In this paper, we develop an efficient Legend...

FIGURE 2. Divergence of the arbitrary grid Legendre Galerkin method for improperly imposed initial conditions. Time-stable boundary conditions for nite-diierence schemes solving hyperbolic systems: methodology and application to high-order compact schemes, JCP 111, 220, (1994). 25 To initialize the problem, we must construct an approximation to the initial condition f(x), based on the grid poin...

An adaptive mesh refinement method for solving optimal control problems is developed. The method employs orthogonal collocation at Legendre-Gauss-Radau points, and adjusts both the mesh size and the degree of the approximating polynomials in the refinement process. A previously derived convergence rate is used to guide the refinement process. The method brackets discontinuities and improves sol...

Due to increasing applications of extended surfaces as passive methods of cooling, study of thermal behaviors and development of mathematical solutions to nonlinear thermal models of extended surfaces have been the subjects of research in cooling technology over the years. In the thermal analysis of fin, various methods have been applied to solve the nonlinear thermal models. This paper focuses...

In this paper, we developed a new three-step method for numerical solution of third order ordinary differential equations. Interpolation and collocation methods were used by choosing interpolation points at steps using power series, while step points, combination powers series perturbation terms gotten from the Legendre polynomials, giving rise to polynomial degree All analysis on derived shows...

We target here to solve numerically a class of nonlinear fractional two-point boundary value problems involving left- and right-sided derivatives. The main ingredient the proposed method is recast problem into an equivalent system weakly singular integral equations. Then, Legendre-based spectral collocation developed for solving transformed system. Therefore, we can make good use advantages Gau...

A method is presented for direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using global collocation at Legendre-Gauss-Radau (LGR) points. A key feature of the method is that it provides an accurate way to map the KKT multipliers of the nonlinear programming problem to the costates of the optimal control problem. More precisely...

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 ...

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration ordinary differential equations (ODEs) is presented. Its derivation based on integral form equation. The approach enables enhancing accuracy established Runge–Kutta while retaining same number stages. We demonstrate that, with proposed approach, Gauss–Legendre and Lobatto IIIA can be derived that th...