**Introduction**

To
solve optimally the problems in science engineering, many works have been
proposed to study the stability for the nonlinear system [1]. Different strong
scheme have been established to solve the nonlinear problems as, Adomian Decomposition
Method (ADM) [2-8], Laplace transform combined with Padé approximation [9-14], Padé
Sumudu Adomian Decomposition Method (PSADM) [15] homotopy perturbation method
[16,17] Modified decomposition method [18-20] have been obtained to approximate
the analytical solutions. In the present paper, we analyze the behaviour of the
Padé Sumudu Adomian Decomposition Methods solution (PSADM) [15] in the case of
nonlinear Schrödinger equations [21] and KdV-Burgers Equations [22]. It can be
seen in many literatures that the Sumudu Adomian Decomposition Methods (SADM)
and Laplace Adomian Decomposition Methods (LADM) give similar results, the
Sumudu transform present some advantage in calculation because have unit
preserving properties (S[1] = 1). The Padé approximations have been used to
control the convergence of the series solution. The function P_{[L/M]}[.],
called double Padé approximation [15] can also be use for Laplace Adomian
decomposition method, and obtain the new solution. In this paper we analyze the
behaviour of the Padé Sumudu Adomian Decomposition Methods Solution and
provided some criteriums for the choice of the best PSADMs.

**Padé Approximation**

The [*L,M*]-order
Padé approximation of the function *f*
denote by *P _{x}^{[L,M]}*
[

*f*], is the quotient of two polynomials

*R*(

_{L}*x*) and

*Q*(

_{M}*x*) of degrees

*L*and

*M*, respectively:

**Remark 1****:** *The [L,M]-order Padé approximation of the
function f(x) is in the form:*

If

**Definition
1:** *Let f be function of two variables x and t.
We defined two dimensional Padé approximation P _{[L/M]}[f](x,t) of the
function f as*

* *

P_{[L/M]}
[*f*] (*x,t*) = P_{x}^{[L,M] }[P_{t}^{[L,M]}
[f]] (*x,t*), (2)

* *

*where P _{t}*

^{[L,M] }[

*f*] (

*x, t*)

*denote the*[

*L,M*]

*-order Padé approximation of f(x, t) with respect to the variable t, and P*[

_{x}*L,M*]

*[*

*f*]

*(x, t) denote the*[

*L,M*]

*-order Padé approximation of f*(

*x, t*)

*with respect to the variable x.*

* *

*If M=L, we
will denote the diagonal Padé approximation of order M by P*_{[M/M]}* *[*f*]*
*(*x, t*)*, and called *[*M,M*]*-order Padé approximation or M Padé
approximation of f*(*x, t*)*.*

**PSADM Procedure [15]**

By replacing the Sumudu transform by Laplce transform and using the same procedure we will obtain the Laplace Adomian Decomposition Methods instead of Sumudu Adomian decomposition method.

We consider the PDE in the form as following:

*L _{t}u*(

*x,t*)

*+L*(

_{x}u*x,t*)

*+R*(

*u*(

*x,t*))

*+G*(

*u*(

*x,t*))

*=F*(

*x,t*)

*(3)*

with u(*x,0*)=h(*x*) the initial condition, L*x* the highest order differential respect
to *x*, *L _{t}* the first order differential respect to

*t*,

*G*(

*u*(

*x,y*)) is the nonlinear term,

*F*(

*x,t*) is the inhomogeneous term, and

*R*the remaining linear terms of lower order derivative.

The procedure of PSADM for solving (3) can be write as follows.

**Step 1:** Take the
Sumudu transform to the equation (3) and apply the differentiation property of
Sumudu transform to obtain

*S _{t}*[

*u*(

*x,t*)](

*v*)=

*h*(

*x*)+

*v.S*[-

_{t}*L*(

_{x}u*x,t*)-

*R*(

*u*(

*x,t*))-

*G*(

*u*(

*x,t*))+

*F*(

*x,t*)](

*v*).

*(4)*

**Step 2:** Apply the
inverse of the Sumudu transform to the above equation to obtain

*u*(*x,t*)=S* _{v}^{-1}*[

*h*(

*x*)](

*t*)+S

*[*

_{v}^{-1}*v.St*[-

*L*(

_{x}u*x,t*)-

*R*(

*u*(

*x,t*))-

*G*(

*u*(

*x,t*))+

*F*(

*x*,

*t*)](

*v*)](

*t*).. (5)

**Step 3:** Use Adom*ian decomposition method to decomposite the
nonlinear function* G(*u*) and the* u*, respectively, as

**Step 4:** Write the
equation in the form

**Step 5:** Deduct the SADM approximation solution *u _{SADM}*=

*u*(

*x,t,j*):

*u*(*x,t,j*)=_{u0}+*u*_{1}+…+ *uj*.

