ON DOUBLE SINE AND COSINE TRANSFORMS,LIPSCHITZ AND ZYGMUND CLASSES

We consider complex-valued functions f ∈ L^1 （R^2＋）, where R＋ ：= [0,∞）, and prove sufficient conditions under which the double sine Fourier transform fss and the double cosine Fourier transform fcc belong to one of the two-dimensional Lipschitz classes Lip（a,β） for some 0 〈 α,β ≤ 1; or to one of the Zygmund classes Zyg（α,β） for some 0 〈 α,β ≤ 2. These sufficient conditions are best possible in the sense that they are also necessary for nonnegative-valued functions f ∈ L^1 （R^2＋）.

Fast High Order Algorithm for Three-Dimensional Helmholtz Equation Involving Impedance Boundary Condition with Large Wave Numbers

Acoustic fields with impedance boundary conditions have high engineering applications, such as noise control and evaluation of sound insulation materials, and can be approximated by three-dimensional Helmholtz boundary value problems. Finite difference method is widely applied to solving these problems due to its ease of use. However, when the wave number is large, the pollution effects are still a major difficulty in obtaining accurate numerical solutions. We develop a fast algorithm for solving three-dimensional Helmholtz boundary problems with large wave numbers. The boundary of computational domain is discrete based on high-order compact difference scheme. Using the properties of the tensor product and the discrete Fourier sine transform method, the original problem is solved by splitting it into independent small tridiagonal subsystems. Numerical examples with impedance boundary conditions are used to verify the feasibility and accuracy of the proposed algorithm. Results demonstrate that the algorithm has a fourth- order convergence in and -norms, and costs less CPU calculation time and random access memory.

Exact Analysis of Non-Linear Fractionalized Jeffrey Fluid.A Novel Approach of Atangana-Baleanu Fractional Model

It is a very difficult task for the researchers to find the exact solutions to mathematical problems that contain non-linear terms in the equation.Therefore,this article aims to investigate the viscous dissipation(VD)effect on the fractional model of Jeffrey fluid over a heated vertical flat plate that suddenly moves in its own plane.Based on the Atangana-Baleanu operator,the fractional model is developed from the fractional constitutive equations.VD is responsible for the non-linear behavior in the problem.Upon taking the Laplace and Fourier sine transforms,exact expressions have been obtained for momentum and energy equations.The influence of relative parameters on fluid flow and temperature distribution is shown graphically.As special cases,and for the sake of correctness,the corresponding results for second-grade fluid and Newtonian viscous fluid are also obtained.It is interesting to note that fractional parameterαprovides more than one line as compared to the classical model.This effect represents the memory effect in the fluid which is not possible to elaborate by the classical model.It is also worth noting that the temperature profile of the generalized Jeffrey fluid rises for higher values of Eckert number which is due to the enthalpy difference of the boundary layer.

Stokes' First Problem for an MHD Burgers Fluid

This work is related to the flow of a magnetohydrodynamic Burgers fluid.The flow of an incompressible conducting Burgers fluid in the presence of a uniform transverse magnetic field over a plate that is moved suddenly is considered.By the application of the Laplace and Fourier sine transforms techniques,the exact analytical expressions for the velocity field and associated shear stress are determined in simple forms.They are written as a sum of steady-state and transient solutions.The graphical results are plotted for different values of indispensable parameters and some interesting results are concluded.The corresponding solutions for the hydrodynamic Burgers fluid appear as the limiting cases of the obtained solutions.