In this paper, we consider the second-kind Chebyshev polynomials (SKCPs) for the numerical solution of the fractional optimal control problems (FOCPs). Firstly, an introduction of the fractional calculus and properties of the shifted SKCPs are given and then operational matrix of fractional integration is introduced. Next, these properties are used together with the Legendre-Gauss quadrature fo...

In this paper, we present a numerical method for solving Hallen’s integral equation based on radial basis functions (RBFs). This method will represent the solution of Hallen’s integral equation by interpolating the radial basis functions based on Legendre-Gauss-Lobatto(LGL) nodes and weights. The numerical results show that the proposed method for Hallen’s integral equation is very accurate and...

This paper reports a new spectral collocation algorithm for solving time-space fractional partial differential equations with subdiffusion and superdiffusion. In this scheme we employ the shifted Legendre Gauss-Lobatto collocation scheme and the shifted Chebyshev Gauss-Radau collocation approximations for spatial and temporal discretizations, respectively. We focus on implementing the new algor...

A Table Errata is submitted concerning a formula in terms of a sum of two Gauss hypergeometric functions for the Ferrers function of the second kind (associated Legendre function of the second kind “on the cut”). This error occurs on p. 167 of Magnus, Oberhettinger & Soni (1966) Formulas and Theorems for the Special Functions of Mathematical Physics (third enlarged edition, Springer-Verlag, New...

A new boundary integral formulation of the second kind for exterior Stokes flow is introduced. The formulation is stable, complete, singularity-free, and natural for bodies of complicated shape and topology. We prove an existence and uniqueness result for the formulation for arbitrary flows and illustrate its performance via several numerical examples using a Nyström method with Gauss–Legendre ...

Barycentric interpolation is arguably the method of choice for numerical polynomial interpolation. The polynomial interpolant is expressed in terms of function values using the so-called barycentric weights, which depend on the interpolation points. Few explicit formulae for these barycentric weights are known. In [H. Wang and S. Xiang, Math. Comp., 81 (2012), 861–877], the authors have shown t...

Fractional directional integrals are the extensions of the Riemann-Liouville fractional integrals from oneto multi-dimensional spaces and play an important role in extending the fractional differentiation to diverse applications. In numerical evaluation of these integrals, the weakly singular kernels often fail the conventional quadrature rules such as Newton-Cotes and Gauss-Legendre rules. It ...