Exact discretization by Fourier transforms
Introduction
Differential and integral equations of integer and non-integer orders are very important for the theory of systems and processes in continua, media and fields. Equations with derivatives and integrals of non-integer orders [1], [2], [3], [4], [5], [6], [7], [8] are actively used to describe processes and systems with power-law nonlocality and memory [12], [13], [14], [15], [16], [17], [18]. As it was shown in [14], [36], [37], [38], [39], the differential equations with derivatives of non-integer orders are directly connected with discrete models of systems with long-range interactions of power-law type. Interconnection between the equations of these discrete systems (lattices) and the fractional differential equations is proved by special transform operator that includes the Fourier series and integral transforms [36], [37], [38], [39]. This approach has been applied to discrete models of fractional nonlocal continua and fields (for example, see [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58]). In works [38], [39], we propose a lattice fractional calculus, which demonstrates that infiniteness of series of discrete operators is a characteristic property of discretization of derivatives of integer and non-integer orders.
Infinite series of fractional differences and the corresponding derivatives of non-integer orders have been first proposed by Grünwald [27] and by Letnikov [28] in 1897 and 1898 respectively. Now there are other types of fractional differences, which are proposed by Kuttner [29], Cargo and Shisha [30], Diaz and Osler [31], Ortigueira and Coito [32], Ortigueira [33], [34], Tarasov [35], [36], [37], Ortigueira, Coito and Trujillo [42] and other. The Grünwald–Letnikov differences [1], [27], [28] and other type of fractional differences, which are proposed in [29], [30], [31], [32], [33], [34], [35], [36], [37], cannot be considered an exact discretization of corresponding derivatives, since the difference have unusual characteristic algebraic properties, which do not coincide with the properties of differential operators. It is easy to see that for integer values of orders, these differences do not have the same algebraic properties as the integer-order derivatives.
It is well-known that the standard finite difference of integer order cannot be considered as an exact discretization of the derivative of this order [86]. The problems of exact discretization of differential equations of integer orders have been formulated in [87], [88], [89], [90], [91] (see also [92], [93], [94] and [95], [96], [97], [98], [99]). Mickens proved that for differential equations there are ”locally exact” finite-difference discretization, where the local truncation errors are zero. A main disadvantage of the Mickens discretization of the integer-order derivatives is that these differences strongly depend on the form of the considered differential equation and the parameters of these equations. In addition, the Mickens differences do not have the same algebraic properties as derivative operators of integer orders. A detailed discussion of the Mickens approach is given in Section 2.
Our approach to exact discretization is based on the principle of universality and the algebraic correspondence principle. The differences, which are exact discretization of derivatives, have a property of universality if they do not depend on the form of differential equations and the parameters of these equations. An algebraic correspondence means that an exact discretization of derivatives satisfies the same algebraic relations as the operators of differentiation. In the exact discretization of derivatives, we should have exact discrete analogs of algebraic properties that are performed for derivatives. The exact discrete analog of derivatives should have the same basic characteristic properties as the operators of differentiation. The Leibniz rule is a characteristic property of the derivative operators of integer orders [20], [26]. Therefore the exact discretization of these operators must obey this rule. The Leibniz rule should be the main characteristic property of exact discrete analogs of derivative. The second important algebraic property of the exact discretization is the semi-group property. For example, the second-order difference should be equal to the repeated action of the first-order differences. The third important algebraic property of the exact discretization is that the exact difference operators of power-law functions should give the same expression as an action of derivatives.
We propose new approach to exact discretization that is based on new difference operators, which can be considered as an exact discretization of derivatives of integer (and non-integer) orders. These differences do not depend on the form of differential equations and the parameters of these equations. Using these differences, we can get an exact discretization of differential equation of integer and non-integer orders. The suggested approach to exact discretization allows us to obtain difference equations that exactly correspond to the differential equations. We consider not only an exact correspondence between the equations, but also exact correspondence between solutions. We demonstrate that exact discrete analogs of solutions of differential equations are solutions of the corresponding difference equations.
In this paper, a new type of differences of integer and non-integer orders, which are represented by infinite series, are suggested. The suggested differences are derived by discretization of the Riesz differentiation and integration of non-integer and integer orders. The procedure of discretization is proposed by analogy with the procedure of Weyl quantization that preserves the properties of algebraic structures.
