Learn More
A new formula expressing explicitly the derivatives of Bernstein polynomials of any degree and for any order in terms of Bernstein polynomials themselves is proved, and a formula expressing the Bernstein coefficients of the general-order derivative of a differentiable function in terms of its Bernstein coefficients is deduced. An application of how to use(More)
It is well known that spectral methods (tau, Galerkin, collocation) have a condition number of $O(N^{4})$ ( $N$ is the number of retained modes of polynomial approximations). This paper presents some efficient spectral algorithms, which have a condition number of $O(N^{2})$ , based on the Jacobi–Galerkin methods of second-order elliptic equations in one and(More)
We are concerned with linear and nonlinear multi-term fractional differential equations (FDEs). The shifted Chebyshev operationalmatrix (COM) of fractional derivatives is derived and used together with spectral methods for solving FDEs. Our approach was based on the shifted Chebyshev tau and collocation methods. The proposed algorithms are applied to solve(More)
A new explicit formula for the integrals of Bernstein polynomials of any degree for any order in terms of Bernstein polynomials themselves is derived. A fast and accurate algorithm is developed for the solution of high even-order boundary value problems (BVPs) with two point boundary conditions but by considering their integrated forms. The(More)
This paper is concerned with fast spectral-Galerkin Jacobi algorithms for solving one- and two-dimensional elliptic equations with homogeneous and nonhomogeneous Neumann boundary conditions. The paper extends the algorithms proposed by Shen (SIAM J Sci Comput 15:1489–1505, 1994) and Auteri et al. (J Comput Phys 185:427–444, 2003), based on Legendre(More)
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(More)