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radial basis functions method for numerical solution of the modified equal width equation

INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2010年第7期87卷 1569-1577页

作者： Dereli, Yilmaz Anadolu Univ Dept Math Eskisehir Turkey

The numerical solution of the modified equal width equation is investigated by using meshless method based on collocation with the well-known radial basis functions. Single solitary wave motion, two solitary waves interaction and three solitary waves interaction are studied. Results of the meshless methods with different radial basis functions are presented.

关键词： MEW equation solitary waves radial basis functions method

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radial basis functions method for valuing options: A multinomial tree approach

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2017年 319卷 97-107页

作者： Zhang, Shengliang Zhejiang Int Studies Univ Sch Sci & Technol Hangzhou Zhejiang Peoples R China

From the view point of probability, this study presents a theoretical framework to show the convergence of the RBFs method for valuing options. It will be proved to be equivalent to a multinomial tree approach, in which the underlying variable can move from its initial value to an infinity of possible values of the next time step. Specially, the probability of a move in a short period time follows the normal distribution when using the Gaussian basis kernel, it is a precise simulation of the behavior of the underlying variable, which provides a more reasonable explanation of high-accuracy of the RBFs method. This helps open a new area of research in developing the expected numerical method for derivative securities (in which the underlying asset follows other stochastic process) by using corresponding radial basis kernel. The paper also illustrates the approach by using it to value stock options and its Greek letters. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Black-Scholes radial basis functions method Convergence American option Probability distribution

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Determination of Nonlinear Source Term in an Inverse Convection-Reaction-Diffusion Problem Using radial basis functions method

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IRANIAN JOURNAL OF SCIENCE AND TECHNOLOGY TRANSACTION A-SCIENCE 2019年第A5期43卷 2239-2252页

作者： Pourgholi, Reza Saeedi, Akram Hosseini, Abbas Damghan Univ Sch Math & Comp Sci POB 36715-364 Damghan Iran

In this paper, we propose a meshless method using radial basis functions (RBFs) method based on finite-difference method to solve a nonlinear inverse convection-reaction-diffusion problem with an unknown source function. Indeed, the aim of this paper is to show that the meshless method based on the radial basis functions approach is also suitable for the treatment of the inverse nonlinear partial differential equations. Usually, the matrices obtained from the discretization of the equations are ill-conditioned, especially in higher-dimensional problems. To overcome such difficulties, we use Tikhonov regularization method. We prove that the time discrete scheme is stable using the energy method. The effectiveness of the proposed method is illustrated by numerical examples.

关键词： Nonlinear inverse problem Convection-reaction-diffusion equations radial basis functions method Meshless methods Control function Tikhonov regularization method Energy method Unconditionally stable

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Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions

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MATHEMATICAL AND COMPUTER MODELLING 2006年第11-12期44卷 1160-1168页

作者： Dehghan, Mehdi Tatari, Mehdi Amirkabir Univ Technol Fac Math & Comp Sci Dept Appl Math Tehran Iran

In this work, the method of radial basis functions is used for finding the solution of an inverse problem with source control parameter. Because a much wider range of physical phenomena are modelled by nonclassical parabolic initial-boundary value problems, theoretical behavior and numerical approximation of these problems have been active areas of research. The radial basis functions (RBF) method is an efficient mesh free technique for the numerical solution of partial differential equations. The main advantage of numerical methods which use radial basis functions over traditional techniques is the meshless property of these methods. In a meshless method, a set of scattered nodes are used instead of meshing the domain of the problem. The results of numerical experiments are presented and some comparisons are made with several well-known finite difference schemes. (c) 2006 Elsevier Ltd. All rights reserved.

关键词： radial basis functions method scattered data control function temperature overspecification parabolic equation

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Modeling the effect of processing parameters on temperature history in Directed Energy Deposition: an analytical and finite element approach

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RAPID PROTOTYPING JOURNAL 2024年第2期30卷 338-349页

作者： Ghasempour-Mouziraji, Mehran Afonso, Daniel Hosseinzadeh, Saman Goulas, Constantinos Najafizadeh, Mojtaba Hosseinzadeh, Morteza Ganji, D. D. de Sousa, Ricardo Alves Univ Aveiro TEMA Ctr Mech Technol & Automat Mech Engn Dept Aveiro Portugal Babol Noshirvani Univ Technol Dept Mech Engn Babol Iran Univ Twente Dept Design Prod & Management Enschede Netherlands Univ Salento Dept Innovat Engn Lecce Italy Islamic Azad Univ Dept Engn Ayatollah Amoli Branch Amol Iran Babol Noshirvani Univ Technol Dept Mech Engn Babol Iran

PurposeThe purpose of this paper is to assess the feasibility of analytical models, specifically the radial basis function method, Akbari-Ganji method and Gaussian method, in conjunction with the finite element method. The aim is to examine the impact of processing parameters on temperature ***/methodology/approachThrough analytical investigation and finite element simulation, this research examines the influence of processing parameters on temperature history. Simufact software with a thermomechanical approach was used for finite element simulation, while radial basis function, Akbari-Ganji and Gaussian methods were used for analytical modeling to solve the heat transfer differential *** accuracy of both finite element and analytical methods was validated with about 90%. The findings revealed direct relationships between thermal conductivity (from 100 to 200), laser power (from 400 to 800 W), heat source depth (from 0.35 to 0.75) and power absorption coefficient (from 0.4 to 0.8). Increasing the values of these parameters led to higher temperature history. On the other hand, density (from 7,600 to 8,200), emission coefficient (from 0.5 to 0.7) and convective heat transfer (from 35 to 90) exhibited an inverse relationship with temperature ***/valueThe application of analytical modeling, particularly the utilization of the Akbari-Ganji, radial basis functions and Gaussian methods, showcases an innovative approach to studying directed energy deposition. This analytical investigation offers an alternative to relying solely on experimental procedures, potentially saving time and resources in the optimization of DED processes.

