The solution of some engineering problems with iterative differential method

The Differential Quadrature Method (DQM) was first described by Bellman et al. (1971). Shu and Richards (1990) created an algorithm that generalizes DQM and introduces Generalized DQM (GDQM). In this thesis, various engineering problems were solved by using GDQM by combining the Newton-Raphson Iteration Method. This algorithm, called Iterative DQM (I-DQM), unlike the classical DQM approach, the solution for multiple boundary conditions at the same point can be applied directly to the point where the condition is required without any additional procedures. Numerical simulations were performed by using I-DQM in the solutions of Linear and Nonlinear Ordinary Differential Equations (OED). The results showed that the developed solution algorithm is very successful for OED problems. The major deflection problem of beams, which are difficult and difficult to solve in the mechanical field, is defined using a nonlinear OED. In order to test the accuracy of the method more reliably, the Large Deflection Problem was solved with the I-DQM algorithm. In order to compare the sensitivity of the I-DQM, the same problem was also addressed using the Combining Method (CM). Numerical results have shown that the Large Deflection Problem can be easily solved with high sensitivity by using I-DQM compared with previous studies and BM. Nonlinear Partial Differential Equations (PDE) are commonly used to define engineering problems. For this reason, I-DQM solutions of various linear and nonlinear PDE problems are provided. When the results were examined, it was seen that high precision solutions were obtained with an easier approach than classical DQM. In order to examine the sensitivity of the method, the Burgers Equation which is widely used in method tests in literature is solved with I-DQM. In the initial numerical analyzes, the initial predictions of Newton-Raphson Iteration were randomly selected with the classical approach. As it is known, in the iterative methods, the initial estimated value is very important in the convergence rate and accuracy of the results. Based on this information, a curve fitting was made using the boundary conditions of the problem discussed and this curve function was taken into consideration as the initial prediction. This new approach is called Curve Fitting Initial Guess (CFIG) for I-DQM. In the Burgers Equation solutions using CFIG, dt = 0,01 steps and 7 iterations with sufficient accuracy were obtained. The results are compared with other solutions of the Burgers Equation in the literature. I-DQM analysis using CFIG showed that for the first time in the literature, the Burgers Equation with a DQM algorithm can be solved with sufficient precision at a step rate of dt = 0,01. In addition, in order to better evaluate the sensitivity of GDQM approach to problems where multiple boundary conditions should be entered at the same point, the linear beam vibration problem has been addressed for different boundary conditions. In order to compare the results, the same problem was solved by using the Rayleigh-Ritz Method. The results of both methods were compared with the results of other studies in the literature and it was found that DQM provided a solution with very high sensitivity.