Cleary, this series (numerical) solution obtained by RDTM converged to exact solution for sufficiently small 't'.

The comparison of RDTM solution of coupled problem of SRLWE (1) with the exact solution z(x, t) of (2) at t= 0.3 and t= 0.1 is given below.

Table 2: Comparison of RDTM solution of coupled problem of SRLWE (1) with the exact solution z(x, t) of (2) at t = 0.3 and t = 0.1.

The comparison of RDTM solution of coupled problem of SRLWE (1) with the exact solution ph(x, t) of (2) at t= 0.3 and t= 0.1 is given below

Table 3: Comparison of RDTM solution of coupled problem of SRLWE (1) with the exact solution ph(x, t) of (2) at t = 0.3 and t = 0.1.

Table 4: Comparison of RDTM solution of coupled problem of SRLWE (1) with the exact solution z(x, t) of (2) at t= 0.3 and t= 0.1.

Table 5: Comparison of RDTM solution of coupled problem of SRLWE (1) with the exact solution ph(x, t) of (2) t= 0.3 and t= 0.1.

One can see in above Examples 1 and 2 that corresponding to t= 0.3, using four terms RDTM, when x was increased from 0 to 10, the solution converged faster to the actual solution as compared with three terms RDTM.

To demonstrate the accuracy of RDTM, we calculate the first 20 terms of the exact solution u x, t at Eq.

comparisons of absolute errors for Eq.(4) using 20-terms of the ADM [16], 7 iterations of the VIM [34] and 20 iterations of the RDTM. These results reveal that, with less and an easier computation, the RDTM is effective, accurate and convenient, has the same results as ADM and better than the other method.

In this paper, we consider the non-linear shock wave equation that describes the flow of air for finding an analytic solution via Reduced Differential Transform Method (RDTM).

The absolute error (25) for solving Shock wave equation (4) by 20-terms of ADM, 7 iterations of VIM and 20 iterations of RDTM