On theories of natural decomposition method applied to system of nonlinear differential equations in fluid mechanics

Abstract

In shallow waters, the Wu-Zhang (WZ) system describes the (1+1)-dimensional dispersive long wave in two horizontal directions, which is important for the engineering community. This paper presents proofs for various theorems and shows that the natural decomposition method (NDM) solves systems of linear and nonlinear ordinary and partial differential equations under proper initial conditions, such as the Wu-Zhang system. We use a combination of two methods, namely the natural transform method to deal with the linear terms and the Adomian decomposition method to deal with the nonlinear terms. Several examples of linear and nonlinear systems (ODEs and PDEs) are given, including the Wu-Zhang (WZ) system. The present approach, which has numerous applications in the science and engineering fields, is a great alternative to the many existing methods for solving systems of differential equations. It also holds great promise for additional real-world applications.

AMS Classification: 35A02, 35A22, 34M25, 35C10, 35F25, 35E15.

Introduction

Due to its numerous applications in physics, chemistry, mechanics, electricity, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena, and aerodynamics, solving nonlinear systems of differential equations has been the subject of extensive research for decades, see (Dundar et al. ¹ ). Numerous scholars have utilized a variety of diverse ways to obtain approximations of this model’s solutions, but in our study, we use two famous techniques (Adomian decomposition, natural transform), which have never been used before on the Wu-Zhang system of nonlinear ordinary and partial differential equations (Wazwaz, ² Adomian, ³ Adomian ⁴ ). There are various marine activities and biochemical processes that take place above and below the ocean’s surface. Therefore, the purpose of our work is to apply a new alternative and powerful schemes to the WZ system in order to develop exact solutions of the problem in discussion that would be useful for those processes and activities, as well as to add more physical qualities to the model, (see Krishnan ⁵ ). The generalized KdV equation is a crucial model for a number of physical phenomena, such as shallow-water waves close to the critical value of surface tension and waves in a nonlinear LC circuit with mutual inductance between neighboring inductors. An LC circuit is a type of electric circuit that consists of a capacitor and an inductor coupled together. It is also known as a tuned circuit, tank circuit, or resonant circuit. Because there is no universal solution for this class of models, even if the fifth order KdV equation’s exact solution was discovered for the unique case of solitary waves, (see Qasim and Ali, ⁶ Miah et al. ⁷ ). One of the crucial characteristics of many models is the dispersion of water waves. The frequency dispersion is described by this feature. In another sense, it refers to the movement of waves with various wave lengths and phase speeds.

One of the nonlinear partial differential equations and the Wu-Zhang system are discussed in this study. In two horizontal orientations and on shallow waters, this system describes (1 + 1)-dimensional dispersive long waves, (see Miah et al. ⁷ ). Mathematical models are commonly used by physicists, engineers, computer scientists, and mathematicians. The use of statistical models is particularly useful in the natural sciences and engineering, as well as in the social sciences. As a result of that, obtaining accurate or approximate solutions to different types of equations in physics and applied mathematics is still a significant issue. Various powerful mathematical methods, such as the Adomian decomposition method, the natural decomposition method (Rawashdeh, ⁸ Rawashdeh and Maitama, ⁹ Obeidat and Bentil, ¹⁰ Obeidat and Bentil, ¹¹ Obeidat and Bentil ¹² ), the reduced differential transform method (RDTM) (Rawashdeh, ¹³ Rawashdeh, ¹⁴ Rawashdeh and Obeidat, ¹⁵ Rawashdeh ¹⁶ ), the natural transform method (Belgacem and Silambarasan, ¹⁷ Hussain and Belgacern, ¹⁸ Khan and Khan ¹⁹ ), The Summudu transform method, (Belgacem et al. ²⁰ ) Laplace decomposition method (Spiegel ²¹ ) are still useful tools for solving these equations. By using the natural Adomian decomposition method, we don’t have to worry about discretization, linearization, or thinking of any restrictive assumptions like with differential transform method and Homotopy perturbation method. The NADM algorithm greatly reduces the size of computational work and avoids round-off errors. The current mechanism consists of combining the natural transformation with the Adomian decomposition. These methods are both new to this century, and have been used in different ways for different purposes. In addition, Khan et al. (2016) found that the use of social media has a significant impact on our social lives. Some properties and applications of natural transformations are given in Hussain and Belgacern ¹⁸ under the name N-transform. In addition, researchers solved the unsteady fluid flow problem over a plane wall, and they showed how it is related to the Laplace and Sumudu transforms. Elementary properties such as first shift, change of scale, transform of derivatives (first and second) and integrals of N-(natural) can be found very important in mathematics.

