In this paper, An effective and simple numerical method is proposed for solving systems of integral equations using radial basis functions (RBFs). We present an algorithm based on interpolation by radial basis functions including multiquadratics (MQs), using Legendre-Gauss-Lobatto nodes and weights. Also a theorem is proved for convergence of the algorithm. Some numerical examples are presented...

A new set of modified Legendre rational functions which are mutually orthogonal in L2(0,+∞) is introduced. Various projection and interpolation results using the modified Legendre rational functions are established. These results form the mathematical foundation of related spectral and pseudospectral methods for solving partial differential equations on the half line. A spectral scheme using th...

Abstract. Let p > 3 be a prime, and let m be an integer with p ∤ m. In the paper we solve some conjectures of Z.W. Sun concerning Pp−1 k=0 2k k 3 /mk (mod p2), Pp−1 k=0 2k k 4k 2k /mk (mod p) and Pp−1 k=0 2k k 2 4k 2k /mk (mod p2). In particular, we show that P p−1 2 k=0 2k k 3 ≡ 0 (mod p2) for p ≡ 3, 5, 6 (mod 7). Let {Pn(x)} be the Legendre polynomials. In the paper we also show that P[ p 4 ]...

In this manuscript a new method is introduced for solving fractional differential equations. The fractional derivative is described in the Caputo sense. The main idea is to use fractional-order Legendre wavelets and operational matrix of fractional-order integration. First the fractional-order Legendre wavelets (FLWs) are presented. Then a family of piecewise functions is proposed, based on whi...