Numerical approximate solutions for variable-order Riesz fractional equations via adaptive method

This paper presents the numerical approximate solutions for variable-order Riesz fractional diffusion equation (RFDE) and Riesz fractional advection-dispersion equation (RFADE) with source term on a finite domain via adaptive method. To deal the numerical approximate solution of Riesz spatial fractional calculus, two numerical methods are provided. Numerical results are included to illustrate the results obtained.


Introduction
Fractional calculus has obtained considerable popularity and importance as generalizations of integer order differential equations during the past three decades or so, it enables the description of the memory and hereditary properties of different substances.Fractional calculus can model price volatility in finance, in hydrology to model fast spreading of pollutants, the generation of fractional Brownian motion as a representation of aquifer material with long-range correlation structure, the particle motions in a heterogeneous environment and long particle jumps of the anomalous diffusion in physics [1,2,5,7,12,14].Using the concept of variable-order fractional integration and differentiation [11], some diffusion processes in response to temperature changes may be better described.Generally speaking, there is a little difficult that we gain exact analytic solutions of fractional differential equation, and so the approximate and numerical techniques can be introduced and used.Multifractional models have been considered in the representation of heterogeneous local behaviors, the solutions of the models are defined in fractional Besov spaces of variable-order on R n [9].Kikuchi et al. [8] investigated the conditions for which general pseudodifferential operators on fractional Sobolev spaces of variable-order on R n form a Feller semigroup which has a transition density.Lin et al. [10] established an equality between the variable-order Riemann-Liouville fractional derivative and its Grünwald-Letnikov expansion.Using this relationship, they defined and obtained some properties of the operator (− d 2 dx 2 ) α(x,t) and devised an explicit finite difference approximation scheme for a corresponding variable-order nonlinear fractional diffusion equation.Meerschaert et al. [13] presented practical approximate methods to solve one dimensional fractional advection-dispersion equations with variable coefficients on a finite domain.Existing results of Riesz fractional diffusion and advection-dispersion equations on the numerical methods through a selected uniform grid, we are unaware of any other published work on discussion for the fractional advection-dispersion equation with adaptive method, which may lead to change the convergence rate and the steady state of the approximate solution error and is the generation of a grid that is adapted to the problem such that a given error criterion is fulfilled by the solution on this grid [16], since recent research devotes to the stable or asymptotically stable rates and as far as possible reducing between solutions estimation error.Harrison [6] suggested that the domain to be resolved with the highest resolution could be kept to a minimum in order to reduce the computer spatial and temporal requirements significantly.To design the adaptive method, ones should intuitively use a larger step size when the estimate is far from the optimum and a smaller step size as it approaches the optimum.So far, there is a few research on such approximate treatment for Riesz fractional equation, this motivates us to investigate generally computationally approximate techniques for solving variable-order Riesz fractional equations with source term.This paper is organized as follows: Section 2 provides some background material from fractional calculus.Section 3 presents the numerical approximate solution for RFDE with variable-order, and the numerical approximate solution for RFADE with variable-order is obtained in Section 4. Two examples are included to illustrate the results obtained in Section 5. Section 6 discusses the present work.

Preliminaries
We provide some background material used throughout the remaining sections of the present paper.Definition 2.1.[15] Let n is a positive integer, Riesz fractional operator for n Lemma 2.1.[4] For n − 1 < α(x,t) ≤ n, a function u(x,t) defined on the infinite interval x ∈] − ∞, ∞[×]0, T ], the following equality holds

