In this paper, a Jacobi-collocation spectral method is developed for Volterra integral equations of second kind with a weakly singular kernel. We use some function transformation and variable transformations to change the equation into a new Volterra integral equation defined on the standard interval [−1, 1], so that the solution of the new equation possesses better regularity and the Jacobi or...

We investigate the interlacing of the zeros of linear combinations pn+ aqm with the zeros of the components pn and qm, where {pn} ∞ n=0 and {qm} ∞ m=0 are different sequences of Jacobi polynomials. The results we prove hold when pn and qm are Jacobi polynomials P (α,β) n (x) and P (α,β) m (x) for certain values of α and β with m = n or m = n−1. Numerical counterexamples are given in situations ...

One of the interesting features in the development of analysis in the twentieth century is the remarkable growth, in various directions, of the theory of orthogonal functions. Two brilliant achievements on the threshold of this century—Fejér's paper on Fourier series and Fredholm's papers on integral equations—have been acting as a powerful inspiring source of attraction, inviting analysts to d...

We introduce an orthogonal system on the half line, induced by Jacobi polynomials. Some results on the Jacobi rational approximation are established, which play important roles in designing and analyzing the Jacobi rational spectral method for various differential equations, with the coefficients degenerating at certain points and growing up at infinity. The Jacobi rational spectral method is p...

Many limits are known for hypergeometric orthogonal polynomials that occur in the Askey scheme. We show how asymptotic representations can be derived by using the generating functions of the polynomials. For example, we discuss the asymptotic representation of the Meixner-Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Laguerre polynomials.

and Applied Analysis 3 nonlinear third-order differential equations. Section 5 gives the corresponding results for those obtained in Sections 2, 3, and 4, but for the fifth-order differential equations. In Section 6, we present some numerical results exhibiting the accuracy and efficiency of our numerical algorithms. 2. SJG Method for Third-Order Differential Equations with Constant Coefficient...