Generalized Jacobi spectral Galerkin method for fractional pantograph differential equation

This work is concerned with the extension of the Jacobi spectral Galerkin method to a class of nonlinear fractional pantograph differential equations. First, the fractional differential equation is converted to a nonlinear Volterra integral equation with weakly singular kernel. Second, we analyze the existence and uniqueness of solutions for the obtained integral equation. Then, the Galerkin method is used for solving the equivalent integral equation. The error estimates for the proposed method are also investigated. Finally, illustrative examples are presented to confirm our theoretical analysis.     

变系数非线性方程的Jacobi椭圆函数展开解

把Jacobi椭圆函数展开法扩展并应用到求解变系数的非线性演化方程,比较方便地得到新的解析解 关键词： Jacobi椭圆函数 变系数非线性方程 类椭圆余弦波解 类孤子解     

Coupled intervals in the discrete calculus of variations: necessity and sufficiency

In this work we study nonnegativity and positivity of a discrete quadratic functional with separately varying endpoints. We introduce a notion of an interval coupled with 0, and hence, extend the notion of conjugate interval to 0 from the case of fixed to variable endpoint(s). We show that the nonnegativity of the discrete quadratic functional is equivalent to each of the following conditions: The nonexistence of intervals coupled with 0, the existence of a solution to Riccati matrix equation and its boundary conditions. Natural strengthening of each of these conditions yields a characterization of the positivity of the discrete quadratic functional. Since the quadratic functional under consideration could be a second variation of a discrete calculus of variations problem with varying endpoints, we apply our results to obtain necessary and sufficient optimality conditions for such problems. This paper generalizes our recent work in [R. Hilscher, V. Zeidan, Comput. Math. Appl., to appear], where the right endpoint is fixed.     

Singularity conjecture and constructions of Jacobi forms

This paper verifies the singularity conjecture for Jacobi forms with higher degree in some typical cases, and gives constructions for the Jacobi cusp forms whose Fourier coefficients can be expressed by some kind of Rankin-typeL-series.     

On Hardy’s Theorem on SU(1, 1)

The classical Hardy theorem asserts that ■ and its Fourier transform ■ can not both be very rapidly decreasing.This theorem was generalized on Lie groups and also for the Fourier-Jacobi transform.However,on SU(1,1)there are infinitely many"good"functions in the sense that ■ and its spherical Fourier transform ■ both have good decay. In this paper,we shall characterize such functions on SU(1,1).     

A New Family of Cyclic Difference Sets with Singer Parameters in Characteristic Three

We construct a new family of cyclic difference sets with parameters ((3 ^d – 1)/2, (3 ^{d – 1} – 1)/2, (3 ^{d – 2} – 1)/2) for each odd d. The difference sets are constructed with certain maps that form Jacobi sums. These new difference sets are similar to Maschietti's hyperoval difference sets, of the Segre type, in characteristic two. We conclude by calculating the 3-ranks of the new difference sets.     

Toda Chains in the Jacobi Method

We use the Jacobi method to construct various integrable systems, such as the Stäckel systems and Toda chains, related to various root systems. We find canonical transformations that relate integrals of motion for the generalized open Toda chains of types B _n, C _n, and D _n.     

Lame Function and Its Application to Some Nonlinear Evolution Equations

In this paper, based on the Lame function and Jacobi elliptic function, the perturbation method is appliedto some nonlinear evolution equations to derive their multi-order solutions.     

On characterization of Poisson and Jacobi structures

We characterize Poisson and Jacobi structures by means of complete lifts of the corresponding tensors: the lifts have to be related to canonical structures by morphisms of corresponding vector bundles. Similar results hold for generalized Poisson and Jacobi structures (canonical structures) associated with Lie algebroids and Jacobi algebroids.     

Rankin-Cohen brackets on pseudodifferential operators

Pseudodifferential operators that are invariant under the action of a discrete subgroup Γ of SL(2,R) correspond to certain sequences of modular forms for Γ. Rankin-Cohen brackets are noncommutative products of modular forms expressed in terms of derivatives of modular forms. We introduce an analog of the heat operator on the space of pseudodifferential operators and use this to construct bilinear operators on that space which may be considered as Rankin-Cohen brackets. We also discuss generalized Rankin-Cohen brackets on modular forms and use these to construct certain types of modular forms.