Filtering in Legendre spectral methods

We discuss the impact of modal filtering in Legendre spectral methods, both on accuracy and stability. For the former, we derive sufficient conditions on the filter to recover high order accuracy away from points of discontinuity. Computational results confirm that less strict necessary conditions appear to be adequate. We proceed to discuss a instability mechanism in polynomial spectral methods and prove that filtering suffices to ensure stability. The results are illustrated by computational experiments.

     

A Computationally Hybrid Method for Solving a Famous Physical Problem on an Unbounded Domain

In this paper, a hybrid method based on the collocation and Newton-Kantorovich methods is used for solving the nonlinear singular Thomas-Fermi equation. At first, by using the Newton-Kantorovich method, the nonlinear problem is converted to a sequence of linear differential equations, and then, the fractional order of rational Legendre functions are introduced and used for solving linear differential equations at each iteration based on the collocation method. Moreover, the boundary conditions of the problem by using Ritz method without domain truncation method are satisfied. In the end, the obtained results compare with other published in the literature to show the performance of the method, and the amounts of residual error are very small, which indicates the convergence of the method.     

Pedal curves of frontals in the Euclidean plane

In this paper, we will give the definition of the pedal curves of frontals and investigate the geometric properties of these curves in the Euclidean plane. We obtain that pedal curves of frontals in the Euclidean plane are also frontals. We further discuss the connections between singular points of the pedal curves and inflexion points of frontals in the Euclidean plane.     

An effective approach for numerical solution of linear and nonlinear singular boundary value problems

In this study, an effective approach is presented to obtain a numerical solution of linear and nonlinear singular boundary value problems. The proposed method is constructed by combining reproducing kernel and Legendre polynomials. Legendre basis functions are used to get the kernel function, and then the approximate solution is obtained as a finite series sum. Comparison of numerical results is made with the results obtained by other methods available in the literature. Furthermore, efficiency and accuracy of the method are demonstrated in tabulated results and plotted graphs. The numerical outcomes demonstrate that our method is very effective, applicable, and convenient.     

Diameter estimates for submanifolds in manifolds with nonnegative curvature

Given a closed connected manifold smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we estimate the intrinsic diameter of the submanifold in terms of its mean curvature field integral. On the other hand, for a compact convex surface with boundary smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we can estimate its intrinsic diameter in terms of its mean curvature field integral and the length of its boundary. These results are supplements of previous work of Topping, Wu-Zheng and Paeng.     

Classification of ideal submanifolds of real space forms with type number ≤2

Roughly speaking, an ideal immersion of a Riemannian manifold into a real space form is an isometric immersion which produces the least possible amount of tension from the ambient space at each point of the submanifold. The main purpose of this paper is to completely classify all non-minimal ideal submanifolds of real space forms with type number $\le 2$.     

On the frame bundle adapted to a submanifold

Let M be a submanifold of a Riemannian manifold . M induces a subbundle of adapted frames over M of the bundle of orthonormal frames . The Riemannian metric g induces a natural metric on . We study the geometry of a submanifold in . We characterize the horizontal distribution of and state its correspondence with the horizontal lift in induced by the Levi–Civita connection on N. In the case of extrinsic geometry, we show that minimality is equivalent to harmonicity of the Gauss map of the submanifold M with a deformed Riemannian metric. In the case of intrinsic geometry we compute the curvatures and compare this geometry with the geometry of M.     

CONSTANT ELASTICITY OF VARIANCE MODEL AND ANALYTICAL STRATEGIES FOR ANNUITY CONTRACTS

The constant elasticity of variance（CEV） model was constructed to study a defined contribution pension plan where benefits were paid by annuity. It also presents the process that the Legendre transform and dual theory can be applied to find an optimal investment policy during a participant＇s whole life in the pension plan. Finally, two explicit solutions to exponential utility function in the two different periods （before and after retirement） are revealed. Hence, the optimal investment strategies in the two periods are obtained.     

基于拟Legendre多项式求解一类分数阶微分方程

本文基于移位的Legendre多项式构造一类新的正交拟Legendre多项式求解一类分数阶微分方程.用阶数随所求未知函数的微分的阶数而变化的拟Legendre多项式逼近未知函数;利用分数阶积分的性质推导拟Legendre多项式的积分算子阵,结合算子矩阵的思想和Tau方法,将问题转化为求解代数方程组的问题.最后,给出数值算例证明该方法的有效性.     

Study on convergence of hybrid functions method for solution of nonlinear integral equations

In this article, we present the uniform convergence analysis and accuracy estimation of hybrid functions (HFs) method for finding the solution of nonlinear Volterra and Fredholm integral equations. The properties of HFs which consist of block-pulse functions (BPFs) and Legendre polynomials are used to reduce the solution of nonlinear integral equations to the solution of algebraic equations. The superiority and accuracy of the HFs method to BPF and Legendre polynomial methods are illustrated through some numerical examples.