The primary purpose of this study is to construct truncated solutions for fractional Bagley-Torvik problems (FBT) by developing a novel method including newly defined Clique wavelets and collocation points. Clique wavelets are defined by utilizing Clique polynomials on $[0,1]$. The convergence of this method is investigated and supported by illustrative examples through tables and figures. As a result, the efficiency and effectiveness of the method is proved by theorems and examples.