The variable order wave equation plays a major role in acoustics, electromagnetics, and fluid dynamics. In this paper, we consider the space-time variable order fractional wave equation with variable coefficients. We propose an effective numerical method for solving the aforementioned problem in a bounded domain. The shifted Jacobi polynomials are used as basis functions, and the variable-order...

We observe that the exact connection coefficient relations transforming modal coefficients of one Jacobi Polynomial class to the modal coefficients of certain other classes are sparse. Because of this, when one of the classes corresponds to the Chebyshev case, the Fast Fourier Transform can be used to quickly compute modal coefficients for Jacobi Polynomial expansions of class (α, β) when 2α an...

We show asymptotic expansions of the eigenfunctions certain perturbations Jacobi operator in a bounded interval, deducing equiconvergence results between with respect to associated orthonormal basis and cosine basis. Several for pointwise convergence then follow.

This paper analyzes a method for solving the thirdand fifth-order differential equations with constant coefficients using a Jacobi dual-Petrov–Galerkin method, which is more reasonable than the standard Galerkin one. The spatial approximation is based on Jacobi polynomials P (α,β) n with α, β ∈ (−1, ∞) and n is the polynomial degree. By choosing appropriate base functions, the resulting system ...

The generalized Bernstein basis in the space Πn of polynomials of degree at most n, being an extension of the q-Bernstein basis introduced recently by G.M. Phillips, is given by the formula (see S. Lewanowicz & P. Woźny, BIT 44 (2004), 63–78) Bn i (x;ω| q) := 1 (ω; q)n [ n i ] q x (ωx−1; q)i (x; q)n−i (i = 0, 1, . . . , n). We give explicitly the dual basis functions Dn k (x; a, b, ω| q) for th...

Exact values are derived for some matrix elements of Lagrange functions, i.e. orthonormal cardinal functions, constructed from orthogonal polynomials. They are obtained with exact Gauss quadratures supplemented by corrections. In the particular case of Lagrange-Laguerre and shifted Lagrange-Jacobi functions, sum rules provide exact values for matrix elements of 1/x and 1/x as well as for the ki...

We give explicitly the recurrence coefficients of a nonsymmetric semi-classical sequence of polynomials of class s = 1. This sequence generalizes the Jacobi polynomial sequence, that is, we give a new orthogonal sequence { P̂ (α,α+1) n (x,μ) } , where μ is an arbitrary parameter with (1−μ) > 0 in such a way that for μ = 0 one has the well-known Jacobi polynomial sequence { P̂ (α,α+1) n (x) } , n≥ 0.