A novel class of fractionally orthogonal quasi-polynomials and new fractional quadrature formulas
Introduction
In classical numerical integration theory [1], [2] the primary objective is to design a quadrature formulaequipped with a desired set of properties. In the current paper, both the integrand f and the weight function g are assumed to be real-valued function of real argument, i.e. and . The weight function g is usually assumed to be non-negative. We refer to the points at which the integrand f is evaluated as grid points. Without loss of generality we assume that . Real numbers are called quadrature coefficients. Quadrature formula (1) using grid points is said to be of order n.
Several groups of quadrature formulas can be distinguished. Among the most widely utilized ones are those of Newton–Cotes and Gauss types. Newton–Cotes quadratures use fixed and pre-specified grid values . Consequently, a Newton–Cotes formula of order n has exactly adjustable parameters (), and can be made accurate for integrands equal to polynomial functions of order up to n. Gaussian quadratures, on the other hand, allow for free selection of both grid points and quadrature coefficients . Thus, all parameters are chosen freely and, consequently, a Gaussian integration formula can be made exact for all polynomial integrands of order or less.
The simultaneous evaluation of both and requires solving a systems of non-linear equations. This system is not easily solvable in most cases. However, it turns out that the grid points can be chosen as zeros of polynomials belonging to suitably chosen orthogonal sets. Once the evaluation points are determined, the quadrature coefficients are easily found. For further details regarding orthogonal polynomials and classical Gaussian quadrature formulas we refer to [2], [3].
Fractional calculus is mathematical theory of non-integer order differentiation and integration. In recent decades fractional calculus is increasingly utilized in description of different natural phenomena, as well as in solving various engineering problems. A detailed survey of numerous fields of application of fractional calculus is beyond the scope of the present paper. A recent accounts can be found in [4], [5], [6], [7]. The primary operator of fractional calculus is fractional integral of order where stands for the well-known Euler’s Gamma function. This operator is known as the (Left) Riemann–Liouville fractional integral of order . For integer values of () formula (2) reduces to the well-known Cauchy repeated integration formula. Thus, for integer values of the integration order the fractional integral becomes the classical, repeated or n-fold integral. When tends to zero, reduces to the identity operator (the limit, however, must be taken in the weak, distributional sense). We refer to [8], [9], [10], [11] for further details on fractional calculus, alternative definitions of fractional integral and differential operators, and related concepts.
Numerous methods have been proposed for numerical evaluation of fractional integrals. A method based on finite memory approximation and Grünwald–Letnikov definition has been presented in [8]. A method based on discrete approximation of (2) is presented in [12], with similar methods generalizing and extending classical Trapezoidal rule proposed in [13], [14], [15]. Some novel numerical approximation methods based on Haar wavelet approximation theory are presented in [16], [17]. Efficient numerical methods for fractional differential equations have also been investigated in [18], with related numerical issues considered in [19]. For recent methods for evaluation of fractional derivatives we refer to [20], [21].
A plethora of methods for practical realization of fractional order systems has been developed. Each of these methods can be seen as an approximation of the infinite dimensional fractional order system by a suitable finite dimensional “rational” approximation, i.e. in a form of FIR or IIR filters of sufficiently high order. A survey of analog approximations is presented in [22]. Examples of digital approximations can be found in [23], [24]. Optimal rational approximations of fractional order systems, and fractional integrators in general, with special emphasis on retaining accurate steady state gain are proposed in [25]. Rational approximations by means of fractional Laguerre basis have been discussed in [26], [27].
Classical quadrature formulas assume that the integrand is well described by a polynomial of sufficiently high order. It has been suggested recently that some of the functions appearing in fractional calculus and, in particular, some of the functions emerging as solutions to fractional differential equations are better approximated using fractional power seriesknown as the fractional Taylor expansion [28], which is also utilized in [29]. A related research is presented also [30], [31]. It seems reasonable, therefore, to seek numerical integration schemes which are particularly suitable for such functions. In the sequel, a polynomial in , with ,is denoted as a quasi-polynomial of order n with commensurate power . We will also use a shorter, but slightly less precise term -polynomial of order n.