The current research work is presented as follows: In section two, we introduce the basic concepts of the natural transformation including some properties and definitions. Also, we give proofs for important theorems of the natural transform, which can be used later to give exact solutions to well-known systems of ODEs and PDEs in section three. Section four is devoted to solve some applications of linear and nonlinear systems of both ODEs and PDEs using the proposed method. Finally, we give the conclusion of our work in section five.

Natural Adomian decomposition method

This section provides an overview of the natural transform’s definitions, literature, and relationships to the Sumudu and Laplace transformations. The idea of the natural transform method (NTM), established by many researchers, is an integral operator that takes a function and converts it (by integration) into another function, see (Belgacem and Silambarasan, ¹⁷ Hussain and Belgacern, ¹⁸ Khan and Khan ¹⁹ ).Unlike when utilizing the differential transform technique and Homotopy Perturbation Method (HPM), we do not need to linearize, discretize, or make any restrictive assumptions while using the natural Adomian decomposition method (NADM). The size of the computing task is then significantly reduced by the NADM computational approach, and round-off error is avoided. We suggest that readers read up on the history of the general integral transform, the Laplace, Sumudu, and natural transform methods, as well as the method’s associated properties for any particular function $\varphi (\eta ),\eta \in \mathbb{R}$ .

• Definition 2.1 Consider $\varphi \left(\eta \right)$ to be a piece-wise continuous function over $\mathbb{R}$ and $C_1,C_2,p,q>0$ where $p<q$ . Suppose that $A=\{\varphi \left(\eta \right):\left|\varphi \left(\eta \right)\right|<C_1{e}^{p\eta }{\chi }_{(x_2,\infty )}(\eta )+C_2{e}^{q\eta }{\chi }_{(-\infty ,{\eta }_1)}(\eta )\}.$ So, $\left|\varphi \left(\eta \right)\right|\le C_1{e}^{p\eta }$ for $\eta \to \infty$ that is, $\eta >{\eta }_2$ and $\left|\varphi \left(\eta \right)\right|\le C_2{e}^{q\eta }$ for $\eta \to -\infty$ that is, $\eta <{\eta }_1$ .

Note that for any $\varphi \left(\eta \right)$ in the class $A$ with $r,w>0$ we have:

$$\begin{array}{c}\multicolumn{1}{c}{\left|{\int }_{-\infty }^{\infty }{e}^{-r\eta }\varphi \left(\eta w\right)d\eta \right|\le C_1{\int }_0^{\infty }{e}^{-r\eta }{e}^{p\left|\eta w\right|}d\eta +C_2{\int }_{-\infty }^0{e}^{-r\eta }{e}^{q\left|\eta w\right|}d\eta }\\ =C_1{\int }_{{\eta }_2}^{\infty }{e}^{(\mathrm{pw}-r)\eta }d\eta +C_2{\int }_{-\infty }^{{\eta }_1}{e}^{(qw-r)\eta }d\eta .\hfill \end{array}$$

The above is convergent if $pw-r<0$ and $qw-r>0$ , thus $pw<r<qw$ and so $p<\frac{r}{w}<q$ . Hence, $\varphi \left(\eta \right)$ is of exponential order.

Then, the natural transformation (N-transformation) is given as:

$$\aleph \left(\varphi \left(\eta \right)\right)=L(r,w)={\int }_{-\infty }^{\infty }{e}^{-r\eta }\varphi (w\eta )d\eta ,r,w>0,$$

(2.1)

where $\aleph$ is the N-transformation of $\varphi \left(\eta \right)$ and $r,w$ are the N-transformation parameters. Note equation (2.1) can be written as,

$$\aleph \left(\varphi \left(\eta \right)\right)={\aleph }^{+}\left(\varphi \left(\zeta \right)\right)+{\aleph }^{-}\left(\varphi \left(\eta \right)\right)=L^{+}(r,w)+L^{-}(r,w),$$

where,

$${\aleph }^{+}\left(\varphi \left(\eta \right)\right)=L^{+}(r,w)={\int }_0^{\infty }{e}^{-r\zeta }\varphi (w\eta )d\eta ,r,w\in (0,\infty ).$$

