Computational methods for integral equations
Computational methods for integral equations
On the coefficients of integrated expansions of ultraspherical polynomials
SIAM Journal on Numerical Analysis
Spectral integration and two-point boundary value problems
SIAM Journal on Numerical Analysis
Boundary Layer Resolving Pseudospectral Methods for Singular Perturbation Problems
SIAM Journal on Scientific Computing
An efficient spectral method for ordinary differential equations with rational function coefficients
Mathematics of Computation
Analysis and Application of Fourier--Gegenbauer Method to Stiff Differential Equations
SIAM Journal on Numerical Analysis
Spectrally Accurate Solution of Nonperiodic Differential Equations by the Fourier--Gegenbauer Method
SIAM Journal on Numerical Analysis
Integration Preconditioning of Pseudospectral Operators. I. Basic Linear Operators
SIAM Journal on Numerical Analysis
Proceedings of the fourth international conference on Spectral and high order methods (ICOSAHOM 1998)
On the A-Stability of Runge--Kutta Collocation Methods Based on Orthogonal Polynomials
SIAM Journal on Numerical Analysis
Journal of Computational and Applied Mathematics
Parameter Optimization and Reduction of Round Off Error for the Gegenbauer Reconstruction Method
Journal of Scientific Computing
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Integration Preconditioning Matrix for Ultraspherical Pseudospectral Operators
SIAM Journal on Scientific Computing
Journal of Computational and Applied Mathematics
Journal of Computational Physics
Pseudospectral integration matrix and boundary value problems
International Journal of Computer Mathematics
Mathematics and Computers in Simulation
Journal of Computational Physics
Optimal Gegenbauer quadrature over arbitrary integration nodes
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
On the optimization of Gegenbauer operational matrix of integration
Advances in Computational Mathematics
Hi-index | 7.29 |
We introduce a hybrid Gegenbauer (ultraspherical) integration method (HGIM) for solving boundary value problems (BVPs), integral and integro-differential equations. The proposed approach recasts the original problems into their integral formulations, which are then discretized into linear systems of algebraic equations using Gegenbauer integration matrices (GIMs). The resulting linear systems are well-conditioned and can be easily solved using standard linear system solvers. A study on the error bounds of the proposed method is presented, and the spectral convergence is proven for two-point BVPs (TPBVPs). Comparisons with other competitive methods in the recent literature are included. The proposed method results in an efficient algorithm, and spectral accuracy is verified using eight test examples addressing the aforementioned classes of problems. The proposed method can be applied on a broad range of mathematical problems while producing highly accurate results. The developed numerical scheme provides a viable alternative to other solution methods when high-order approximations are required using only a relatively small number of solution nodes.