In this paper, the problem of numerical evaluation of fractional integral with unit upper integration limitis considered. By means of simple scaling of the upper integration limit t, the general problem of evaluating (2) can easily be reduced to the problem of evaluating (5). In particular, it can readily be shown that
In order to evaluate (5), a family of orthogonal sets of quasi-polynomials with arbitrary commensurate powers is designed first. The orthogonality is understood in the sense of fractional integration of order with upper limit equal to 1,where denotes orthogonal -polynomial of order n. The symbol in the upper right index simply denotes the order of the fractional integration used to define orthogonality.
Second, using designed orthogonal quasi-polynomials, a family of Gaussian quadrature formulais proposed. A formula of order n can be chosen from this family such that it is accurate for any -polynomial integrand of order or less. Both quadrature coefficients and grid points are functions of the integration order and the commensurate power . Thus, the commensurate power can be chosen in such a way that the integration error is minimized, even if the considered class of integrands can not be represented exactly by a quasi-polynomial of a finite order.
The rest of the paper is organized as follows. Interpolation by means of quasi-polynomials is considered in the following Section 2. Generalized Hermitean quadratures are considered and an error estimate is provided by Theorem 1. A short comment on the quasi-polynomial Lagrange interpolation is also given. Hermitean and Gaussian quasi-polynomial quadratures are developed in Section 3. The quadrature error has been investigated in Theorem 2. The main results of the paper are presented in Sections 4 A new class of orthogonal quasi-polynomials, 5 The proposed novel fractional Gaussian quadratures, presenting a new class of orthogonal polynomials and related novel Gaussian quadrature formulas, respectively. The existence of the proposed orthogonal quasi-polynomial families is established by Theorem 3, and the localization of their roots by Theorem 4. Concluding remarks are given in Section 6.
Section snippets
Interpolation by means of quasi-polynomials
Let us consider the problem of approximating a mapping with known function values and values of the first derivatives at distinct node points , such that
Consider the auxiliary functionwhere and . It is not difficult to see that is a -polynomial of order n, with zeros located at interpolation nodes (). More precisely,
Hermitean quadratures
By means of a classical Hermitean formula, the value of an integral of a function is evaluated numerically by exact integration of the corresponding Hermitean interpolating polynomial. In the present paper, a generalization of this approach to the case of fractional integration is presented. The proposed formulas are obtained by exact fractional integration of the corresponding Hermitean quasi-polynomial.
Consider a real-valued mapping . Assume that values of f and its first derivative
A new class of orthogonal quasi-polynomials
Quasi-polynomials orthogonal with respect to the fractional integration operator will be considered next. A procedure for explicit calculation of quasi-polynomial coefficients will be presented first in the following Section 4.1. An existence results will be presented in the subsequent Section 4.2, together with a convenient result claiming that all roots of such quasi-polynomials are simple and located within . Finally, some examples of orthogonal quasi-polynomials are presented in
The proposed novel fractional Gaussian quadratures
Considerations of the last two sections enable us to construct a continuum of approximate fractional integration formulas. First, by means of (6), each numerical integration problem is first reduced to the equivalent problem in which the upper terminal is 1. Second, given any positive real , a quadrature formulais designed in the following way
- 1.
Choose such that the integrand under consideration is naturally approximated by a -polynomial. Choose n so that the
Conclusions
A novel class of fractionally orthogonal quasi-polynomials has been developed and investigated. These quasi-polynomials are then utilized for development of suitable Gaussian quadratures for numerical evaluation of fractional integrals. The proposed quadratures are highly efficient and flexible, since the commensurate order of the interpolating quasi-polynomials can be selected freely and fine-tuned for a particular class of problems. As a secondary result, novel class of Lagrangian and
Acknowledgments
The authors wish to express their gratitude to prof. dr Gradimir V. Milovanović, Member of the Serbian Academy of Science and Arts, on his numerous suggestions and insights that considerably contributed to our work.
M.R. Rapaić and T.B. Šekara were partially supported by the Ministry of Education, Science and Technological Development of Republic of Serbia, Grants Nos. TR32018 and TR33013. (MRR) and TR33020 (TBŠ).
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