(2.2)

Moreover,

$${\aleph }^{-1}\left[L(r,w)\right]=\varphi \left(\eta \right)=\frac{1}{2\pi i}{\int }_{c-i\infty }^{c+i\infty }{e}^{\frac{r\eta }{w}}L(r,w)\mathrm{dr}.$$

(2.3)

Thus, equation (2.2) is the natural transformation and equation (2.3) is the inverse natural transformation of $L(r,w)$ .

Properties of interest:

The following are a few of the fundamental properties of N-transformation, see Rawashdeh ¹⁶ :

1. ${\aleph }^{+}\left[C\right]=\frac{C}{r}$ .

2. ${\aleph }^{+}\left[{\eta }^m\right]=\frac{m!w^m}{r^{m+1}}$ , where $m\ge 0.$

3. ${\aleph }^{+}\left[e^{c\eta }\right]=\frac{1}{(r-\mathrm{cw})}$ .

4. If ${\varphi }^{\left(i\right)}\left(\eta \right)$ is the ${\mathrm{i}}^{\mathrm{th}}$ derivative of the function $\varphi \left(\eta \right)$ , then its N-transformation given by:

$${\aleph }^{+}\left[{\varphi }^{\left(i\right)}\left(\eta \right)\right]=L_i(r,w)=\frac{r^i}{w^i}L(r,w)-\sum _{j=0}^{i-1}\frac{r^{i-(j-1)}}{w^{i-j}}{\varphi }^{\left(j\right)}\left(0\right).$$

Adomian polynomials evaluations

In this section, we provide methodical algebraic computations of the Adomian polynomials. For many years, Taylor series based on Adomian polynomials were utilized to readily break down complex nonlinear expressions into smaller, more integrable terms. The following decomposition series can be used to represent the unknown linear function $w$ during the process:

$$w=\sum _{k=0}^{\infty }{w}_k.$$

(2.4)

Here the terms $w_k,k\ge 0$ can be evaluated as a recursive formula. For the nonlinear components, such as; $w^2,w^3,\cos \left(w\right),e^w,w{w}_x,w_x^2$ etc. $N\left(w\right)$ can easily be presented as the so-called Adomian polynomials $C_k$ , which are:

$$N\left(w\right)=\sum _{k=0}^{\infty }{C}_k(w_0,w_1,....,w_k),$$

(2.5)

and the $C_k$ of the nonlinear part $N\left(w\right)$ can be easily evaluated as:

$$C_k=\frac{1}{k!}\frac{d^k}{d{\sigma }^k}{\left[N\left(\sum _{i=0}^k{\sigma }^i{w}_i\right)\right]}_{\sigma =0},k=0,1,2,....$$

(2.6)

Now, the general formula in equation (2.6) can easily be simplified as follows:

Let $N\left(w\right)$ be the nonlinear function. Then, by using equation (2.6), we obtain the following result by the definition of Adomian polynomials:

$$\begin{array}{c}{C}_0=N\left(w_0\right)\hfill \\ C_1=w_{1}N\prime \left(w_0\right)\hfill \\ C_2=w_{2}N\prime \left(w_0\right)+\frac{1}{2!}{w}_1^{2}N″\left(w_0\right)\hfill \\ C_3=w_{3}N\prime \left(w_0\right)+w_1{w}_{2}N″\left(w_0\right)+\frac{1}{3!}{w}_1^{3}N″\prime \left(w_0\right)\hfill \\ C_4=w_{4}N\prime \left(w_0\right)+(\frac{1}{2!}{w}_2^2+w_1{w}_3)N″\left(w_0\right)\hfill \\ +\frac{1}{2!}{w}_1^2{w}_{2}N″\prime \left(w_0\right)+\frac{1}{4!}{w}_1^4{N}^{(4)}\left(w_0\right).\hfill \end{array}$$

(2.7)

The rest of these components can be easily accomplished in the same way. The above polynomials in equation (2.7) give two important observations. First, $C_0$ depends only on $w_0$ , $C_1$ depend only on $w_0$ and $w_1$ , $C_2$ depend only on $w_0$ , $w_1$ , and $w_2$ , etc.

