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Search Results (4)

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Keywords = rational Gauss quadrature

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15 pages, 6045 KiB  
Article
Numerical Simulation Based on Interpolation Technique for Multi-Term Time-Fractional Convection–Diffusion Equations
by Xindong Zhang, Yan Chen, Leilei Wei and Sunil Kumar
Fractal Fract. 2024, 8(12), 687; https://doi.org/10.3390/fractalfract8120687 - 23 Nov 2024
Viewed by 694
Abstract
This paper introduces a novel approach for solving multi-term time-fractional convection–diffusion equations with the fractional derivatives in the Caputo sense. The proposed highly accurate numerical algorithm is based on the barycentric rational interpolation collocation method (BRICM) in conjunction with the Gauss–Legendre quadrature rule. [...] Read more.
This paper introduces a novel approach for solving multi-term time-fractional convection–diffusion equations with the fractional derivatives in the Caputo sense. The proposed highly accurate numerical algorithm is based on the barycentric rational interpolation collocation method (BRICM) in conjunction with the Gauss–Legendre quadrature rule. The discrete scheme constructed in this paper can achieve high computational accuracy with very few interval partitioning points. To verify the effectiveness of the present discrete scheme, some numerical examples are presented and are compared with the other existing method. Numerical results demonstrate the effectiveness of the method and the correctness of the theoretical analysis. Full article
(This article belongs to the Section Numerical and Computational Methods)
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16 pages, 8303 KiB  
Article
High-Order Numerical Approximation for 2D Time-Fractional Advection–Diffusion Equation under Caputo Derivative
by Xindong Zhang, Yan Chen and Leilei Wei
Fractal Fract. 2024, 8(8), 474; https://doi.org/10.3390/fractalfract8080474 - 13 Aug 2024
Cited by 1 | Viewed by 1177
Abstract
In this paper, we propose a novel approach for solving two-dimensional time-fractional advection–diffusion equations, where the fractional derivative is described in the Caputo sense. The discrete scheme is constructed based on the barycentric rational interpolation collocation method and the Gauss–Legendre quadrature rule. We [...] Read more.
In this paper, we propose a novel approach for solving two-dimensional time-fractional advection–diffusion equations, where the fractional derivative is described in the Caputo sense. The discrete scheme is constructed based on the barycentric rational interpolation collocation method and the Gauss–Legendre quadrature rule. We employ the barycentric rational interpolation collocation method to approximate the unknown function involved in the equation. Through theoretical analysis, we establish the convergence rate of the discrete scheme and show its remarkable accuracy. In addition, we give some numerical examples, to illustrate the proposed method. All the numerical results show the flexible application ability and reliability of the present method. Full article
(This article belongs to the Section Numerical and Computational Methods)
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12 pages, 323 KiB  
Article
Pole Allocation for Rational Gauss Quadrature Rules for Matrix Functionals Defined by a Stieltjes Function
by Jihan Alahmadi, Miroslav Pranić and Lothar Reichel
Axioms 2023, 12(2), 105; https://doi.org/10.3390/axioms12020105 - 19 Jan 2023
Viewed by 1500
Abstract
This paper considers the computation of approximations of matrix functionals of form F(A):=vTf(A)v, where A is a large symmetric positive definite matrix, v is a vector, and f is a [...] Read more.
This paper considers the computation of approximations of matrix functionals of form F(A):=vTf(A)v, where A is a large symmetric positive definite matrix, v is a vector, and f is a Stieltjes function. The functional F(A) is approximated by a rational Gauss quadrature rule with poles on the negative real axis (or part thereof) in the complex plane, and we focus on the allocation of the poles. Specifically, we propose that the poles, when considered positive point charges, be allocated to make the negative real axis (or part thereof) approximate an equipotential curve. This is easily achieved with the aid of conformal mapping. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
15 pages, 440 KiB  
Article
Rational Transformations for Evaluating Singular Integrals by the Gauss Quadrature Rule
by Beong In Yun
Mathematics 2020, 8(5), 677; https://doi.org/10.3390/math8050677 - 1 May 2020
Cited by 3 | Viewed by 2171
Abstract
In this work we introduce new rational transformations which are available for numerical evaluation of weakly singular integrals and Cauchy principal value integrals. The proposed rational transformations include parameters playing an important role in accelerating the accuracy of the Gauss quadrature rule used [...] Read more.
In this work we introduce new rational transformations which are available for numerical evaluation of weakly singular integrals and Cauchy principal value integrals. The proposed rational transformations include parameters playing an important role in accelerating the accuracy of the Gauss quadrature rule used for the singular integrals. Results of some selected numerical examples show the efficiency of the proposed transformation method compared with some existing transformation methods. Full article
(This article belongs to the Section E: Applied Mathematics)
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