In this paper, numerical solution of partial differential equations of the so-called hyperbolic telegraph, which has different applications in many fields such as engineering and physics, is investigated. The numerical solutions of telegraph equation defined by Caputo fractional derivative and by Atangana–Baleanu fractional derivative are obtained by Dufort–Frankel difference scheme method. It is important to investigate the solution of this equation defined by these two fractional derivatives and to compare these solutions. Studying this problemfor the derivatives of different fractional order makes this problem different from previous studies. The originality of this problem is illustrated by initially considering two types of problems with both the Caputo and Atangana– Baleanu fractional derivatives. In addition, the approximate solution of these two problems with the Dufort–Frankel difference scheme method and their comparison indicate the originality of this study. Difference schemes are constructed for this equation defined by Caputo and Atangana–Baleanu fractional derivatives. Stability estimates are given for this difference scheme method. The error analysis is calculated by comparing the exact solutions of these two problems, which are defined by both Caputo and Atangana–Baleanu fractional derivatives. Present results show that this method is effective and suitable for these equations defined by Caputo and Atangana–Baleanu fractional derivative. From the simulations obtained using the Matlab program, it can be seen that the Dufort–Frankel difference scheme method is suitable for both types of problems and has approximate solutions close to the exact solution.