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Unsupervised neural networks for solving troesch＇s problem

Chinese Physics B 2014年第1期23卷 546-556页

作者： Muhammad Asif Zahoor Raja Department of Electrical Engineering COMSATS Institute of Information Technology Attock Campus Attock Pakistan

In this study, stochastic computational intelligence techniques are presented for the solution of troesch's boundary value problem. The proposed stochastic solvers use the competency of a feed-forward artificial neural network for mathematical modeling of the problem in an unsupervised manner, whereas the learning of unknown parameters is made with local and global optimization methods as well as their combinations. Genetic algorithm (GA) and pattern search (Ps) techniques are used as the global search methods and the interior point method (IPM) is used for an efficient local search. The combination of techniques like GA hybridized with IPM (GA-IPM) and Ps hybridized with IPM (Ps-IPM) are also applied to solve different forms of the equation. A comparison of the proposed results obtained from GA, Ps, IPM, Ps-IPM and GA-IPM has been made with the standard solutions including well known analytic techniques of the Adomian decomposition method, the variational iterational method and the homotopy perturbation method. The reliability and effectiveness of the proposed schemes, in term of accuracy and convergence, are evaluated from the results of statistical analysis based on sufficiently large independent runs.

关键词： troesch’s problem artificial neural network genetic algorithm hybrid methods

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Wavelet multiresolution interpolation Galerkin method for nonlinear boundary value problems with localized steep gradients

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Applied Mathematics and Mechanics(English Edition) 2022年第6期43卷 863-882页

作者： Xiaojing LIU Youhe ZHOU Jizeng WANG Key Laboratory of Mechanics on Disaster and Environment in Western China(Lanzhou University) the Ministry of EducationCollege of Civil Engineering and MechanicsLanzhou UniversityLanzhou 730000China

The wavelet multiresolution interpolation for continuous functions defined on a finite interval is developed in this study by using a simple alternative of transformation *** wavelet multiresolution interpolation Galerkin method that applies this interpolation to represent the unknown function and nonlinear terms independently is proposed to solve the boundary value problems with the mixed Dirichlet-Robin boundary conditions and various nonlinearities,including transcendental ones,in which the discretization process is as simple as that in solving linear problems,and only common two-term connection coefficients are *** matrices are independent of unknown node values and lead to high efficiency in the calculation of the residual and Jacobian matrices needed in Newton’s method,which does not require numerical integration in the resulting nonlinear discrete *** validity of the proposed method is examined through several nonlinear problems with interior or boundary *** results demonstrate that the proposed wavelet method shows excellent accuracy and stability against nonuniform grids,and high resolution of localized steep gradients can be achieved by using local refined multiresolution *** addition,Newton’s method converges rapidly in solving the nonlinear discrete system created by the proposed wavelet method,including the initial guess far from real solutions.

关键词： wavelet multiresolution interpolation transcendental nonlinearity localized steep gradient singularly perturbed boundary value problem troesch’s problem

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维普期刊数据库

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