Difference methods for solving boundary value problems for fractional differential equations

Taukenova, F. I.; Shkhanukov-Lafishev, M. Kh.

doi:10.1134/s0965542506100149

Cited by 58 publications

(24 citation statements)

References 5 publications

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“…( , ) 0 1 (where h is the mesh spacing in the geometrical variable) and associate the problem of calculating the pressure field (22), (25) with the differential-difference problem of the form…”

Section: A Technique To Derive An Approximate Solution Of the Boundarmentioning

confidence: 99%

Nonclassical mathematical model in geoinformatics to solve dynamic problems for nonequilibrium nonisothermal seepage fields

Bulavatsky

2011

Cybern Syst Anal

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Section: A Technique To Derive An Approximate Solution Of the Boundarmentioning

confidence: 99%

Nonclassical mathematical model in geoinformatics to solve dynamic problems for nonequilibrium nonisothermal seepage fields

Bulavatsky

2011

Cybern Syst Anal

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“…Also, several algorithms have been developed to solve FDEs numerically such as fractional Adams-Moulton methods, explicit Adams multistep methods, fractional difference method, decomposition method, variational iteration method, least squares finite element solution, extrapolation method and the Kansa method which is meshless, easy-to-use and has been used to handle a broad range of partial differential equation models [6-8, 11, 12, 22, 29, 32]. The existence of at least one solution of fractional two-point boundary value problems can be seen in [3,25,31,32,34].…”

Section: Introductionmentioning

confidence: 99%

“…There are many applications of fractional derivative and fractional integration in several complex systems such as physics, chemistry, fluid mechanics, viscoelasticity, signal processing, mathematical biology, and bioengineering and various applications in many branches of science and engineering could be found [1-5, 16, 19-21, 24-27, 31, 33, 34]. Boundary value problems of fractional order occur in the description of many physical processes of stochastic transport and in the investigation of liquid filtration in a strongly porous (fractal) medium [32]. Also, boundary value problems with integral boundary conditions constitute a very interesting and important class of problems.…”

Section: Introductionmentioning

confidence: 99%

Quadratic spline solution for boundary value problem of fractional order

Zahra

Elkholy

2011

Numer Algor

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Fractional differential equations are widely applied in physics, chemistry as well as engineering fields. Therefore, approximating the solution of differential equations of fractional order is necessary. We consider the quadratic polynomial spline function based method to find approximate solution for a class of boundary value problems of fractional order. We derive a consistency relation which can be used for computing approximation to the solution for this class of boundary value problems. Convergence analysis of the method is discussed. Four numerical examples are included to illustrate the practical usefulness of the proposed method.Keywords Boundary value problems of fractional order · Riemann-Liouville fractional derivative · Caputo fractional derivative · Quadratic polynomial spline · Error bound

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“…There are three principal FPDEs: space FPDEs, time FPDEs and space-time FPDEs. Taukenova and Lafishev [24] studied the difference schemes for solving time fractional diffusion equations in one-and multi-dimensional domains, they also proved the stability and convergence of their schemes. In [25], the authors investigated the spectral tau method for solving space FPDEs.…”

Section: Introductionmentioning

confidence: 99%

Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method

Doha

Bhrawy²,

Ezz-Eldien

2013

Open Physics

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Abstract:In this paper, a class of fractional diffusion equations with variable coefficients is considered. An accurate and efficient spectral tau technique for solving the fractional diffusion equations numerically is proposed. This method is based upon Chebyshev tau approximation together with Chebyshev operational matrix of Caputo fractional differentiation. Such approach has the advantage of reducing the problem to the solution of a system of algebraic equations, which may then be solved by any standard numerical technique. We apply this general method to solve four specific examples. In each of the examples considered, the numerical results show that the proposed method is of high accuracy and is efficient for solving the timedependent fractional diffusion equations.PACS (2008)

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Difference methods for solving boundary value problems for fractional differential equations

Cited by 58 publications

References 5 publications

Nonclassical mathematical model in geoinformatics to solve dynamic problems for nonequilibrium nonisothermal seepage fields

Nonclassical mathematical model in geoinformatics to solve dynamic problems for nonequilibrium nonisothermal seepage fields

Quadratic spline solution for boundary value problem of fractional order

Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method

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