uniformly Jacobi-weighted Sobolev space, which play important role in numerical analysis of spectral and triangle spectral element methods for differential equations on complex geometries. As an example, a model problem is considered.

Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces

Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces are investigated. Some results on orthogonal projections and interpolations are established. Explicit expressions describing the dependence of approximation results on the parameters of Jacobi polynomials are given. These results serve as an important tool in the analysis of numerous quadratures and numerical methods for differential and integral equations.