The meshless Kansa method for time-fractional higher order partial differential equations with constant and variable coefficients

Haq, Sirajul; Hussain, Manzoor

doi:10.1007/s13398-018-0593-x

Cited by 14 publications

(3 citation statements)

References 34 publications

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“…Dehgan [14] studied numerical solution of nonlinear Klein-Gordon equation, whereas Khattak [15] obtained numerical solution of nonlinear PDEs using meshfree collocation method. Recently, Hussain and their co-worker used the meshless RBFs for various classes of fractional PDEs [16][17][18]. In this article, we experienced the application of RBFs meshless method for numerical solution of boundary value problems.…”

Section: Introductionmentioning

confidence: 99%

Approximate Solution of Second Order Singular Perturbed and Obstacle Boundary Value Problems Using Meshless Method Based on Radial Basis Functions

Ali

Haq

Ullah

et al. 2022

J Nonlinear Math Phys

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In this article, a meshless numerical technique based on radial basis functions (RBFs) is proposed for the solution of singular perturbed, obstacle, and second-order boundary value problems. First, the unknown function and their derivatives are approximated by RBFs which reduces the given problem into a system of algebraic equations which is easy to solve. The shape parameter involved in RBFs is chosen by the hit and trial method. Despite this, the convergence of the scheme is briefly discussed numerically. The nonlinear terms are linearized by quasi-linearization technique. The main objective of this paper is to show that the meshless RBFs-based method is convenient for various classes of boundary value problems. Efficiency and performance of the proposed technique are examined by calculating absolute error norms. Obtained accurate results confirm applicability and efficiency of the method.

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Section: Introductionmentioning

confidence: 99%

Approximate Solution of Second Order Singular Perturbed and Obstacle Boundary Value Problems Using Meshless Method Based on Radial Basis Functions

Ali

Haq

Ullah

et al. 2022

J Nonlinear Math Phys

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“…Fallah et al (2019) [ 29 ] used the Kansa approach for solving seepage problems. Haq and Hussain (2019) [ 30 ] utilized the Kansa method to solve time-fractional higher-order partial differential equations with constant and variable coefficients. Jankowska and Karageorghis (2019) [ 31 ] used the Kansa approach for the numerical solution of second- and fourth-order nonlinear boundary-value problems in two dimensions.…”

Section: Introductionmentioning

confidence: 99%

Kansa Method for Unsteady Heat Flow in Nonhomogenous Material with a New Proposal of Finding the Good Value of RBF’s Shape Parameter

Popczyk

Dziatkiewicz

2021

Materials

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New engineering materials exhibit a complex internal structure that determines their properties. For thermal metamaterials, it is essential to shape their thermophysical parameters’ spatial variability to ensure unique properties of heat flux control. Modeling heterogeneous materials such as thermal metamaterials is a current research problem, and meshless methods are currently quite popular for simulation. The main problem when using new modeling methods is the selection of their optimal parameters. The Kansa method is currently a well-established method of solving problems described by partial differential equations. However, one unsolved problem associated with this method that hinders its popularization is choosing the optimal shape parameter value of the radial basis functions. The algorithm proposed by Fasshauer and Zhang is, as of today, one of the most popular and the best-established algorithms for finding a good shape parameter value for the Kansa method. However, it turns out that it is not suitable for all classes of computational problems, e.g., for modeling the 1D heat conduction in non-homogeneous materials, as in the present paper. The work proposes two new algorithms for finding a good shape parameter value, one based on the analysis of the condition number of the matrix obtained by performing specific operations on interpolation matrix and the other being a modification of the Fasshauer algorithm. According to the error measures used in work, the proposed algorithms for the considered class of problem provide shape parameter values that lead to better results than the classic Fasshauer algorithm.

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“…It has been extensively used to model anomalous diffusion in transport processes through complex and disordered systems with/without fractal media [59]. Numerical solution of some TFAD models has been reported by FDM [46], RBFs‐based methods [39, 58, 60–62], MLS method [59], and Sinc‐Legendre collocation method [63]. Therefore, in this study we obtain accurate numerical solution of TFBS and TFAD equations via HRBFs approximation method while utilizing the proposed optimal parameters selection algorithm.…”

Section: Introductionmentioning

confidence: 99%

A hybrid radial basis functions collocation technique to numerically solve fractional advection–diffusion models

Hussain¹,

Haq

2020

Numerical Methods Partial

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In this work, we propose a hybrid radial basis functions (RBFs) collocation technique for the numerical solution of fractional advection-diffusion models. In the formulation of hybrid RBFs (HRBFs), there exist shape parameter (c*) and weight parameter () that control numerical accuracy and stability. For these parameters, an adaptive algorithm is developed and validated. The proposed HRBFs method is tested for numerical solutions of some fractional Black-Sholes and diffusion models. Numerical simulations performed for several benchmark problems verified the proposed method accuracy and efficiency. The quantitative analysis is made in terms of L ∞ , L 2 , L rms , and L rel error norms as well as number of nodes N over space domain and time-step t. Numerical convergence in space and time is also studied for the proposed method. The unconditional stability of the proposed HRBFs scheme is obtained using the von Neumann methodology. It is observed that the HRBFs method circumvented the ill-conditioning problem greatly, a major issue in the Kansa method.

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The meshless Kansa method for time-fractional higher order partial differential equations with constant and variable coefficients

Cited by 14 publications

References 34 publications

Approximate Solution of Second Order Singular Perturbed and Obstacle Boundary Value Problems Using Meshless Method Based on Radial Basis Functions

Approximate Solution of Second Order Singular Perturbed and Obstacle Boundary Value Problems Using Meshless Method Based on Radial Basis Functions

Kansa Method for Unsteady Heat Flow in Nonhomogenous Material with a New Proposal of Finding the Good Value of RBF’s Shape Parameter

A hybrid radial basis functions collocation technique to numerically solve fractional advection–diffusion models

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