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 equation Banach Spaces of Solutions of the Helmholtz Equation in the Plane Time Decay and Regularity of Solutions to the Wave Equation Mapping Properties of a Projection Related to the Helmholtz Equation Herglotz Wave Functions are the entire solutions of the Helmholtz equation which have L2-Far-Field-Pattern. The Ces\'aro operator $\mathcal{C}_{\alpha}$ is defined by \begin{equation*} (\mathcal{C}_{\alpha}f)(x) = \int_{0}^{1}t^{-1}f\left( t^{-1}x \right)\alpha (1-t)^{\alpha -1}\,dt~, \end{equation*} where $f$ denotes a function on $\mathbb{R}$. On Global Finite Energy Solutions of the Camassa-Holm Equation We consider the Camassa-Holm equation with data in the energy norm H1(R1). Sharp Global Well-Posedness for a Higher Order Schrodinger Equation It is transferred to a boundary value problem for analytic functions and then further reduced to a singular integral equation, the unique solvability and an effective method of solution for which are established. This paper provides a functional equation satisfied by the dichromatic sum function of rooted outer-planar maps. By the equation, the dichromatic sum function can be found explicitly. By means of our inequality and the method of Lyapunov function we study the stability of two kinds of large scale differential systems with time lag and the stability of a higher-order differential equation with time lag. The Alternating Segment Crank-Nicolson scheme for one-dimensional diffusion equation has been developed in [1], and the Alternating Block Crank-Nicolson method for two-dimensional problem in [2]. In this paper for the two-dimensional diffusion equation, the net region is divided into bands, a special kind of block. The errors estimation of the Chebyshev spectral-difference method for two-dimensional vorticity equation We construct a Chebyshev spectral-difference scheme for solving two-dimensional vorticity equation with semi-homogeneous boundary conditions. The chebyshev spectral method with a restraint operator for burgers equation For a differential equation, a theoretical proof of the relationship between the symmetry and the one-parameter invariant group is given: the relationship between symmetry and the group-invariant solution is presented. As an application, some solutions of theKdV equation are discussed. Exact envelope wave solution to nonlinear Schr?dinger equation with dissipative term

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