The method of approximate particular solutions for the time-fractional diffusion equation with a non-local boundary condition

Yan, Liang; Yang, Fenglian

doi:10.1016/j.camwa.2015.04.030

Cited by 23 publications

(6 citation statements)

References 40 publications

(65 reference statements)

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“…To the best of our current knowledge, the meshless schemes, including Kansa method [14], method of fundamental solutions [15], method of particular solutions [16,17], element-free Galerkin method [18], local point interpolation [19], and boundary knot method [20], are widely used to approximate a large class of partial differential equations in science and engineering fields. As reported in the literatures, the MPS has been applied to solve the Navier-Stokes problem [21], wave propagation problem [22], and time-fractional diffusion problem [23]. Despite the effectiveness of the MPS, there are some disadvantages such as the ill-conditioned collocation matrix, the uncertainty of the shape parameters, and difficulties in deriving the closed-form particular solutions for general differential operators, and for more details, please refer to [12,17,[24][25][26] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Method of Particular Solutions for Second-Order Differential Equation with Variable Coefficients via Orthogonal Polynomials

Xie

Zhang

Jiang

et al. 2023

Journal of Function Spaces

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In this paper, with classic Legendre polynomials, a method of particular solutions (MPS, for short) is proposed to solve a kind of second-order differential equations with a variable coefficient on a unit interval. The particular solutions, satisfying the natural Dirichlet boundary conditions, are constructed with orthogonal Legendre polynomials for the variable coefficient case. Meanwhile, we investigate the a-priori error estimates of the MPS approximations. Two a-priori error estimations in H 1 - and L ∞ -norms are shown to depict the convergence order of numerical approximations, respectively. Some numerical examples and convergence rates are provided to validate the merits of our proposed meshless method.

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Section: Introductionmentioning

confidence: 99%

Method of Particular Solutions for Second-Order Differential Equation with Variable Coefficients via Orthogonal Polynomials

Xie

Zhang

Jiang

et al. 2023

Journal of Function Spaces

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“…In this paper, a meshless discretization technique based on MQ RBF and its modified form i.e integrated MQ RBF is applied for the numerical solution of elliptic PDEs with nonlocal boundary conditions (NBC). In the last decades, RBF-based collocation technique for the solution of nonlocal boundary value problems has been vital research area in many branches of science and hence was successfully applied to solve such problems numerically [20][21][22]. One-dimensional hyperbolic equation with integral boundary condition was studied in [23].…”

Section: Introductionmentioning

confidence: 99%

Meshless Analysis of Nonlocal Boundary Value Problems in Anisotropic and Inhomogeneous Media

Muhammad¹,

Ahsan²,

Ahmad³

et al. 2020

Preprint

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In this work, meshless methods are applied for the solution of two-dimensional steady-state heat conduction problems with nonlocal multi-point boundary conditions (NMBC). These meshless procedures are based on multiquadric radial basis function (MQ RBF) and its modified version (i.e. integrated MQ RBF). The proposed meshless methods which were recently published in \cite{Reutskiy2016} is compared with standard collocation method (i.e. Kansa's method). Three different sorts of solution domain are considered in which Dirichlet boundary condition is specified on some part of the boundary and is related to the unknown function values at a discrete set of interior points. The influence of NMBC on the accuracy and condition number of the system matrix associated to the proposed methods is investigated. The sensitivity of the shape parameter is also analyzed in the proposed methods. Performance of the proposed approaches in terms of accuracy and efficiency is confirmed on the benchmark problems.

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“…Such fractional models are used extensively by many experts to explain their complicated structures easily, simplified the controlling design without any loss of hereditary behaviors as well as create nature issues closely understandable for these phenomena. Accordingly, fractional-order derivatives provide more accurate models of realism problems than integer-order derivatives; they are actually found to be a suitable tool to describe certain physical and engineering problems including advection–diffusion models, dynamical mathematical models, electrical circuits models and networks models (El-Ajou et al , 2015; Abu Arqub et al , 2015; Zhao and Deng, 2015; Ray, 2016; Chen et al , 2016; Yana and Yang, 2015; El-Ajou et al , 2015; Ortigueira and Machado, 2003; Raja et al , 2015; Raja et al , 2017; Ray, 2007; Ray et al , 2008; Ray and Gupta, 2016a, 2016b; Ray, 2013; Ray and Gupta, 2014). Developing analytical and numerical methods for the solutions of time-fractional PDEs is a very important task owing to their practical interest.…”

Section: Introductionmentioning

confidence: 99%

Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm

Arqub

2018

HFF

185

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Purpose The purpose of this study is to introduce the reproducing kernel algorithm for treating classes of time-fractional partial differential equations subject to Robin boundary conditions with parameters derivative arising in fluid flows, fluid dynamics, groundwater hydrology, conservation of energy, heat conduction and electric circuit. Design/methodology/approach The method provides appropriate representation of the solutions in convergent series formula with accurately computable components. This representation is given in the W(Ω) and H(Ω) inner product spaces, while the computation of the required grid points relies on the R(y,s) (x, t) and r(y,s) (x, t) reproducing kernel functions. Findings Numerical simulation with different order derivatives degree is done including linear and nonlinear terms that are acquired by interrupting the n-term of the exact solutions. Computational results showed that the proposed algorithm is competitive in terms of the quality of the solutions found and is very valid for solving such time-fractional models. Research limitations/implications Future work includes the application of the reproducing kernel algorithm to highly nonlinear time-fractional partial differential equations such as those arising in single and multiphase flows. The results will be published in forthcoming papers. Practical implications The study included a description of fundamental reproducing kernel algorithm and the concepts of convergence, and error behavior for the reproducing kernel algorithm solvers. Results obtained by the proposed algorithm are found to outperform in terms of accuracy, generality and applicability. Social implications Developing analytical and numerical methods for the solutions of time-fractional partial differential equations is a very important task owing to their practical interest. Originality/value This study, for the first time, presents reproducing kernel algorithm for obtaining the numerical solutions of some certain classes of Robin time-fractional partial differential equations. An efficient construction is provided to obtain the numerical solutions for the equations, along with an existence proof of the exact solutions based upon the reproducing kernel theory.

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The method of approximate particular solutions for the time-fractional diffusion equation with a non-local boundary condition

Cited by 23 publications

References 40 publications

Method of Particular Solutions for Second-Order Differential Equation with Variable Coefficients via Orthogonal Polynomials

Method of Particular Solutions for Second-Order Differential Equation with Variable Coefficients via Orthogonal Polynomials

Meshless Analysis of Nonlocal Boundary Value Problems in Anisotropic and Inhomogeneous Media

Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm

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