The exact solution of fractional telegraph partial differential equation of nonlocal boundary value problem is obtained. The theorem of stability estimates is presented for this equation. Difference schemes are constructed for both the implicit finite difference scheme and the Dufort–Frankel finite difference scheme (DFFDS). The stability of these difference schemes for this problem are proved. With the help of these methods, numerical solutions of the fractional telegraph differential equation defined by Caputo fractional derivative for fractional orders \(\alpha=0.1, 0.5, 0.9\) are calculated. Numerical results are compared with the exact solution and the accuracy and effectiveness of the proposed methods are investigated.