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The main purpose of this work is to develop a spectrally accurate collocation method for solving weakly singular integral equations of the second kind with nonsmooth solutions in high dimensions. The proposed spectral collocation method is based on a multivariate Jacobi approximation in the frequency space. The essential idea is to adopt a smoothing transformation for the spectral collocation method to circumvent the curse of singularity at the beginning of time. As such, the singularity of the numerical approximation can be tailored to that of the singular solutions. A rigorous convergence analysis is provided and confirmed by numerical tests with nonsmooth solutions in two dimensions. The results in this paper seem to be the first spectral approach with a theoretical justification for high-dimensional nonlinear weakly singular Volterra type equations with nonsmooth solutions.
相似文献Volleyball premier league (VPL) simulating some phenomena of volleyball game has been presented recently. This powerful algorithm uses such racing and interplays between teams within a season. Furthermore, the algorithm imitates the coaching procedure within a game. Therefore, some volleyball metaphors, including substitution, coaching, and learning, are used to find a better solution prepared by the VPL algorithm. However, the learning phase has the largest effect on the performance of the VPL algorithm, in which this phase can lead to making the VPL stuck in optimal local solution. Therefore, this paper proposed a modified VPL using sine cosine algorithm (SCA). In which the SCA operators have been applied in the learning phase to obtain a more accurate solution. So, we have used SCA operators in VPL to grasp their advantages resulting in a more efficient approach for finding the optimal solution of the optimization problem and avoid the limitations of the traditional VPL algorithm. The propounded VPLSCA algorithm is tested on the 25 functions. The results captured by the VPLSCA have been compared with other metaheuristic algorithms such as cuckoo search, social-spider optimization algorithm, ant lion optimizer, grey wolf optimizer, salp swarm algorithm, whale optimization algorithm, moth flame optimization, artificial bee colony, SCA, and VPL. Furthermore, the three typical optimization problems in the field of designing engineering have been solved using the VPLSCA. According to the obtained results, the proposed algorithm shows very reasonable and promising results compared to others.
相似文献A hybrid analytical-intelligent approach is proposed for fuzzy reliability analysis of the composite beams reinforced by zinc oxide (ZnO) nanoparticle. The fuzzy reliability index corresponding to buckling failure mode of nanocomposite beam under thickness-direction external voltage is computed based on three-levels: (1) fuzzy analysis, (2) reliability analysis and (3) analytical buckling analysis. In fuzzy analysis level, an improved gravitational search algorithm has been applied to determine uncertainty interval for membership levels of reliability index. The adaptive formulation with a dynamical self-adjusting process is used for reliability analysis level based on conjugate first-order reliability method (FORM). The self-adjusting term in conjugate sensitivity vector is used to satisfy the sufficient descent condition for controlling instability of FORM formula while the proposed conjugate scalar factor is computed less than the original conjugate FORM, thus it may be provided with the efficient results for the convex problem. The new and previous sensitivity vectors obtained by conjugate and steepest descent vectors dynamically adjusted the proposed conjugate factor. In the buckling analysis level, an exponential theory in conjunction with the method of energy is utilized. Fuzzy random variables including applied voltage, the volume fraction of ZnO, thickness of beam, spring constant and shear constant of the foundation are considered in studied nanocomposite beam. Survey results indicated that the proposed method can provide stable and acceptable fuzzy membership functions for parametric study. Moreover, the ratio of length to thickness and spring constant of foundation are the more sensitive parameters which affect fuzzy reliability index significantly.
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