Differentiation by integration using orthogonal polynomials, a survey

通过微分求积分的探索

Exploring Integration by Differentiation

本工作验证并扩展了通过微分求积分的方法，证明其与经典积分规则兼容，并将其应用于经典积分、多变量情形、单纯形体积计算及Jackson的q-积分。

A Fractional Calculus on Arbitrary Time Scales: Fractional Differentiation and Fractional Integration

We introduce a general notion of fractional (noninteger) derivative for functions defined on arbitrary time scales. The basic tools for the time-scale fractional calculus (fractional differentiation and fractional integration) are then developed. As particular cases, one obtains the usual time-scale Hilger derivative when the order of differentiation is one, and a local approach to fractional calculus when the time scale is chosen to be the set of real numbers.

Sobolev-Orthogonal Systems with Tridiagonal Skew-Hermitian Differentiation Matrices

We introduce and develop a theory of orthogonality with respect to Sobolev inner products on the real line for sequences of functions with a tridiagonal, skew-Hermitian differentiation matrix. While a theory of such L2-orthogonal systems is well established, Sobolev orthogonality requires new concepts and their analysis. We characterise such systems completely as appropriately weighed Fourier transforms of orthogonal polynomials and present a number of illustrative examples, inclusive of a Sobolev-orthogonal system whose leading N coefficients can be computed in $\mathcal{O}(N \log N)$ operati…

Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation

A solution to the more than 300-years old problem of geometric and physical interpretation of fractional integration and differentiation (i.e., integration and differentiation of an arbitrary real order) is suggested for the Riemann-Liouville fractional integration and differentiation, the Caputo fractional differentiation, the Riesz potential, and the Feller potential. It is also generalized for giving a new geometric and physical interpretation of more general convolution integrals of the Volterra type. Besides this, a new physical interpretation is suggested for the Stieltjes integral.

Generalized fractional operator representations of Jacobi type orthogonal polynomials

The aim of this paper is to apply generalized operators of fractional integration and differentiation involving Appell's function $F_{3}(:)$ due to Marichev-Saigo-Maeda (MSM), to the Jacobi type orthogonal polynomials. The results are expressed in terms of generalized hypergeometric function. Some of the interesting special cases of the main results also established.

Fractional Powers of the Differentiation and Integration Operators and its Application in Accelerator Physics and Technology

In this paper we present a solution to a fractional integral of the order 3/2 with the use of a novel method. The integral arises during solving the Biot-Savart equation to find the exact analytical solution for the magnetic field components of a solenoid. We solved the integral by cutting the branch line in order to have an analytic function inside the integral instead of multi-valued operation.