This paper is concerned with the second order linear differential equationd 2ydx 2-x λ y=0, which is called the generalized Airy equation of order λ , where the constant λ≥ 0 (not necessary pos...This paper is concerned with the second order linear differential equationd 2ydx 2-x λ y=0, which is called the generalized Airy equation of order λ , where the constant λ≥ 0 (not necessary positive integer) satisfies (-1) λ =± 1 . The properties of its two special linearly independent real-valued solutions, called the generalized Airy function of order λ , are systematically investigated.展开更多
Several improvements are made for existing asymptotic expansions for the axisymmetric toroidal shells. The new expansions are numerically satisfactory and satisfy the accuracy of the theory of thin shells. All of t...Several improvements are made for existing asymptotic expansions for the axisymmetric toroidal shells. The new expansions are numerically satisfactory and satisfy the accuracy of the theory of thin shells. All of them are expressed in terms of generalized Airy functions , instead of Bessel or Airy function for the homogeneous and Lommel function for the particular solutions, respectively, as in the existing work. In this paper, three particular solutions are given, one of which is just the solution obtaine d by Tumarkin(1959) and Clark(1963).展开更多
We study the derivative operator of the generalized spherical mean S^γt. By considering a more general multiplier m^Ωγ,b=Vn-2/2+γ(|ξ|)|ξ|^bΩ(ξ') and finding the smallest γ such that m^Ωγ,b is an Hp mult...We study the derivative operator of the generalized spherical mean S^γt. By considering a more general multiplier m^Ωγ,b=Vn-2/2+γ(|ξ|)|ξ|^bΩ(ξ') and finding the smallest γ such that m^Ωγ,b is an Hp multiplier, we obtain the optimal range of exponents (γ,β,p)to ensure the H^p(R^n) boundedness of a^βS^γ1f(x). As an application, we obtain the derivative estimates for the solution for the Cauchy problem of the wave equation on H^p(R^n) spaces.展开更多
文摘This paper is concerned with the second order linear differential equationd 2ydx 2-x λ y=0, which is called the generalized Airy equation of order λ , where the constant λ≥ 0 (not necessary positive integer) satisfies (-1) λ =± 1 . The properties of its two special linearly independent real-valued solutions, called the generalized Airy function of order λ , are systematically investigated.
文摘Several improvements are made for existing asymptotic expansions for the axisymmetric toroidal shells. The new expansions are numerically satisfactory and satisfy the accuracy of the theory of thin shells. All of them are expressed in terms of generalized Airy functions , instead of Bessel or Airy function for the homogeneous and Lommel function for the particular solutions, respectively, as in the existing work. In this paper, three particular solutions are given, one of which is just the solution obtaine d by Tumarkin(1959) and Clark(1963).
基金the National Natural Science Foundation of China (Grant Nos. 1177138& 11371316, 1147128&11601456).
文摘We study the derivative operator of the generalized spherical mean S^γt. By considering a more general multiplier m^Ωγ,b=Vn-2/2+γ(|ξ|)|ξ|^bΩ(ξ') and finding the smallest γ such that m^Ωγ,b is an Hp multiplier, we obtain the optimal range of exponents (γ,β,p)to ensure the H^p(R^n) boundedness of a^βS^γ1f(x). As an application, we obtain the derivative estimates for the solution for the Cauchy problem of the wave equation on H^p(R^n) spaces.
