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The present paper introduces the distributed-order (DO) Bessel fractional derivative for study of the Euler–Poisson–Darboux (EPD) equation including the spatial Riesz fractional derivative (RFD). For this purpose, we discretize the integral term of the DO fractional derivative and approximate the RFD derivative. We thereafter apply an implicit difference method (IDM) for numerical analysis and solvability of the reduced system from the IDM is discussed. The stability and convergent theorems are stated and the numerical tests are given to show efficiency of the proposed technique and to verify good agreement with the theoretical concepts. • We introduced the concept of the distributed order fractional derivative for the Bessel derivative. • We showed numerical analysis of the DO fractional Bessel derivative and the RFD for the EPD equation on bounded domains. • We applied an implicit difference method for numerical analysis. [ABSTRACT FROM AUTHOR]