Substitute all the values of and in the Equation (2.7), we get
This is an exact solution of the given system of nonlinear partial differential Equations (2.1) and (2.2). We have verified this through the substitution, which is identical to the solution obtained by R. E. Bellman using the method of differential quadrature . Let we change the initial conditions to
From the recursive relation (2.9), (2.10) and above initial conditions, we get
Also, and are calculated as
and so on. Substitute all the values of and in Equation (2.7), we get
(The shock occurs at ). This is an approximate solution of given system of equations.
From the examples above, we can clearly say that we can calculate and when explicitly solutions exist for given initial functions. More importantly, the methodology    does have potential application to the system of nonlinear partial differential equations and clearly in the case of stochastic parameters as well. The given system of equation has a unique solution for the given boundary conditions.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
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