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 = n = 0 A n , v u y = n = 0 B n , u v x = n = 0 C n , v v y = n = 0 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

n = 0 u n ( x , y , t ) = f ( x , y ) + L t 1 [ 1 s L t [ n = 0 A n + n = 0 B n ] ]

n = 0 v n ( x , y , t ) = g ( x , y ) + L t 1 [ 1 s L t [ n = 0 C n + n = 0 D n ] ]

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

u 0 ( x , y , t ) = f ( x , y ) , u n + 1 ( x , y , t ) = L t 1 [ 1 s L t [ n = 0 A n + n = 0 B n ] ] , n 0. (9)

v 0 ( x , y , t ) = g ( x , y ) , v n + 1 ( x , y , t ) = L t 1 [ 1 s L t [ n = 0 C n + n = 0 D n ] ] , n 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 ( x , y ) = g ( x , y ) = x + y (11)

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

u 0 ( x , y , t ) = v 0 ( x , y , t ) = x + y

then u 1 ( x , y , t ) , v 1 ( x , y , t ) can be calculate as

u 1 ( x , y , t ) = L t 1 [ 1 s L t [ A 0 + B 0 ] ] = L t 1 [ 1 s L t [ u 0 u 0 x + v 0 u 0 y ] ] = L t 1 [ 1 s L t [ ( x + y ) + ( x + y ) ] ] = 2 t ( x + y )

Similarly,

v 1 ( x , y , t ) = L t 1 [ 1 s L t [ C 0 + D 0 ] ] = L t 1 [ 1 s L t [ u 0 v 0 x + v 0 v 0 y ] ] = 2 ( x + y ) t

Also, u 2 ( x , y , t ) and v 2 ( x , y , t ) are calculated as

u 2 ( x , y , t ) = L t 1 [ 1 s L t [ A 1 + B 1 ] ] = L t 1 [ 1 s L t [ ( u 0 u 1 x + u 1 u 0 x ) + ( v 0 u 1 y + v 1 u 0 y ) ] ] = L t 1 [ 1 s L t [ ( 2 t ( x + y ) + 2 t ( x + y ) ) + ( 2 t ( x + y ) + 2 t ( x + y ) ) ] ] = L t 1 [ 1 s L t [ 8 t ( x + y ) ] ] = 2 t 2 ( x + y )

Similarly,

v 2 ( x , y , t ) = 4 t 2 ( x + y )

Substitute all the values of u 0 , u 1 , u 2 , and v 0 , v 1 , v 2 , in the Equation (2.7), we get

u ( x , y , t ) = ( x + y ) + 2 t ( x + y ) + 4 t 2 ( x + y ) +

v ( x , y , t ) = ( x + y ) + 2 t ( x + y ) + 4 t 2 ( x + y ) +

This implies,

u ( x , y , t ) = ( x + y ) [ 1 + 2 t + 4 t 2 + ]

v ( x , y , t ) = ( x + y ) [ 1 + 2 t + 4 t 2 + ]

u ( x , y , t ) = x + y 1 2 t

v ( x , y , t ) = x + y 1 2 t

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 ( x , y ) = x 2 , g ( x , y ) = y (12)

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

u 0 ( x , y , t ) = x 2 , v 0 ( x , y , t ) = y

u 1 ( x , y , t ) = L t 1 [ 1 s L t [ A 0 + B 0 ] ] = 2 x 3 t

Similarly,

v 1 ( x , y , t ) = L t 1 [ 1 s L t [ C 0 + D 0 ] ] = y t

Also, u 2 ( x , y , t ) and v 2 ( x , y , t ) are calculated as

u 2 ( x , y , t ) = L t 1 [ 1 s L t [ A 1 + B 1 ] ] = 5 x 4 t

Similarly,

v 2 ( x , y , t ) = y t 2 , u 3 ( x , y , t ) = 14 x 5 t 3

and so on. Substitute all the values of u 0 , u 1 , u 2 , and v 0 , v 1 , v 2 , in Equation (2.7), we get

u ( x , y , t ) = x 2 ( 1 + 2 t x + 5 t 2 x 2 + 14 x 2 t 2 + )

v ( x , y , t ) = y ( 1 + t + t 2 + ) = y 1 t

(The shock occurs at t = 1 4 x ). 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 ( x , y , t ) and v ( x , y , t ) 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

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