2006
DOI: 10.1016/j.sigpro.2006.02.009
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Numerical treatment of an initial-boundary value problem for fractional partial differential equations

Abstract: This paper deals with numerical solutions to a partial differential equation of fractional order. Generally this type of equation describes a transition from anomalous diffusion to transport processes. From a phenomenological point of view, the equation includes at least two fractional derivatives: spatial and temporal. In this paper we proposed a new numerical scheme for the spatial derivative, the so called RieszFeller operator. Moreover, using the finite difference method, we show how to employ this scheme … Show more

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Cited by 28 publications
(20 citation statements)
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“…In fact, the fractional Laplacian operator ðÀDÞ a=2 has already found numerous applications in many fields of physics, mathematics, mechanical engineering, biology, electrical engineering, control theory and finance [3,10,11]. One can find interesting properties and interpretations of fractional calculus in [14,15,17], which also give a useful mathematical tool for modeling many process in nature.…”
Section: Discussionmentioning
confidence: 99%
“…In fact, the fractional Laplacian operator ðÀDÞ a=2 has already found numerous applications in many fields of physics, mathematics, mechanical engineering, biology, electrical engineering, control theory and finance [3,10,11]. One can find interesting properties and interpretations of fractional calculus in [14,15,17], which also give a useful mathematical tool for modeling many process in nature.…”
Section: Discussionmentioning
confidence: 99%
“…where ν > 0 is the coefficient of the kinematics viscosity of the fluid, R α x is the the Riesz fractional derivative of order α and 0 < α ≤ 2, α ̸ = 1. The Riesz fractional derivative and its generalizations are used to model phenomenae as random walk models and anomalous diffusion in the form of fractional partial differential equations (FPDEs) [2]- [7]. Numerical solutions for FPDEs where Riesz derivative is defined on unbounded domains include for example the work in [8] and [9].…”
Section: Introductionmentioning
confidence: 99%
“…Fractional calculus in mathematics is a natural extension of integer-order calculus and gives a useful mathematical tool for modeling many processes in nature. One of these processes, in which fractional derivatives have been successfully applied, is called diffusion [1]. Fractional derivatives have recently been applied to many problems in physics [2][3][4][5][6][7][8], finance [9,10], and hydrology [11].…”
Section: Introductionmentioning
confidence: 99%
“…For simplicity, we only consider fractional derivatives in space and the index  is assumed to lie within the range 1 2    and therefore 2. m  By applying the Laplace transformation on equations (1) and (3, 4) with consideration of initial condition, we obtain…”
Section: Introductionmentioning
confidence: 99%