Method of approximate particular solutions for constant- and variable-order fractional diffusion models

Fu, Zhuo; Ling, Leevan

doi:10.1016/j.enganabound.2014.09.003

Cited by 104 publications

(35 citation statements)

References 45 publications

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“…We will also investigate the boundedness of fractional multilinear integral operators with rough kernels T A 1 ;A 2 ;:::;A k ;˛o n the generalized weighted Nikolskii-Morrey spaces, see for example, [30]. These results may be applicable to some problems of partial differential equations; see for example [6,7,19,20,26,28,30,43].…”

Section: Introduction and Resultsmentioning

confidence: 99%

Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

Akbulut

Hasanov

2016

Open Mathematics

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Abstract:In this paper, we study the boundedness of fractional multilinear integral operators with rough kernels T A 1 ;A 2 ;:::;A k ;˛, which is a generalization of the higher-order commutator of the rough fractional integral on the generalized weighted Morrey spaces M p;' .w/. We find the sufficient conditions on the pair .' 1 ; ' 2 / with w 2 A p;q which ensures the boundedness of the operators T ;˛a re given in terms of Zygmund-type integral inequalities on .' 1 ; ' 2 / and w, which do not assume any assumption on monotonicity of ' 1 .x; r/; ' 2 .x; r/ in r.

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Section: Introduction and Resultsmentioning

confidence: 99%

Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

Akbulut

Hasanov

2016

Open Mathematics

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“…Finite element methods have been introduced in [11][12][13] to obtain numerical solutions of FDEs; also the numerical treatments based on finite difference methods have been developed in [14][15][16]. Moreover, several spectral techniques were designed for such equations (see for instance, [17][18][19][20][21][22][23][24][25][26][27]). …”

Section: Introductionmentioning

confidence: 99%

Efficient Spectral Collocation Algorithm for a Two-Sided Space Fractional Boussinesq Equation with Non-local Conditions

2015

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A spectral shifted Legendre Gauss-Lobatto collocation method is developed and analyzed to solve numerically one-dimensional two-sided space fractional Boussinesq (SFB) equation with non-classical boundary conditions. The method depends basically on the fact that an expansion in a series of shifted Legendre polynomials PL,n(x), x ∈ [0, L] is assumed, for the function and its space-fractional derivatives occurring in the two-sided SFB equation. The Legendre-Gauss-Lobatto quadrature rule is established to treat the non-local conservation conditions, and then the problem with its non-local conservation conditions is reduced to a system of ordinary differential equations (ODEs) in time. Thereby, the expansion coefficients are then determined by reducing the two-sided SFB with its boundary and initial conditions to a system of ODEs for these coefficients. This system may be solved numerically in a step-by-step manner by using implicit Runge-Kutta method of order four. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented.Mathematics Subject Classification. 34A08, 65M70, 33C45.

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“…Using the interval [ min , max ] = [6,7] in variable shape parameter, the error functions obtained by constant, exponential, linear, and random shape parameter strategies are shown in Figure 3(b). To demonstrate the reliability of this interval, the error functions obtained by exponential and random variable shape parameter in the intervals [0.5, 1.5], [2,3], [4,5], and [6,7] are plotted in Figure 4. The errors demonstrate that using the proposed interval [6,7], obtained by the proposed algorithm, results in better accuracy.…”

Section: One-and Two-dimensional Interpolationmentioning

confidence: 99%

“…Radial basis function methods are the tools for interpolating a multivariate data set, approximating a function, and solving partial differential equations [1][2][3][4][5]. A radial basis function, say ( ), is a continuous univariate function that has been realized by composition with the Euclidean norm in R .…”

Section: Introductionmentioning

confidence: 99%

Selection of an Interval for Variable Shape Parameter in Approximation by Radial Basis Functions

Biazar

Hosami

2016

Advances in Numerical Analysis

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In radial basis function approximation, the shape parameter can be variable. The values of the variable shape parameter strategies are selected from an interval which is usually determined by trial and error. As yet there is not any algorithm for determining an appropriate interval, although there are some recipes for optimal values. In this paper, a novel algorithm for determining an interval is proposed. Different variable shape parameter strategies are examined. The results show that the determined interval significantly improved the accuracy and is suitable enough to count on in variable shape parameter strategies.

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Method of approximate particular solutions for constant- and variable-order fractional diffusion models

Cited by 104 publications

References 45 publications

Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

Efficient Spectral Collocation Algorithm for a Two-Sided Space Fractional Boussinesq Equation with Non-local Conditions

Selection of an Interval for Variable Shape Parameter in Approximation by Radial Basis Functions

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