**Step 6:** The [*L,M*]-order
PSADM solution *uPSADM*=*u*(*x,t,j,*[*L,M*]) is given by

*u*(*x,t,j,* [*L,M*])=*P[L/M]* [*u _{SADM}*]
(

*x,t*),

if
*L=M*, we denote *M*-PSADM solution by

*u*(*x,t,j,M*)=P_{[M/M] }[*u _{SADM}*] (

*x,t*).

**Remark 2****:** *In step 5 we obtain
the Sumudu Adomian Decomposition Method (SADM), instead of Sumudu transform we
can use other integral transform like Laplace transform to obtain in step 5 the
Laplace Adomian Decomposition Method (LADM). The Sumudu Adomian Decomposition
Method and Laplace Adomian Decomposition Methods give similar result. The
Sumudu transform due to the unit preserving properties (S(1)=1, provided some
advantages in calculation.** *

For
different type of Padé approximation and different order of the Padé approximation
we will analyse the behaviour of the PADM solutions.

**Example 1 **

In
the first case of the following example, we will show that for different type
of Padé approximation:

*u*(*x, t, j*, [L,M]) , for L > M,

*u*(*x, t, j*, [L,M]), for L = M, and

*u*(*x, t, j*, [L,M]), for L < M, we have
different solutions and one of them is more accurate.

In the second case of the following example, we will
show that for diagonal Padé approximation *u*(*x,t,j,* [L,M]), for L = M, we can
increase the accuracy of the method by increasing or reducing de value of M
accordingly to the topologie of the solution.

** Case 1: **Consider the
equation:

We can easily deduice the SADM solution [15]:

The algorithm is coded by the symbolic computation
software Mathematica. We know *u*(*x,t*) = e^{i(x+t) }is the exact
solution for the Problem.

**Figure (1a)** and **Figure** **(1b)** show the real part and imaginary part of SADM solution *u _{SADM} *= u(

*x,t,*15) in Domain

*D*= [0,2] × [0,2].

For different values of L and M we plot different
orders of the PSADM solutions to see the behaviour of the methods.

**Figure** **(1c)** and **Figure** **(1d)** show
respectively, the real part and imaginary part of PSADM solution *u _{PSADM}* =

*u*(

*x,t*,15,[7,0]) in Domain

*D*= [0,2] × [0,2].

**Figure** **(1e) **and **Figure** **(1f)** show
respectively, the real part and imaginary part of PSADM solution *u _{PSADM}* =

*u*(

*x,t*,15,[1,7]) in Domain

*D*= [0,2] × [0,2].

**Figure** **(2a)** and **Figure** **(2b)** show
respectively, the real part and imaginary part of PSADM solution *u _{PSADM}* =

*u*(

*x*,

*t*,15,[7,1]) in Domain

*D*= [0,2] × [0,2].

**Figure** **(2c)** and **Figure** **(2d)** show
respectively, the real part and imaginary part of PSADM solution *u _{PSADM}* =

*u*(

*x,t,*15,[7,7]) or in short u(

*x,t*,15,7) in Domain

*D*= [0,2] × [0,2].

**Figure** **(2e)** and** Figure** **(2f)** show the
real part and imaginary part of the exact solution *u*(*x,t*) in Domain* D* = [0,2] × [0,2].

Figure 1: SADM and PSADM solutions using 15 terms.

Figure 2: (a)-(d) PSADM solutions using 15 terms, (e) and (f) exact solutions.

In domain *D*
= [0, 2] × [0, 2] we can see that the *u*(*x, t*, 15, [1, 7]) and *u*(*x,
t*, 15, [7, 1]) are not smooth compare to the other solutions. Next we plots
the absolute errors.

**Figure** **(3a)** and **Figure** **(3b)** show
respectively the absolute error for real part and imaginary part of the Sumudu
Adomian Decomposition solution *u*(*x, t,* 15).

**Figure** **(3c)** and** Figure** **(3d)** show
respectively the absolute error for real part and imaginary part of the Padé
Sumudu Adomian Decomposition solution *u*(*x, t*, 15, [7, 0]).

**Figure** **(3e)** and **Figure** **(3f)** show
respectively the absolute error for real part and imaginary part of the Sumudu
Adomian Decomposition solution *u*(*x, t*, 15).

Now let see the behaviour of the SADM, PSADM solutions
in domain *D* = [0, 10] × [0, 10].

**Figure (4a)** and **Figure** **(4b)** show the real part and imaginary part of SADM solution *u _{SADM}* = u(

*x, t*, 15) in Domain

*D*= [0, 10] × [0, 10].