关键词： Directed energy deposition Finite element simulation Akbari-Ganji method radial basis functions method

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Numerical techniques for solving system of nonlinear inverse problem

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ENGINEERING WITH COMPUTERS 2018年第3期34卷 487-502页

作者： Pourgholi, Reza Tabasi, S. Hashem Zeidabadi, Hamed Damghan Univ Sch Math & Comp Sci POB 36715-364 Damghan Iran

In this paper, based on the cubic B-spline finite element (CBSFE) and the radial basis functions (RBFs) methods, the inverse problems of finding the nonlinear source term for system of reaction-diffusion equations are studied. The approach of the proposed methods are to approximate unknown coefficients by a polynomial function whose coefficients are determined from the solution of minimization problem based on the overspecified data. In fact, this work considers a comparative study between the cubic B-spline finite element method and radial basis functions method. The stability and convergence analysis for these problems are investigated and some examples are given to illustrate the results.

关键词： Cubic B-spline Finite element method radial basis functions method Inverse problems Stability analysis Convergence analysis Tikhonov regularization method Ill-posed problems Noisy data

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A radial basis FUNCTION BASED FRAMES STRATEGY FOR BYPASSING THE RUNGE PHENOMENON

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2016年第4期38卷 A2262-A2282页

作者： Piret, Cecile Michigan Technol Univ Houghton MI 49931 USA

Similarly to polynomials, smooth radial basis function (RBF) interpolants converge exponentially fast to analytic functions on a one dimensional bounded domain but are also vulnerable to the Runge phenomenon [R. Platte, IMA J. Numer. Anal. 31 (2014), pp. 1578-1597]. A common topic in the study of RBFs has been to find and match an optimal node sets and an RBF shape parameter;the location of the nodes is critical to prevent the Runge phenomenon from occur ring, and small variations in the shape parameter can radically change the interpolation accuracy. However, even a finely tuned combination of shape parameter and node distribution leads to numerical unstability in finite precision arithmetic as the number of nodes increases. In [D. Huybrechs, SIAM J. Numer. Anal. 47 (2010), pp. 4326-4355], [B. Adcock, D. Huybrechs, and J. Martin-Vaquero, Found. Comput. Math.,14 (2014), pp. 635-687.], Huybrechs et al. showed that it is always possible, using the Fourier extensions method, to stably approximate an analytic nonperiodic function on a bounded domain with at least a superalgebraic accuracy, even on an equispaced grid, despite a crippling ill-conditioning. In this study, we develop an RBF-based frames method, which builds upon the Fourier extensions method. Minute modi fi cations to the standard RBF algorithm, including a clear methodology for determining the RBF shape parameter, are provided. We observe identical convergence and stability patterns with our and the Fourier extensions methods and expose their fundamental connection by showing that the Fourier extensions method is a limiting case of our algorithm.

关键词： frames radial basis functions method RBF Fourier extensions method Fourier continuation method Runge phenomenon

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Dynamics of Convective Thermal Explosion in Porous Media

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INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS 2020年第6期30卷

作者： Allali, Karam Joundy, Youssef Taik, Ahmed Volpert, Vitaly Univ Hassan II Casablanca FST Mohammedia Lab Math & Applicat POB 146 Casablanca 20650 Morocco Mohammed VI Polytech Univ Complex Syst Engn & Human Syst Benguerir Morocco Univ Claude Bernard Lyon1 Inst Camille Jordan UMR 5208 CNRS F-69622 Villeurbanne France Univ Lyon1 Univ Lyon INRIA Inst Camille Jordan 43Bd 11 Novembre 1918 F-69622 Villeurbanne France Peoples Friendship Univ Russia RUDN Univ 6 Miklukho Maklaya St Moscow 117198 Russia

In this paper, we study complex dynamics of the interaction between natural convection and thermal explosion in porous media. This process is modeled with the nonlinear heat equation coupled with the nonstationary Darcy equation under the Boussinesq approximation for a fluid-saturated porous medium in a rectangular domain. Numerical simulations with the radial basis functions method (RBFM) reveal complex dynamics of solutions and transitions to chaos after a sequence of period doubling bifurcations. Several periodic windows alternate with chaotic regimes due to intermittence or crisis. After the last chaotic regime, a final periodic solution precedes transition to thermal explosion.

关键词： Porous media convective thermal explosion period doubling crisis intermittency chaos radial basis functions method

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Numerical analysis of parameters in a laminated beam model by radial basis functions

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CMC-COMPUTERS MATERIALS & CONTINUA 2005年第1期2卷 39-49页

作者： Hon, YC Ling, L Liew, KM City Univ Hong Kong Hong Kong Hong Kong Peoples R China Nanyang Technol Univ Singapore 2263 Singapore

In this paper we investigate a thermal driven Micro-Electfical-Mechanical system which was originally designed for inkjet printer to precisely deliver small ink droplets onto paper. In the model, a tiny free-ended beam of metal bends and projects ink onto paper. The model is solved by using the recently developed radial basis functions method. We establish the accuracy of the proposed approach by comparing the numerical results with reported experimental data. Numerical simulations indicate that a light (low composite mass) beam is more stable as it does not oscillate much. A soft (low rigidity) beam results in a higher rate of deflection, when compared to a high rigidity one. Effects caused by the values of physical parameters are also studied. Finally, we give a prediction on the optimal time for the second current pulse which results in maximum rate of second deflection of the beam.

关键词： Micro-Electrical-Mechanical System thermal driven radial basis functions method

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