v n ( x , y , t ) (7)

be the solution of given system of Equations (2.1), (2.2) in series form. Also we can decompose the nonlinear terms appeared in given system by using adomian polynomials, namely

$u{u}_{x}=\underset{n=0}{\overset{\infty }{\sum }}{A}_{n},\text{\hspace{0.17em}}\text{\hspace{0.17em}}v{u}_{y}=\underset{n=0}{\overset{\infty }{\sum }}{B}_{n},\text{\hspace{0.17em}}\text{\hspace{0.17em}}u{v}_{x}=\underset{n=0}{\overset{\infty }{\sum }}{C}_{n},\text{\hspace{0.17em}}\text{\hspace{0.17em}}v{v}_{y}=\underset{n=0}{\overset{\infty }{\sum }}{D}_{n}$ (8)

where ${A}_{n}$,${B}_{n}$,${C}_{n}$ and ${D}_{n}$ are adomian polynomials [16]. From the Equations (2.5), (2.6), (2.7) and (2.8), we get

$\underset{n=0}{\overset{\infty }{\sum }}{u}_{n}\left(x,y,t\right)=f\left(x,y\right)+{L}_{t}^{-1}\left[\frac{1}{s}{L}_{t}\left[\underset{n=0}{\overset{\infty }{\sum }}{A}_{n}+\underset{n=0}{\overset{\infty }{\sum }}{B}_{n}\right]\right]$

$\underset{n=0}{\overset{\infty }{\sum }}{v}_{n}\left(x,y,t\right)=g\left(x,y\right)+{L}_{t}^{-1}\left[\frac{1}{s}{L}_{t}\left[\underset{n=0}{\overset{\infty }{\sum }}{C}_{n}+\underset{n=0}{\overset{\infty }{\sum }}{D}_{n}\right]\right]$

Comparing the both sides of above system of equations, we get the following recursive relations

${u}_{0}\left(x,y,t\right)=f\left(x,y\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{n+1}\left(x,y,t\right)={L}_{t}^{-1}\left[\frac{1}{s}{L}_{t}\left[\underset{n=0}{\overset{\infty }{\sum }}{A}_{n}+\underset{n=0}{\overset{\infty }{\sum }}{B}_{n}\right]\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\ge 0.$ (9)

${v}_{0}\left(x,y,t\right)=g\left(x,y\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{v}_{n+1}\left(x,y,t\right)={L}_{t}^{-1}\left[\frac{1}{s}{L}_{t}\left[\underset{n=0}{\overset{\infty }{\sum }}{C}_{n}+\underset{n=0}{\overset{\infty }{\sum }}{D}_{n}\right]\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\ge 0.$ (10)

Note that the solution of (2.1), (2.2) can exhibit a shock phenomenon for finite t; we select f(x, y) and g(x, y) such that the shock occurs for a value of t far from our region of interest. Let

$f\left(x,y\right)=g\left(x,y\right)=x+y$ (11)

Therefore from the recursive relation (2.9) and (2.10), we get

${u}_{0}\left(x,y,t\right)={v}_{0}\left(x,y,t\right)=x+y$

then ${u}_{1}\left(x,y,t\right)$,${v}_{1}\left(x,y,t\right)$ can be calculate as

$\begin{array}{c}{u}_{1}\left(x,y,t\right)={L}_{t}^{-1}\left[\frac{1}{s}{L}_{t}\left[{A}_{0}+{B}_{0}\right]\right]\\ ={L}_{t}^{-1}\left[\frac{1}{s}{L}_{t}\left[{u}_{0}{u}_{0x}+{v}_{0}{u}_{0y}\right]\right]\\ ={L}_{t}^{-1}\left[\frac{1}{s}{L}_{t}\left[\left(x+y\right)+\left(x+y\right)\right]\right]\\ =2t\left(x+y\right)\end{array}$

Similarly,

${v}_{1}\left(x,y,t\right)={L}_{t}^{-1}\left[\frac{1}{s}{L}_{t}\left[{C}_{0}+{D}_{0}\right]\right]={L}_{t}^{-1}\left[\frac{1}{s}{L}_{t}\left[{u}_{0}{v}_{0x}+{v}_{0}{v}_{0y}\right]\right]=2\left(x+y\right)t$