**Figure** **(4c)** and **Figure** **(4d)** show
respectively, the real part and imaginary part of PSADM solution *u _{PSADM}* = u(

*x, t*, 15, [7, 0]) in Domain

*D*= [0, 10] × [0, 10].

**Figure** **(4e)** and** Figure** **(4f)** show
respectively, the real part and imaginary part of PSADM solution *u _{PSADM}*

_{ }= u(

*x, t*, 15, [1, 7]) in Domain

*D*= [0, 10] × [0, 10].

**Figure** **(5a)** and **Figure** **(5b)** show
respectively, the real part and imaginary part of PSADM solution *u _{PSADM} *= u(

*x, t,*15, [7, 1]) in Domain

*D*= [0, 10] × [0, 10].

**Figure** **(5c)** and **Figure** **(5d)** show
respectively, the real part and imaginary part of PSADM solution *u _{PSADM} *= u(

*x, t*, 15, [7, 7]) or in short u(

*x, t*, 15, 7) in Domain

*D*= [0, 10] × [0, 10].

**Figure** **(5e)** and **Figure** **(5f)** show the
real part and imaginary part of the exact solution u(*x, t*) in Domain *D* = [0,
10] × [0, 10].

Figure 4: SADM and PSADM solutions using 15 terms.

Figure 5: (a)-(d) PSADM solutions using 15 terms, (e) and (f) exact solutions.

We can see in this example, in domain *D* = [0, 2] × [0, 2] the SADM behave well,
but in domain *D* = [0, 10] × [0, 10]
the SADM give bad result. Only the u(x, t, 15, 7) approaching well the exact
solution in the domain *D* = [0, 2] × [0,
2] and *D* = [0, 10] × [0, 10]. Even in
domain

*D* = [0, 2] × [0,
2] the graph of absolute error show the [7, 7]-order PSADM (in short 7-PASDM)
solution is better than SADM solution and other type of PSADM solutions. The
solution can be perfom by choosing different value of M.

**Remark 3**

*The real
part and imaginary part of the exact solution are bounded,*

*and the real
part and imaginary part of the SADMs are not bounded, in this case in better to
use diagonal Padé approximatin to make bounded the Approximate solution.*

**Case 2:**** **Consider
the equation:

We can easily deduce the SADM solution [15]:

And the [L,M] order PSADM solution:

*u _{PSADM}* = u(

*x, t, j*, [

*L,M*]) = P

*[*

_{[L/M]}*u*(

*x, t, j*)] (

*x, t*), (14)

The algorithm is coded by the symbolic computation
software Mathematica.

We know *u*(*x, t*) = e^{3i(x-t)} is the exact
solution of the Problem.

**Figure **(6c) and **Figure** (6d) show respectively the
absolute error for real part and imaginary part of the PSADM solution *u*(*x,
t*, 20, 7).

**Figure** (6e) and **Figure** (6f) show respectively the
absolute error for real part and imaginary part of the PSADM solution *u*(*x,
t*, 20, 8).

**Figure** (6a) and** Figure** (6b) show respectively the
absolute error for real part and imaginary part of the SADM solution *u*(*x,
t*, 20).

The graph of the absolute errors show that the SADM solution give better approximation than 7-order PSADM solution, and the 8-order PSADM solution give better approximation than SADM solution. The solution can be perform by choosing different value of M.

**Example 2**

**Case 1:** Consider
the equation:

Using the SADMs, we can deduice:

The solution is given by:

The Sumudu Adomian Decomposition solution is given by

Then the [M,N]-oder Padé Sumudu Adomian Decomposition solution is given by

By using the symbolic computation software
Mathematica.

The **figure** **(7a)**, **Figure** **(7b)**, **Figure** **(7c)** and **Figure** **(7d)** show respectively the curve of
SADM solution *u*(*x, t*, 3), PSADM solution *u _{PSADM}*
=

*u*(

*x, t*, 3, [2,2]),

*u*=

_{PSADM}*u*(

*x, t,*3, [1, 2]) and the exact solution u = u(x, t), in domain

*D*= [-0.5, 0.5] × [0, 0.1]. The exact solution is given by

*u*(

*x, t*) = -2sech

^{2}(

*x*-4

*t*)

Figure 7: (a) SADM solution, (b) and (c) PSADM solutions using 3 terms, (d) exact solutions.

**Figure** **(8a)**,** Figure** **(8b)**, **Figure** **(8c)** and **Figure** **(8d)** show respectively the curve of
SADM solution *u*(*x, t*, 3), PSADM solution *u _{PSADM}*
=

*u*(

*x, t,*3, [1, 2]) and the exact solution

*u*=

*u*(

*x, t*), in domain

*D*= [-1, 1] × [0, 1]

Figure 8: (a) SADM solution, (b) and (c) PSADM solutions using 3 terms, (d) exact solutions.

