A localized differential quadrature method (LDQM) is introduced for buckling analysis of axially functionally graded nonuniform columns with elastic restraints. Weighting coefficients of differential quadrature discretization are obtained making use of neighboring points in forward and backward type schemes for the reference grids near the beginning and end boundaries of the physical domain, respectively, and central type scheme for the reference grids inside the physical domain. Boundary conditions are directly implemented into weighting coefficient matrices, and there is no need to use fictitious points near the boundaries. Compatibility equations are not required because the governing differential equation is discretized only once for each reference grid using neighboring points and variation of flexural rigidity is taken to be continuous in the axial direction. A large case of columns having different variations of cross-sectional profile and modulus of elasticity in the axial direction are considered. The results for nondimensional critical buckling loads are compared to the analytical and numerical results available in the literature. Some new results are also given. Comparison of the results shows the potential of the LDQM for solving such generalized eigenvalue problems governed by fourth-order variable coefficient differential equations with high accuracy and less computational effort.

The Differential Quadrature Method (DQM) was first proposed by Bellman and his associates [1, 2]. Since 1970s, DQM has been applied many areas in engineering problems. Bert and Malik [3] have mentioned examples to deal with the high order differential equations of Euler beam, whose governing equation is a forth order one with double boundary conditions at each boundary. The main difficulty for such forth order problems as Euler beams is that there are multiple boundary conditions but only one variable at each boundary. Applying multiple boundary condition at the same location is a big problem that to be dealt with in DQM. For this reason, Jang et al. [4] have proposed a-point approximation. A point very adjacent to the real boundary has been inserted to impose the second condition on there. Bert and Malik [3] have applied the-technique in the solution of linear structural problems. For differential equations involving more than one boundary condition at one point, the DQM has no generally effective technique to solve them without use to the current δ-point technique.

SUMMARYNowadays, most of the ordinary differential equations (ODEs) can be solved by modelica-based approaches, such as Matlab/Simulink, Dymola and LabView, which use simulation technique (ST). However, these kinds of approaches restrict the users in the enforcement of conditions at any instant of the time domain. This limitation is one of the most important drawbacks of the ST. Another method of solution, differential quadrature method (DQM), leads to very accurate results using only a few grids on the domain. On the other hand, DQM is not flexible for the solution of non-linear ODEs and it is not so easy to impose multiple conditions on the same location. For these reasons, the author aims to eliminate the mentioned disadvantages of the simulation technique (ST) and DQM using favorable characteristics of each method in the other. This work aims to show how the combining method (CM) works simply by solving some non-linear problems and how the CM gives more accurate results compared with those of other methods.

SUMMARYThe Blasius and Sakiadis equation was solved earlier with different numerical methods. In this study, it was solved by using the generalized iterative differential quadrature method (GIDQM). And more than one condition are imposed at the same point without using any higher-order polynomial or -point approximation in GIDQM although it is one of the most important drawbacks in the differential quadrature method (DQM). Procedure is started with an initial guess value and true results are obtained by iterations. More grid points are used. Hence, the solution of the Blasius equation is calculated precisely and showed good agreements when compared with other works.

Most ordinary differential equations (ODEs) can be solved using simulation technique (ST), which, however, requires the enforcement of conditions at any time domain, this limitation is the main drawback. Another numerical method, the differential quadrature method (DQM) or the integral quadrature method (IQM), leads to very accurate results using only a few grids on the domain, which, however, is not flexible for nonlinear ODEs and is not easy to impose multiple conditions on the same location. For these reasons, ST is combined with IQM to eliminate the mentioned disadvantages of the ST and IQM. Numerical examples (nonlinear vibration and boundary value problems) showed good agreements when compared with other works.