A mesh refinement method is described for solving a continuous-time optimal control problem using collocation at Legendre-Gauss-Radau points. The method allows for changes in both the number of mesh intervals and the degree of the approximating polynomial within a mesh interval. First, a relative error estimate is derived based on the difference between the Lagrange polynomial approximation of ...

We describe a technique to analytically compute the multipole moments of a charge distribution confined to a planar triangle, which may be useful in solving the Laplace equation using the fast multipole boundary element method (FMBEM) and for charged particle tracking. This algorithm proceeds by performing the necessary integration recursively within a specific coordinate system, and then trans...

A collocation procedure is developed for the linear and nonlinear Fredholm and Volterra integral equations, using the globally defined B-spline and auxiliary basis functions. The solution is collocated by cubic B-spline and then the integral equation is approximated by the 5-points Gauss–Turán quadrature formula with respect to the Legendre weight function. Combination of these two approaches i...

In this work, Laplace transform-Legendre-wavelet collocation method is adopted for a semi-numerical analysis of predicting transient nonlinear behaviour radiative-convective fin with temperature-variant internal heat generation. The verification the results hybrid shown good agreements direct numerical and approximate analytical s in previous works. Parametric depicts significance model paramet...

This work presents a method to solve boundary value problems based on polynomial approximations and the application of the methods of moments and the Galerkin method. The weighted average residuals are evaluated by improved Gauss-Radau and Gauss-Lobatto quadratures, capable to exactly compute integrals of polynomials of degree 2n and 2n + 2 (where n is the number of internal quadrature points),...

The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicate that the numerical errors (in the infinity norm) will decay exponentially provid...

The idea of `1-minimization is the basis of the widely adopted compressive sensing method for function approximation. In this paper, we extend its application to high-dimensional stochastic collocation methods. To facilitate practical implementation, we employ orthogonal polynomials, particularly Legendre polynomials, as basis functions, and focus on the cases where the dimensionality is high s...

In this paper, we extend the operational matrix method to solve tempered fractional differential equation, via shifted Legendre polynomial. Although is widely used in solving various calculus problems, it yet apply equations defined derivatives. We first derive analytical expression for derivative x p</...

Stable and spectrally accurate numerical methods are constructed on arbitrary grids for partial di erential equations. These new methods are equivalent to conventional spectral methods but do not rely on speci c grid distributions. Speci cally, we show how to implement Legendre Galerkin, Legendre collocation, and Laguerre Galerkin methodology on arbitrary grids. Research supported by AFOSR gran...

The development and production of high performance equipment necessitate the use of passive cooling technology. In this paper, heat transfer study of convective-radiative straight fin with temperature-dependent thermal conductivity under the influence of magnetic field is carried out using Legendre wavelet collocation method. The numerical solution is used to investigate the effects of magnetic...