To investigate the accuracy of the ADM compared with the Sinc method, we choose examples with known solutions that allows for a more complete error analysis. So a
system of balance laws with Riemann type conditions is tested numerically by using the Sinc function methodology, and for comparison purposes, we also solve the same model by ADM. The example reported here is selected to show the convergence of the two schemes. Consider the system
where
T is some small constant. In addition,
with flux function
, such that
with the approximations to Riemann type condition
given by
In
Section 3 when we construct the approximate solution by means of the Sinc methodology, it was necessary that the initial condition belonged to the class of functions
previously mentioned in
Section 3, but we know that the Riemann condition does not belong to this family, so a transformation had to be made to make the primitive condition belong to the family. Since the boundary conditions are non-homogeneous, then the transformation
where
, and
will convert the given boundary conditions to homogeneous conditions, provided that
. Now, after substituting the transformation (
29) into Equation (
28), we get a new system with the unknown
given by
with initial condition
where
. Equation (
30) can be solved for
, and then, using the transformation in (
29), for
. To proceed, since the space and the time domains are
and
, respectively, choose the conformal maps
and
. We solve the above system using ADM subject to the initial conditions
. Integrate the system in (
28) with respect to
t and using the initial conditions
, we arrive at
and,
Substitute the above infinite sum for Adomian polynomials, together with the approximate solutions into Equations (
32) and (
33), exactly as mentioned in
Section 5. Balancing terms in Equations we obtain an approximate solution of the form
Table 1 shows that the method converges for
using Sinc methodology, where the second column reports the supremum norm of the error between the exact solution and the Sinc approximate
solution, while the third column reports the error between the exact solution and the Sinc approximate
solution. The error in the approximate
u-solution using ADM is reported in
Figure 1.
Figure 2 reported the
-solution using ADM for different values of time
t. The error in approximating
show that most of the error concentrates at origin, which caused by approximating Riemann type condition by a smooth function. Through our study, and upon consulting the
Table 1 and
Table 2, we are able to find the values for the order of convergence using Equation (
27), it was found that the order of convergence for the ADM is
, while it is approximately
using the Sinc method. This is in accordance with the theories formulated in
Section 4, which show that the Sinc methodology is faster, better, and more comprehensive in finding an approximate solution. Because of the nature of the issues under discussion, it is noted that the two methods show the symmetry of the approximate solution. A quick look at in calculated approximate solutions for both
and
shows the symmetry of the solutions where the error value at the point
is the same as at
. The main goal of this research paper is to compare the two methods together, but if the reader wants more ways to solve systems from the same context, we cite the following [
22]. It was shown in [
25] that traditional methods, like finite difference method is unstable when the system is of mixed-type when Riemann conditions are involved, which means that the two methods used in this research are an improvement over the previous traditional methods. To present a physical and engineering application for the purposes of explaining the importance of the topic, please review the research paper [
18] where the
system was solved. Finally, what we intend to refer to in a future research in the hope of discussing it from all sides is the importance of the Telegraph system. As an illustration, we shall consider conductors (such as telephone wires or submarine cables) in which the current
may leak to ground. The resultant decrease in current is governed by
, where
is voltage,
is the conductance to ground, and
is the capacitance with the ground. The change in voltage is governed by
, where
is the resistance and
is the inductance of the cable.