New Methods Based On Differential Quadrature Method And Sinc-collocation Method

The differential quadrature method (DQM), proposed in the early seventiesby Bellman and associates(BELLMANå’ŒCASTI 1971, BELLMAN, KASHEFand CASTI 1972), is a numerical technique for the solution of initial and bound-ary value problems. The method has been experimented with and its generalversatility has been established in a variety of physical problems. The principle ofDQM is simple, and DQM has been improved to be both accurate and computa-tionally inexpensive. When the problem has a smooth solution, DQM is beyondof the finite element method and the finite difference method. In a recent com-prehensive study (Malikå’ŒCivan 1995), it has been shown that the DQM standsout in numerical accuracy as well as computational efficiency over the well-knownfinite difference method and the finite element method. When applying DQM toproblems with crack, we might meet difficulty. It may be very difficult for DQMto calculate a problem with discontinuity, because it will generate a lot of errorto approximate a discontinuous function by interpolation polynomials.The domain decomposition method (DDM), proposed in the 1860s by a Ger-man mathematician named H. Schwarz, is introduced to solve PDE in complexregions. There are three kinds of DDM: (1) DDM with overlapping subdomains(DDM-1), (2) DDM with non-overlapping subdomains (DDM-2), (3) DDM withfictitious subdomains (DDM-3). DDM has many advantages, for example, theway of domain decomposition with arbitrariness, the descriptions in mathematicsof problems after domain decomposition with diversification. DDM is no longera pure numerical technique even a class of numerical methods, but collection ofideas and methods to solve problems. By applying these properties, the multipli-cation complexity can be reduced greatly and the efficiency of the DQM can besignificantly increased.Sinc methods have been studied extensively and found to be a very effec-tive technique, particularly for problems with singular solutions and those onunbounded domain. In addition, Sinc function seem to capture oscillating behav-iors in space, hence, are useful to deal with problems characterized by this type ofsolution[14]. [64, 65]provide overviews of the methods based on the Sinc functionfor solving O.D.E., P.D.E. and integral equation. But it is difficult for the tradi-tional Sinc method to solve two dimensional elliptic boundary value problems withthe mixed nonhomogeneous boundary condition[66]. In this paper, we present aSinc-collocation method with boundaries treatment(SCMBT) based on the Sinc-collocation incorporated with the double exponential transformation technique.By our method, we can solve the P. D. E directly no matter what the boundaryconditions are.The double exponential formula (DE formula), which is a quadrature formulabased on the double exponential transformation (DE transformation), was firstproposed by Takshasi and Mori [85]in 1974. The DE formula has been widely used in the last three decades and is now recognized to be one of the most effi-cient quadrature formulas [86, 68]. The use of the DE transformation techniquein the Sinc method yields highly efficient numerical method for interpolation,quadrature, approximation of transformation, differential and partial differentialequations [66, 67, 68, 87].In general, we will use the numerical differentiation when dealing with thedifferential equations. Thus the differential equations can be transformed intoalgebraic equations and then we can get the numerical solutions. But as weall have known, the numerical differentiation process is very sensitive to evena small level of errors. In contrast it is expected that on average the numericalintegration process is much less sensitive to errors. In this paper, based on the Sincmethod we provide a new method using Sinc method based on the interpolationof the highest derivatives (SIHD) for the differential equations. The error in theapproximation of the solution is shown to converge at an exponential rate. Andthe numerical results show that compared with the exiting results, our method isof high accuracy, of good convergence with little computational efforts. And it iseasy to treat nonhomogeneous mixed boundary condition for our method, whichis unlike the traditional Sinc method.The Stokes equations and the Navier-Stokes are concerned much. The Stokesequations describe the stable fluid flow without nonlinear terms and the Navier-Stokes equations describe the fluid flow developing with time. By neglecting theterm of (?)u/(?)t and the nonlinear convection, the Navier-Stokes equations can be trans-formed into the Stokes equations which can be regarded as the approximation ofthe Navier-Stokes equations when the Reynold number is very low or the velocityis very small.There are some difficulties in solving the Navier-Stokes equations numerically.Firstly, there is evidently no transport or other equation for pressure. The momen-tum equation and the continuity equation is intricately coupled since the velocityappears in these two equations. Secondly, satisfying the continuity equation onthe boundary is not automatic, and the condition should be enforced. Thirdly,the equations are nonlinear.In this paper, we will discuss the differential quadrature method and itsapplication. Then will discuss a new sinc collocation method, sinc collocationmethod with boundary conditions treatment. Finally we will discuss a sinc methodbased on approximation of the highest order derivatives.