An Efficient Alternating Segment Parallel Difference Method for the Time Fractional Telegraph Equation

Abstract

The fractional telegraph equation is a kind of important evolution equation, which has an important application in signal analysis such as transmission and propagation of electrical signals. However, it is difficult to obtain the corresponding analytical solution, so it is of great practical value to study the numerical solution. In this paper, the alternating segment pure explicit-implicit (PASE-I) and implicit-explicit (PASI-E) parallel difference schemes are constructed for time fractional telegraph equation. Based on the alternating segment technology, the PASE-I and PASI-E schemes are constructed of the classic explicit scheme and implicit scheme. It can be concluded that the schemes are unconditionally stable and convergent by theoretical analysis. The convergence order of the PASE-I and PASI-E methods is second order in spatial direction and 3-α order in temporal direction. The numerical results are in agreement with the theoretical analysis, which shows that the PASE-I and PASI-E schemes are superior to the classical implicit schemes in both accuracy and efficiency. This implies that the parallel difference schemes are efficient for solving the time fractional telegraph equation.

A New Class of Difference Methods with Intrinsic Parallelism for Burgers–Fisher Equation

“…But, this superiority can only be reflected when the amount of data is large. In case the number of grid points is less than a certain range, the effect of data communication on the cycle can reduce the computational efficiency, and the superiority of parallel computing is not obvious [13,32]. When the amount of data is large, the impact of program loop execution is much greater than that of data communication, and the parallel scheme is more effective.…”

“…e speedup S p � T 1 /T p , and efficiency E p � S p /p (T 1 is the computing time of C-N and T p is the computing time of parallel scheme) are defined [32,33]. Using four cores for this numerical experiment, we fix the time layer as N � 500 and take the number of space grid points as M � 2001, 3001, 4001, 5001, 6001, 7001. e computational results are shown in Table 5.…”

Linear barycentric rational collocation method for solving telegraph equation

“…The sinc‐Chebyshev collocation method 23 is used to solve time fractional order telegraph equation. The alternating segment parallel difference method 24 is proposed for solving the time fractional telegraph equation. The Crank–Nicolson finite difference method 25 is developed to solve the fractional order telegraph equation.…”