Application Of The Differential Quadrature Method To Problems In Engineering Mechanics

The numerical solution of linear and nonlinear partial differential equations plays a prominent role in many areas of engineering and physical sciences. In many cases all that is desired is a moderately accurate solution at a few grid points that can be calculated rapidly. The standard finite difference method currently in use have the characteristic that the solution must be calculated with a large number of mesh points in order to obtain moderately accurate results at the points of interest. Consequently, both the computing time and storage required often prohibit the calculation. Furthermore, the mathematical techniques involved in the finite difference schemes or in the Fourier transform methods, are often quite sophisticated and thus not easily learned or used.The differential quadrature method (DQM) is a numerical solution technique, which has been presented in this thesis. This method is a simple and direct technique, which can be applied in a large number of cases to circumvent the difficulties of programming complex algorithms for the computer, as well as excessive use of storage and computer time. The initial and/or boundary value problems can be solved by this method directly and efficiently. The accuracy of the differential quadrature (DQ) method depends mainly on the accuracy of the weighting coefficient computation, which is a vital key of the method. In this thesis, the technique has been illustrated with the solution of six partial differential equations arising in Heat transfer, Poisson and Torsion problem with accurate weighting coefficient computation and two types of mesh· points distribution (equally spaced and unequally spaced). In all cases, the obtained DQ numerical results are of good accuracy with the exact solutions and hence show the potentiality of the method. It is also shown that the obtained DQ results in this thesis either agree very well or improved than those of some similar published results. This method is a vital alternative to the conventional numerical methods, such as finite difference and finite element methods. It is expected that this technique can be applied in a large number of cases in science and engineering to circumvent both the above-mentioned conventional difficulties.

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