A general framework for proving an order of convergence for set-valued Runge Kutta methods is given in the case of linear differential inclusions, if the attainable set at a given time should be approximated. The set-valued method is interpreted as a (set-valued) quadrature method with disturbed values for the fundamental solution at the nodes of the quadrature method. If the precision of the q...

This paper proposes a direct approach for solving optimal control problems. The time domain is divided into multiple subdomains, and a Lagrange interpolating polynomial using the Legendre–Gauss– Lobatto points is used to approximate the states and controls. The state equations are enforced at the Legendre–Gauss–Lobatto nodes in a nonlinear programming implementation by partial Gauss–Lobatto qua...

in this paper, operational matrices of riemann-liouville fractional integration and caputo fractional differentiation for shifted jacobi polynomials are considered. using the given initial conditions, we transform the fractional differential equation (fde) into a modified fractional differential equation with zero initial conditions. next, all the existing functions in modified differential equ...

Highly-oscillatory integrals are allegedly difficult to calculate. The main assertion of this paper is that that impression is incorrect. As long as appropriate quadrature methods are used, their accuracy increases when oscillation becomes faster and suitable choice of quadrature points renders this welcome phenomenon more pronounced. We focus our analysis on Filon-type quadrature and analyse i...

A quadrature Galerkin scheme with the Bogner–Fox–Schmit element for a biharmonic problem on a rectangular polygon is analyzed for existence, uniqueness, and convergence of the discrete solution. It is known that a product Gaussian quadrature with at least three-points is required to guarantee optimal order convergence in Sobolev norms. In this article, optimal order error estimates are proved f...

In this research, a numerical method by piecewise approximated method for solving fuzzy differential equations is introduced. In this method, the solution by piecewise fuzzy polynomial is present. The base of this method is using fuzzy Taylor expansion on initial value of fuzzy differential equations. The existence, uniqueness and convergence of the approximate solution are investigated. To sho...

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the W (m,m−1) 2 [0, 1] space for calculating Fourier coefficients. Using S. L. Sobolev’s method we obtain new optimal quadrature formulas of such type for N + 1 ≥ m, where N + 1 is the number of the nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We investigate the...

where 5 > 0, a is a constant, and F\(p) is analytic and bounded in the half plane Re(j>) > c. We assume that this condition is satisfied. Whether this condition is also sufficient for the convergence of the w-point quadrature formula to the true value of f(t) in (1), when n tends to infinity, has not been determined. The author makes use here of the fact that the convergence occurs whenever F(p...

Spherical t-designs are point sets XM := {x1, . . . ,xM} ⊂ S2 which provide quadrature rules with equal weights for the sphere which are exact for polynomials up to degree t. In this paper we consider the problem of finding numerical spherical t-designs on the sphere S2 for high polynomial degree t ∈ N. That is, we compute numerically local minimizers of a certain quadrature error At(XM ). The ...

Gauss and Clenshaw–Curtis quadrature, like Legendre and Chebyshev spectral methods, make use of grids strongly clustered at boundaries. From the viewpoint of polynomial approximation this seems necessary and indeed in certain respects optimal. Nevertheless such methods may “waste” a factor of π/2 with respect to each space dimension. We propose new nonpolynomial quadrature methods that avoid th...