We can see the SADM, [2, 2]- order PSADM and [1,
2]-order PSADM solutions behave well in domain* D* = [-0.5, 0.5] × [0, 0.1], but only [1, 2]-order PSADM solution
give better result in domain *D* = [-1,
1] × [0, 1]. The [2, 2]-order PSADM solution in this case in not better than [1,
2]-order PSADM solution. Then the diagonal Padé approximation are not accurate
in this case. It is recommended in case to use [M,N]-order Padé approximation
with M ≠ N.

**Case 2:**** **Consider
the equation:

We know for q = 6, β = -1, and subject to the initial condition

the exact solution is given by:

For k = 0,5, the **figure (9a)**, **Figure**** (9b)**, **Figure**** (9c)**, **Figure**** (9d)**, and **Figure**** (9e)** show
respectively the curve of the SADM solution *uSADM*
= *u*(*x*, *t*, 4), PSADM solutions
*u*(*x,
t*, 4, 2), *u*(*x, t*, 4, [0, 2]), *u*(*x, t*, 4, [2, 0]) and the exact solution
u_{exact} in domain *D *= [0,1] × [0, 2]

Figure 9: (a) SADM solution, (b) and (d) PSADM solutions using 4 terms, (e) exact solutions.

For *k* = 0.5,
the **figures** (10a), (10b), (10c), and
(10d) show respectively the absolute error curve for the SADM solution *u _{SADM}* =

*u*(

*x,t*,4), PSADM solutions

*u*(

*x,t*,4,2),

*u*(

*x,t*,4,[0,2]), and

*u*(

*x,t*,4,[2,0]) in domain

*D*= [0,1] × [0,2]

The SADM and [2, 2]-order PSADM (or in short 2-PSADM)
provide same results. The [0, 2]-order PSADM solution in this case providing
better error than [2, 2]-order PSADM and [2, 0]-order PSADM solutions. The
diagonal Padé approximations are not recommended in this case.

**Remark 4**

The following conditions can help to choose the best PSADM
solution.

**Condition (*)**

If

We comput the new solution by

*u*(*x, t, j,* [L,M]) = P_{[L/M]} [*u _{SADM}*] (

*x, t*),with L < M

Then

**Condition (**)**

** **

If

We comput the new solution by

*u*(*x, t, j*, [L,M]) = P_{[L/M]} [*u _{SADM}*] (

*x, t*),with L >M

then

**Condition (***)**

If we are not in the case mentioning in conditions **(*)** and **(**)**, we comput the new solution by

*u*(*x, t, j,* [M,M]) = *u*(*x, t, j*, M) = P_{[M/M]}
[*u _{SADM}*] (

*x, t*),with L = M

**Conclusion**

**Conclusion**

In
this work, we show the behaviour et of the function *P*_{[L/M]}[.](*x, t*)
using to obtain the Padé Sumudu Adomian Decomposition Methods solution for
nonlinear partial differential equations such as the Schrödinger equations, and
the KdV-Burger's equations. The proposed function provide us a suitable way for
controlling the convergences of series solutions with high accuracy by using
different order of Padé approximation and different type of the Padé
approximation according to the topology of the exact solution *u*(*x,
t*) and the topology of the SADM solution *u*(*x, t, j*). When the
exact solutions are unknown, we have some mathematical approach to obtain more
information about the topology of the exact solution. This approach can be
generalized to investigate more complicated nonlinear partial differential
equations that can only be solved by numerically.

**Acknowledgement**

This research is partially supported by the Foundations of China (Nos. 12071261, 12001539, 11831010, 11871068), the Science Challenge Project (No. TZ2018001), and the national key basic research program (No. 2018YFA0703903). The first author also acknowledges the financial support of the Chinese Scholarship Council (CSC) in Shandong University with grand (CSC No: 2020DFJ001523).

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**Corresponding author**

**Metonou
Richard**, School of Mathematics, Shandong
University, China, E-mail: metonourichard@yahoo.fr

**Weidong Zhao**,
School of Mathematics, Shandong University, China, E-mail: wdzhao@sdu.edu.cn

**Citation**

Richard M and Zhao W. Behaviour analysis of the Padé
sumudu adomian decomposition method solution (2021) Edelweiss Appli Sci Tech 5:
39-45.#### Keywords

Adomian Decomposition Method (ADM), Padé Sumudu
Adomian Decomposition Method (PSADM), Nonlinear Schrödinger equation, Nonlinear
KdV Burger's equation.