<![CDATA[The usefulness of orthogonal polynomials has increasingly been extended to the solution of initial and boundary value problems in recent years. Among these, Chebyshev polynomials—classified into four distinct kinds—are widely employed; however, trial functions in numerical schemes have predominantly relied on polynomials of the second kind, with limited attention to the others. This study applies all four kinds of Chebyshev polynomials as trial functions within the collocation method. Shifted forms of each kind of Chebyshev polynomial were used as trial functions and substituted into the governing differential equations. The resulting equations were then evaluated at selected collocation points within the domain, converting the differential equations into systems of linear equations, which were solved simultaneously using Maple 18.0 software. For each kind of Chebyshev polynomial, approximations of sixth, tenth, and twelfth order were constructed, and the corresponding results were compared with available exact solutions and, where exact solutions were not available, with results from other established numerical methods. Three mathematical problems were considered to validate the effectiveness of the four kinds of Chebyshev polynomials in this framework. Residual equations for each kind of polynomial were obtained at different orders, and the associated constants were also determined for each order, thereby providing a systematic assessment of their performance as trial functions in the collocation technique.]]>
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