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This study introduces a new fractional order Fibonacci wavelet technique proposed for solving the fractional Bagley-Torvik equation (BTE), along with the block pulse functions. To convert the specified initial and boundary value problems into algebraic equations, the Riemann–Liouville (R-L) fractional integral operator is defined, and the operational matrices of fractional integrals (OMFI) are built. This numerical scheme’s performance is evaluated and examined on particular problems to show its proficiency and effectiveness, and other methods that are accessible in the current literature are compared. The numerical results demonstrate that the approach produces extremely precise results and is computationally more decisive than previous methods.