Finite-difference schemes for a diffusion equation with fractional derivatives in a multidimensional domain

Абрашина-Жадаева, Н. Г.; Timoshchenko, Igor

doi:10.1134/s001226611307001x

Cited by 5 publications

(3 citation statements)

References 7 publications

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“…Simple formulae relating the dimension of the fractal with the order of the fractional derivative were obtained in work [2]. At present, partial differential equations with fractional derivatives in time and spatial variables are treated as mathematical models of physical processes [3]- [6].…”

Section: Introductionmentioning

confidence: 99%

“…The mathematical models of such processes are described by partial differential equations with fractional derivatives in time and spatial variables [3]- [5]. In work [6], a numerical modelling of an anomalous diffusion in a multi-dimensional domain is made by means of the approximate factorization method. The Dirichlet initial-boundary value problem for a partial differential equations with fractional derivatives in time and spatial variables, there was studied an implicit scheme based on the approximate factorization method and the stability of the scheme was proved for the considered class of the problems.…”

Section: Introductionmentioning

confidence: 99%

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Difference schemes for partial differential equations of fractional order

Bazzaev¹,

Tsopanov²

2019

Ufimskii Matematicheskii Zhurnal

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Nowadays, fractional differential equations arise while describing physical systems with such properties as power nonlocality, long-term memory and fractal property. The order of the fractional derivative is determined by the dimension of the fractal. Fractional mathematical calculus in the theory of fractals and physical systems with memory and non-locality becomes as important as classical analysis in continuum mechanics.In this paper we consider higher order difference schemes of approximation for differential equations with fractional-order derivatives with respect to both spatial and time variables. Using the maximum principle, we obtain apriori estimates and prove the stability and the uniform convergence of difference schemes.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Difference schemes for partial differential equations of fractional order

Bazzaev¹,

Tsopanov²

2019

Ufimskii Matematicheskii Zhurnal

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“…Sometimes the preconditioned matrix is well structured than the original matrix. Figure(6) corresponding to the chequer-board ordering (odd even ordering) the numbers e B X appears in the element position (7,9) and (8,10) can be annihilated by using a precondition matrix without affecting the structure of the block sub matrices as illustrated in figure (7) and figure (8) The preconditioned system is written in the form P(AU=f) or A p U=f p (15)…”

Section: Figure6chequer-board Ordering;mentioning

confidence: 99%

Precondition for Discretized Fractional Boundary Value Problem

Youssef¹

2014

PAMJ

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Abstract:A precondition that uses the special structure of the algebraic system arising from the discretization of a fractional partial differential equation on the red black ordering grid is introduced. Comparison with the numerical solution of the classical Poisson's equationis considered. A graphical representation for the precondition is illustrated. The performance of our treatment is calculated for different values of the fractional order. The results of the implementation of the SOR and the KSOR with the help of MATLAB are given in compact tables. The value of the relaxation parameter is chosen on the bases of the graphical representation of the behavior of the spectral radius of the iteration matrices. Also, the natural ordering grid is considered.

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