In this paper, we propose the differences which can be considered as exact discretization of the derivatives dn/dxn (), since these differences preserve the following characteristic property of differential operator.
- (1)
The Leibniz rule for -difference
- (2)
The equations of -difference of power-law functions
- (3)
The semigroup property
The suggested differences are represented by infinite series instead of finite series that are the usually used in standard and non-standard (Mickens) differences. The proposed approach to exact discretization allows us to get difference equations that exactly correspond to the differential equations. The suggested discretization of the differential equations is exact for wide class of functions and equations.
Section snippets
Problem of exact discretization of equations
There are various approaches to discretization of differential equation with some approximations. In this paper, we do not consider these approached in details. We consider a problem of an exact discretization of the differential equations of integer and non-integer orders. It should be noted that some approaches to exact discretization have been considered in [87], [88], [89], [90], [91] (see also [92], [93], [94] and [95], [96], [97], [99]).
The problem of exact discretization has been
Discretization of differential and integral operators and criterion of exact discretization
To obtain new type of differences, which are discretization of derivatives of integer and non-integer orders, we use the Fourier series and integral transforms. In order to introduce the notation of these transformations we consider functions f(x) belonging to the space of square integrable functions [59] or the Lizorkin space [1]. We can apply the Fourier integral transform to obtain where . By the next step, we define the function for . Finally
Criterion of exact discretization for differential equations
An exact discretization is a map of differential equation into a discrete (difference) equation, for which the solution f[n] of the discrete equation and the solution f(x) of associated differential equation are the same, i.e. if and only if the discrete function f[n] is exactly equal to the function f(x) for i.e., () for arbitrary values of h > 0.
Let consider an criterion of exact discretization of derivatives. Suppose that is a function space and ADα is a differential
Criterion of exact discretization for differential operators
Principle of universality: The difference operators, which are exact discretization of derivatives of integer or non-integer orders, should not depend on the form of differential equations and the parameters of these equations.
Principle of algebraic correspondence: The difference operators, which are exact discretization of differential operators of integer or non-integer orders, should satisfy the same algebraic characteristic relations as the differential operators.
In some sense, the exact
Exact discretization of Riesz operators of fractional orders
Theory of derivatives and integrals of non-integer orders [1], [2], [3], [4], [5], [6], [7], [8] has a big history [9], [10], [11] and it goes back to the well-known scientist such as Leibniz, Riemann, Liouville, Letnikov, Weyl, Riesz and other. Fractional integro-differential equations are very important to describe processes in continua, media and fields with power-law nonlocality and memory [12], [13], [14], [15], [16], [17], [18]. Fractional integral and differential operators are nonlocal
Exact discretization of differential operators of integer orders
In this section, the suggested approach to exact discretization are applied to derivatives of integer orders. Using the Fourier transforms, we map derivatives of integer orders into the special differences that are represented by infinite series. The differences, which are derived by application of transform operator to the derivative of integer order will be denoted by . We can represent these sets of transformations of the derivatives of integer orders into differences in the form
Differenceability of discrete functions
Let us define a discrete analogue of differentiability of a continuous function of one real variable that will be called ”differenceability”. A differentiable function f(x) of variable is a function whose derivative df(x)/dx exists at each point in its domain.
Definition 11 Discrete function f[n] defined for is said to be of differenceability class if the -differences . . . , are represented by convergent series.
For example, the discrete function f[n] is called
Standard finite differences and exact discretization
Let us consider standard finite differences of integer orders to compare these differences with the suggested -differences of integer orders.
Definition 17 The forward finite difference fΔm of order is given by
The backward finite difference fΔm of order is given by
The central finite difference cΔm of order is given by
Let us give the proposition about these standard
Examples of exact discretization of differential equations and its solutions
In this section we give examples of the suggested approach to exact discretization of differential equations by the -differences. To compare our approach with the standard approach and the Mickens exact discretization, we will present the results in the form of tables (Tables 1 and 2).
Conclusion
In this paper, we propose new approach to exact discretization that is based on new difference operators of integer and non-integer orders, which are represented by infinite series. The suggested differences can be considered as an exact discretization of derivatives of integer and non-integer orders. These differences have a property of universality because they do not depend on the form of differential equations and the parameters of these equations. The suggested differences have the same
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