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A numerical method for solving distributed-order multi-term time-fractional telegraph equations involving
2
Table 3. Comparison of the maximum errors and convergence orders with ∆t= (∆x) for Example 3
∆x = h Reference 30 Reference 31 Proposed method
e(h, ∆t) Rate ∆t e(h, ∆t) Rate ∆t e(h, ∆t) Rate ∆t
0.5 7.43e-2 - 2.45e-2 - 1.34e-7 -
0.25 3.22e-2 0.60 5.47e-3 1.08 4.36e-8 1.99
0.125 8.99e-3 0.92 1.30e-3 1.04 1.21e-8 1.78
0.0625 2.31e-3 0.98 3.20e-4 1.01 2.13e-9 1.73

convergence order for the proposed method. This spatial variable. To approximate the distributed-
table provides a comprehensive overview of the order fractional operator, we applied the mid-
performance of the proposed method in relation point method to discretize the integral term, fol-
to those from previous studies, highlighting the lowed by the use of the finite difference method
accuracy and convergence behavior. to approximate the Caputo fractional derivative
with respect to the time variable. We have
rigorously proved the convergence and stability
of the proposed numerical method. To demon-
strate the high efficiency and accuracy of the
method, we conducted several numerical experi-
ments, comparing the results with those obtained
from other numerical methods found in the litera-
ture. The comparison clearly shows that the pro-
posed method outperforms existing approaches in
terms of both efficiency and performance, making
it a highly effective tool for solving distributed-
Figure 7. The surface of the approximate solution order time-fractional telegraph equations.
u(x, t) with L = 20, M = N = 30 and various choices
of α for Example 3
Acknowledgments
None.

Funding

The work was supported by the University of
Tabriz, Iran (Grant No. 2070).

Conflict of interest
Figure 8. The absolute error functions with L = 20, The authors declare that they have no competing
M = N = 30, and various choices of α for Example 3
interests.

Author contributions
Conceptualization: Safar Irandoust Pakchin, Mo-
hammad Hossein Derakhshan
Data curation: Mohammad Hossein Derakhshan
Investigation: All authors
Methodology: All authors
Figure 9. The absolute error functions with L = 20,
Software: All authors
M = N = 30 and various choices of α for Example 3
at t = 0.5 Writing – original draft: Shahram Rezapour
Writing – review & editing: All authors
6. Conclusion Availability of data

In this study, we analyzed the distributed-order All data analyzed and generated have been in-
multi-dimensional time-fractional telegraph equa- cluded in the manuscript.
tions, incorporating Caputo time- and Riesz
AI tools statement
space-fractional derivatives. The finite difference
method was employed to approximate the Riesz All authors confirm that no AI tools were used in
space-fractional derivative with respect to the the preparation of this manuscript.
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