Search Results (5)

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

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

We propose higher-order adaptive energy-preserving methods for a charged particle system and a guiding center system. The higher-order energy-preserving methods are symmetric and are constructed by composing the second-order energy-preserving methods based on the averaged vector field. In order to overcome the energy drift problem that occurs in the energy-preserving methods based on the average vector field, we develop two adaptive algorithms for the higher-order energy-preserving methods. The two adaptive algorithms are developed based on using variable points of Gauss–Legendre’s quadrature rule and using two different stepsizes. The numerical results show that the two adaptive algorithms behave better in phase portrait and energy conservation than the Runge–Kutta methods. Moreover, it is shown that the energy errors obtained by the two adaptive algorithms can be bounded by the machine precision over long time and do not show energy drift. Full article

In this research, some new and efficient quadrature rules are proposed involving the combination of function and its first derivative evaluations at equally spaced data points with the main focus on their computational efficiency in terms of cost and time usage. The methods are theoretically derived, and theorems on the order of accuracy, degree of precision and error terms are proved. The proposed methods are semi-open-type rules with derivatives. The order of accuracy and degree of precision of the proposed methods are higher than the classical rules for which a systematic and symmetrical ascendancy has been proved. Various numerical tests are performed to compare the performance of the proposed methods with the existing methods in terms of accuracy, precision, leading local and global truncation errors, numerical convergence rates and computational cost with average CPU usage. In addition to the classical semi-open rules, the proposed methods have also been compared with some Gauss–Legendre methods for performance evaluation on various integrals involving some oscillatory, periodic and integrals with derivative singularities. The analysis of the results proves that the devised techniques are more efficient than the classical semi-open Newton–Cotes rules from theoretical and numerical perspectives because of promisingly reduced functional cost and lesser execution times. The proposed methods compete well with the spectral Gauss–Legendre rules, and in some cases outperform. Symmetric error distributions have been observed in regular cases of integrands, whereas asymmetrical behavior is evidenced in oscillatory and highly nonlinear cases. Full article

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