Also, ${u}_{2}\left(x,y,t\right)$ and ${v}_{2}\left(x,y,t\right)$ are calculated as

$\begin{array}{c}{u}_{2}\left(x,y,t\right)={L}_{t}^{-1}\left[\frac{1}{s}{L}_{t}\left[{A}_{1}+{B}_{1}\right]\right]\\ ={L}_{t}^{-1}\left[\frac{1}{s}{L}_{t}\left[\left({u}_{0}{u}_{1x}+{u}_{1}{u}_{0x}\right)+\left({v}_{0}{u}_{1y}+{v}_{1}{u}_{0y}\right)\right]\right]\\ ={L}_{t}^{-1}\left[\frac{1}{s}{L}_{t}\left[\left(2t\left(x+y\right)+2t\left(x+y\right)\right)+\left(2t\left(x+y\right)+2t\left(x+y\right)\right)\right]\right]\\ ={L}_{t}^{-1}\left[\frac{1}{s}{L}_{t}\left[8t\left(x+y\right)\right]\right]\\ =2{t}^{2}\left(x+y\right)\end{array}$

Similarly,

${v}_{2}\left(x,y,t\right)=4{t}^{2}\left(x+y\right)$

Substitute all the values of ${u}_{0},{u}_{1},{u}_{2},\cdots$ and ${v}_{0},{v}_{1},{v}_{2},\cdots$ in the Equation (2.7), we get

$u\left(x,y,t\right)=\left(x+y\right)+2t\left(x+y\right)+4{t}^{2}\left(x+y\right)+\cdots$

$v\left(x,y,t\right)=\left(x+y\right)+2t\left(x+y\right)+4{t}^{2}\left(x+y\right)+\cdots$

This implies,

$u\left(x,y,t\right)=\left(x+y\right)\left[1+2t+4{t}^{2}+\cdots \right]$

$v\left(x,y,t\right)=\left(x+y\right)\left[1+2t+4{t}^{2}+\cdots \right]$

$u\left(x,y,t\right)=\frac{x+y}{1-2t}$

$v\left(x,y,t\right)=\frac{x+y}{1-2t}$

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 [15]. Let we change the initial conditions to

$f\left(x,y\right)={x}^{2},\text{ }g\left(x,y\right)=y$ (12)

From the recursive relation (2.9), (2.10) and above initial conditions, we get

${u}_{0}\left(x,y,t\right)={x}^{2},\text{ }{v}_{0}\left(x,y,t\right)=y$

${u}_{1}\left(x,y,t\right)={L}_{t}^{-1}\left[\frac{1}{s}{L}_{t}\left[{A}_{0}+{B}_{0}\right]\right]=2{x}^{3}t$

Similarly,

${v}_{1}\left(x,y,t\right)={L}_{t}^{-1}\left[\frac{1}{s}{L}_{t}\left[{C}_{0}+{D}_{0}\right]\right]=yt$

Also, ${u}_{2}\left(x,y,t\right)$ and ${v}_{2}\left(x,y,t\right)$ are calculated as

${u}_{2}\left(x,y,t\right)={L}_{t}^{-1}\left[\frac{1}{s}{L}_{t}\left[{A}_{1}+{B}_{1}\right]\right]=5{x}^{4}t$

Similarly,

${v}_{2}\left(x,y,t\right)=y{t}^{2},\text{ }{u}_{3}\left(x,y,t\right)=14{x}^{5}{t}^{3}$

and so on. Substitute all the values of ${u}_{0},{u}_{1},{u}_{2},\cdots$ and ${v}_{0},{v}_{1},{v}_{2},\cdots$ in Equation (2.7), we get

$u\left(x,y,t\right)={x}^{2}\left(1+2tx+5{t}^{2}{x}^{2}+14{x}^{2}{t}^{2}+\cdots \right)$

$v\left(x,y,t\right)=y\left(1+t+{t}^{2}+\cdots \right)=\frac{y}{1-t}$

(The shock occurs at $t=\frac{1}{4x}$ ). This is an approximate solution of given system of equations.

