The Nonlinear partial differential equations (NPDEs) have a big advantage in various fields of science like mathematical, physics, chemistry, biology, mechanics, and engineering. It is important to get a reliable solution and numerical solution for nonlinear systems. In this section we solved the Wu-Zhang Systems of one dimensional with initial condition, the Wu-Zhang system can be written as
where (v) is height water,(u) is the surface speed of the water along x-direction (Mix, S. K. , Laing, Y. C.; Feng , Ja, Anwar; Jameel , Manaa, Saad A.; Easif , R. K., Saeed , Zheng, Xuedong; Chen , Manaa, Saad A.; Easif  , Hosseini, K.; Ansari , Khater, Mostafa M. A.; Attia , Jafari, M. A.; Aminataei, A. ).
2. Successive Approximation Method (S.A.M) 
The successive approximation method from important and active methods to solve partial differential equations, and also is good method for solving any initial value problem
It starts by observing that any solution to (2) must also be a solution to
Thus iterative these steps of solutions, we obtain on solution closer to the accurate solutions of (3), the S.A.M depends on the integral equation (3), as in the following:
This process can be continued to obtain the nth approximate
Therefore determine if closer the solution as n increases. This is done by proving the following:
The sequence converges to a limit , that is:
The limiting function is a solution of (3) on the interval . The solution of (3) is unique. A proof of these results can be depended along the lines of the corresponding proof for ordinary differential equations (Coddington, 1995).
3. Application (S.A.M) to solve of Equation (1)
In this section, we solve the Wu-Zhang systems of one dimensional
With initial condition:
where and are arbitrary constants.
By using S.A.M as follows:
Integrating both sides of Equations (4), (5) with respect to (s), from (0) to (t), we get
Using the initial condition in Equations (8), (9) we obtain:
We substituting initial conditions and in the integral Equations (10), (11) to get a first approximation and
Then this and is substituted again in the integral of (10), (11) after replacing (t) by (s) to get a second approximation and
This process can continued to get the nth approximation
For to solve Equations (16), (17), we use the initial conditions and which are given in Equations (6) and (7) , respectively. by using iterative steps for Equations (16), (17) we can obtain , , and , , , after then substituting these values in the Equations (18) and (19)
We get the following series:
Therefore, the exact solution of Wu-Zhang equation is given by
This solution is convergent to the exact solution
The approximate results for and were compared with exact solution as showing in Table 1 and Table 2. Also we compare figure approximate solution with figure exact solution as showing in Figure 1 and Figure 2. Figure 3, error between approximate solution and exact solution.
Table 1. Comparing exact solutions with approximate solutions (S.A.M) of the Wu-Zhang systems with initial conditions Equations (6), (7) where , , , for .
Table 2. Comparing exact solutions with approximate solutions (S.A.M) of the Wu-Zhang systems with initial conditions Equations (6), (7) where , , , for .
Figure 1. Showing matching approximate solution with exact solution. (USAM) approximate solution, (Uexact) exact solution.
Figure 2. Showing matching approximate solution with exact solution. (VSAM) approximatesolution, (Vexact) exact solution.
Figure 3. (UError) Showing error of the approximate solution for with exact solution, (VError) Showing error of the approximate solution for with exact solution.
In this search, the Successive Approximate Method (S.A.M) was used to get the approximate solution of Wu-Zhang Systems. The results we got from this method are high efficient with big accurate and give a good convergence to the exact solution.
The authors are very grateful to the University of Mosul/College of Computer Sciences and Mathematics for their provided facilities, which helped to improve the quality of this work.
 Liang, Y.C., Feng, D.P., Lee, H.P., Lim, S.P. and Lee, K.H. (2002) Successive Approximation Training Algorithm for Feedforward Neural Networks. Neurocomputing, 42, 311-322.
 Ja, A., Jameel, F. and Al, I. (2019) Soliton Solutions of Wu-Zhang System of Evolution Equations. Numerical and Computational Methods in Science and Engineering an International Journal, 11, 1-11.
 Manaa, S.A., Easif, F.H. and Mahmood, B.A. (2013) Successive Approximation Method for Solving Nonlinear Diffusion Equation with Convection Term. IOSR Journal of Engineering, 3, 28-31. https://doi.org/10.9790/3021-031232831
 Zheng, X., Chen, Y. and Zhang, H. (2003) Generalized Extended Tanh-Function Method and Its Application to (1 + 1)-Dimensional Dispersive Long Wave Equation. Physics Letters A, 311, 145-157. https://doi.org/10.1016/S0375-9601(03)00451-1
 Hosseini, K., Ansari, R. and Gholamin, P. (2011) Exact Solutions of Some Nonlinear Systems of Partial Differential Equations by Using the First Integral Method. Journal of Mathematical Analysis and Applications, 387, 807-814. https://doi.org/10.1016/j.jmaa.2011.09.044
 Khater, M.M.A., Attia, R.A.M. and Lu, D. (2019) Numerical Solutions of Nonlinear Fractional Wu-Zhang System for Water Surface versus Three Approximate Schemes. Journal of Ocean Engineering and Science, 4, 144-148. https://doi.org/10.1016/j.joes.2019.03.002