Numerical approximate solution for RFDE
In this section, we consider the RFDE with 1 < α(x,t) ≤ 2 and the source term where K α(x,t) represents the average fluid velocity, f (x,t) can be used to represent source or sink term, g ∈ L 1 (]a, b[).The spectral representation of the Laplacian operator −∆ for RFDE with 1 < α(x,t) ≤ 2 is defined by and so (3.1) is rewritten as We will apply the L-approximate method to RFDE.The numerical approximate scheme for equation is based on by replacing the the first-order and second-order spatial derivatives with variable-order Riesz fractional derivatives, respectively.For 1 < α(x,t) ≤ 2, the left-handed and right-handed Grünwald-Letnikov fractional derivatives on [a, b] are defined by h N− j .Thus we have the following discrete scale result of du(x l ,t) dt .Theorem 3.1.The numerical approximate solution of RFDE (3.1) satisfies the system of the temporal differential Proof.The second term of the right-hand side of (3.3) can be approximated at x = x l by The third term of the right-hand side of (3.3) can be approximated at x = x l by ) , International Scientific Publications and Consulting Services and so we obtain an approximation of the left-handed fractional derivative (3.3) with 1 < α(x,t) ≤ 2 as ) ) ) . (3.6) Similarly, we have an approximation of the left-handed fractional derivative (3.4) with 1 < α(x,t) ≤ 2 as )) ) . ( Therefore, using Grünwald-Letnikov fractional derivatives definition on [a, b] (3.3) and (3.4), together with the numerical approximation (3.6) and (3.7), we can obtain the desired result, which can be solved by the differential/algebraic system solver [3].
The classical Grünwald weight are defined by [10] The coefficients g We also note that the left-handed fractional derivative of u(x,t) for variable x depends on all function values to the left of the point (x,t), i.e., this derivative is the weighted average of such function values.Similarly, the right-handed fractional derivative of u(x,t) for variable x depends on all function values to the right of this point (x,t).In general, the double-handed derivatives are not equal unless ⌈α(x,t)⌉ is an even integer, in which case, these derivatives become localized and equal.When ⌈α(x,t)⌉ is an odd integer, these derivatives become localized and opposite in sign.The Grünwald definitions for the double-handed spatial fractional derivatives are respectively where M + , M − are positive integers, Note that these normalized weights only depend on the order α(x,t) and the index k.The analytic definitions given by (3.3) and (3.4) are used in the formulation of the fractional PDE, while the Grünwald definitions (3.8) and (3.9) may be used to discretize the FPDE to obtain the numerical approximate solution.Here, we give the improvement form of the Grünwald definition, where λ 1 , λ 1 , . . ., λ M + are the given coefficient and ∑ To obtain a stable explicit Euler method, we state the shifted Grünwald formulae, where |H | = max{h 1 , h 2 , . . ., h M + }, and so we present the following shifted Grünwald estimate to the left-handed fractional derivative The shifted Grünwald estimate defined by (3.10) and (3.11) generally provides a more accurate numerical approximate than the standard Grünwald finite sum estimates obtained from (3.8) and (3.9).By the Shifted Grünwald approximate method (SGAM), we can also obtain the numerical approximate solution for RFDE Similar to Theorem 3.1, we can provide the following discrete scale result of du(x l ,t) dt .
Theorem 3.2.The numerical approximate solution of RFDE (3.1) satisfies the system of the temporal differential equations ) Proof.It is easy to check by using (3.12) and (3.13), we omit the proof process.

Numerical approximate solution for RFADE
We consider the RFADE with 1 < α(x,t) ≤ 2, 0 < β (x,t) ≤ 1 and the source term where K α(x,t) and K β (x,t) represent the dispersion coefficient and the average fluid velocity, f (x,t) can be used to represent source or sink term.The spectral representation of the Laplacian operator −∆ for RFADE is similar to (3.2), We employ L-approximate method for RFADE.For 0 < β (x,t) ≤ 1, the left-handed and right-handed Grünwald-Letnikov fractional derivatives on [a, b] are defined by ) )

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Proof.The second term of the right-hand side of (4.16) can be approximated at x = x l by ) , so we get an approximation of the left-handed fractional derivative (4.16) with 1 < α(x,t) ≤ 2 as Similarly, we have an approximation of the left-handed fractional derivative (4.17) with 1 < α(x,t) ≤ 2 as ( Therefore, using Grünwald-Letnikov fractional derivatives definition on [a, b], (4.16) and (4.17) together with the numerical numerical approximate (4.18) and (4.19), we have the numerical approximate solution for the equation (4.15).
By the Shifted Grünwald approximate method, we can also obtain the numerical approximate solution for RFADE.
Theorem 4.2.The numerical approximate solution of RFADE (4.15) satisfies the system of the temporal differential equations (5.21)The exact solution of (5.21) is u(x,t) = α(x,t)x(1 − x)(1 + t).Table 1 shows that the numerical approximate solution of (5.21) with the numerical scheme (3.5) or (3.14) for different times by using Theorems 3.1 and 3.2, it is seen that the problem exhibits diffusion behaviors that the solution continuously depends on the spatial and temporal variables.
It can be seen that all two numerical approximate solution are in good agreement with the exact solution, and can show that the LM(SGAM) is stable and convergent for solving (5.21).

Conclusion
From a new point of view, with variable step-size of the Riesz space, numerical approximate solution of both the RFDE and RFADE are derived, two numerical approximate methods are provided to deal with the variable-order Riesz fractional calculus on a bounded domain, the RFDE and RFADE are transformed into a system of ordinary differential equations and solved.The approximate schemes for equations are based on by replacing the the first-order and secondorder spatial derivatives with the Riesz fractional derivatives of order α ∈]1, 2] and of order β ∈]0, 1].Numerical results have been presented to demonstrate the effectiveness of the methods.We believe that the proposed methods can be applied to other kinds of variable-order nonlinear fractional differential equations, to nonlinear fractional partial derivative with the other derivative sense and to higher dimensions in the spatial domain.

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Table 1 :
Approximate solution and error at t = 0.5