3. Conclusion

From the examples above, we can clearly say that we can calculate $u\left(x,y,t\right)$ and $v\left(x,y,t\right)$ when explicitly solutions exist for given initial functions. More importantly, the methodology [1] [2] [3] 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.

Cite this paper

Handibag, S.S. (2019) Laplace Decomposition Method for the System of Non Linear PDEs. Open Access Library Journal, 6: e5954. https://doi.org/10.4236/oalib.1105954

References

1. 1. Handibag, S.S. and Karande, B.D. (2012) Application of Laplace Decomposition Method to Solve Linear and Nonlinear Heat Equation. International Journal of Applied Physics and Mathematics, 2, 369-371.
https://doi.org/10.7763/IJAPM.2012.V2.137

2. 2. Handibag, S.S. and Karande, B.D. (2013) Existence the Solutions of Some Fifth-Order Kdv Equation by Laplace Decomposition Method. American Journal of Computational Mathematics, 3, 80-85. https://doi.org/10.4236/ajcm.2013.31013

3. 3. Khan, M. (2010) Application of Laplace Decomposition Method to Solve Nonlinear Coupled Partial Differential Equations. World Applied Sciences Journal, 9, 13-19.

4. 4. Liu, M.-H. and Guan, K.-Y. (2009) The Lie Group Integrality of the Fisher Type Travelling Wave Equation. Acta Mathematicae Sinica, 25, 305-320.
https://doi.org/10.1007/s10255-007-7106-6

5. 5. Wazwaz, A.-M. (2008) The Hirota’s Bilinear Method and the Tanh-Coth Method for Multiple Soliton Solutions of the Sawada-Kotera Kadomtsev-Petviashvili Equation. Applied Mathematics and Computation, 200, 160-166.
https://doi.org/10.1016/j.amc.2007.11.001

6. 6. Borhanifar, A. and Kabir, M.M. (2009) New Periodic and Soliton Solutions by Application of Exp-Function Method for Nonlinear Evolution Equations. Journal of Computational and Applied Mathematics, 229, 158-167.
https://doi.org/10.1016/j.cam.2008.10.052

7. 7. Parkes, E.J. and Duffy, B.R. (1996) An Automated Tanh-Function Method for Finding Solitary Wave Solutions to Non-Linear Evolution Equations. Computer Physics Communications, 98, 288-300.
https://doi.org/10.1016/0010-4655(96)00104-X

8. 8. Fan, E.G. and Hon, Y.C. (2003) Applications of Extended Tanh Method to Special Types of Nonlinear Equations. Applied Mathematics and Computation, 141, 351-358. https://doi.org/10.1016/S0096-3003(02)00260-6

9. 9. Tascan, F. and Bekir, A. (2009) Analytic Solutions of the (2 + 1)-Dimensional Nonlinear Evolution Equations Using the Sine-Cosine Method. Applied Mathematics and Computation, 215, 3134-3139. https://doi.org/10.1016/j.amc.2009.09.027

10. 10. Nadukandi, P., Oate, E. and Garcia, J. (2010) A High-Resolution Petrov-Galerkin Method for the 1D Convection-Diffusion-Reaction Problem. Computer Methods in Applied Mechanics and Engineering, 199, 525-546.
https://doi.org/10.1016/j.cma.2009.10.009

11. 11. Zhou, J.K. (1986) Differential Transform and Its Application for Electrical Circuits. Huazhong University Press, Wuhan.

12. 12. Drazin, P.G. and Johnson, R.S. (1989) Solutions: An Introduction. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9781139172059

13. 13. Liu, X.Q. and Bai, C.L. (2000) Exact Solutions of Some Fifth-Order Nonlinear Equations. Applied Mathematics: A Journal of Chinese Universities, 15, 28-32.
https://doi.org/10.1007/s11766-000-0005-8

14. 14. Adomian, G. (1988) A Review of the Decomposition Method in Applied Mathematics. Journal of Mathematical Analysis and Applications, 135, 501-544.
https://doi.org/10.1016/0022-247X(88)90170-9

15. 15. Bellman, R.E. and Adomian, G. (1985) Partial Differential Equations—New Methods for Their Treatment and Application. Reidel, Dordrecht.
https://doi.org/10.1007/978-94-009-5209-6