Numerical solutions of some mechanical problems by matrix methods

In this thesis study, the matrix methods based on Taylor, Chebyshev, Bernstein, Fibonacci, shifted Legendre, Hermite and Laguerre polynomials together with collocation points are developed to obtain the numerical solutions of the differential and integro differential equations arising in mechanical problems. The fundamental matrix equations of the matrix collocation methods based on special polynomials are adapted to use the basis of the special polynomial directly without converting them into the standard basis. Also, the Taylor-Splitting collocation method based on the truncated Taylor series is developed to solve linear and nonlinear ordinary differential equations. Unlike previous approaches, the fundamental matrix equation and collocation points are reformulated using interval splitting. Thus, oscillation problems that are very difficult to obtain numerical results using the classical matrix collocation methods are easily analyzed. In this study, the methods and error analyses are presented in general form and applied to the physical models from mechanics frequently encountered in the literature. Spring-mass, pendulum models, Euler Bernoulli, Rayleigh, Timoshenko beam and plate models are investigated. The codes developed in the Wolfram Mathematica program are used to solve and analyze the model problems discussed in this thesis study. Numerical examples are given for each model problem, and the results are scrutinized with the help of tables and graphs. Results of the presented methods are compared with studies in the literature and analytical results. After evaluating the results, it is concluded that the proposed methods are practical, simple, and precise in the investigation of the results.