Second, by substituting equation (2.7) into equation (2.5), we get:

$$\begin{array}{c}N\left(w\right)=C_0+C_1+C_2+C_3+...\hfill \\ =N\left(w_0\right)+(w_1+w_2+w_3+...)N\prime \left(w_0\right)\hfill \\ +\frac{1}{2!}(w_1^2+2w_1{w}_2+2w_1{w}_3+w_2^2+...)N″\left(w_0\right)\hfill \\ +\frac{1}{3!}(w_1^3+3w_1^2{w}_2+3w_1^2{w}_3+6w_1{w}_2{w}_3+...)N″\left(w_0\right)+...\hfill \\ =N\left(w_0\right)+(w-w_0)N\prime \left(w_0\right)+\frac{1}{2!}{(w-w_0)}^{2}N″\left(w_0\right)+...\hfill \end{array}$$

One may notice it because the most recent expansion provides the Taylor series as a function rather than a traditional point. The Adomian polynomials provided in equation (2.6) thus unequivocally demonstrate that for any term of the sum of the subscripts of the components. Note that there are several algorithms available for computing the Adomian polynomials of the nonlinear terms.

Natural transform theories derivation

Throughout this section, we will look at some fresh, detailed proofs for various natural transformation-related theorems. In addition, we shall use these theorems to find exact solutions to four systems of ODEs and PDEs, namely, the Wu-Zhang nonlinear systems of linear and nonlinear ordinary and partial differential equations under proper initial conditions.

• Theorem 3.1 Let $\varphi \left(\eta \right)=\cosh \left(\alpha \eta \right)$ . Then its N-transformation is:

$${\aleph }^{+}[\cosh (\alpha \eta )]=\frac{r^2}{(r^2-w^2{\alpha }^2)}.$$

(3.1)

Proof: Using integration by parts twice, we arrive at:

$$\begin{array}{l}{\aleph }^{+}\left[\cosh (\alpha \eta )\right]=\frac{1}{w}{\int }_0^{\infty }{e}^{\frac{-r\eta }{w}}\cosh (\alpha \eta )d\eta \\ =\frac{1}{w}\left(\left[\frac{1}{\alpha }\sinh (\alpha \eta )e^{\frac{-r\eta }{w}}{|}_0^{\infty }\right]+\frac{r}{w\alpha }{\int }_0^{\infty }{e}^{\frac{-r\eta }{w}}\sinh (\alpha \eta )d\eta \right)\\ =+\frac{r}{w^2\alpha }{\int }_0^{\infty }{e}^{\frac{-r\eta }{w}}\sinh (\alpha \eta )d\eta )\\ =\frac{r}{w^2\alpha }\left({\left[\frac{1}{\alpha }\cosh (\alpha \eta )e^{\frac{-r\eta }{w}}\right]}_0^{\infty }+\frac{r}{w\alpha }{\int }_0^{\infty }{e}^{\frac{-r\eta }{w}}\cosh (\alpha \eta )d\eta \right)\\ =\frac{-r}{w^2{\alpha }^2}+\frac{r^2}{w^3{\alpha }^2}{\int }_0^{\infty }{e}^{\frac{-r\eta }{w}}\cosh (\alpha \eta )d\eta \\ =\frac{-r}{w^2{\alpha }^2}+\frac{r^2}{w^2{\alpha }^2}{\aleph }^{+}\left[\cosh (\alpha \eta )\right].\end{array}$$

Which implies:

$$(1-\frac{r^2}{w^2{\alpha }^2}){\aleph }^{+}[\cosh (\alpha \eta )]=\frac{-r}{w^2{\alpha }^2}$$

Therefore,

$${\aleph }^{+}[\cosh (\alpha \eta )]=\frac{r}{(r^2-w^2{\alpha }^2)}.$$

Hence, the proof of Theorem 3.1 is now complete.

• Theorem 3.2 Let $\varphi \left(\eta \right)=\eta \cos \left(\alpha \eta \right)$ . Then its N-transformation is given by:

$${\aleph }^{+}[\eta \cos (\alpha \eta )]=\frac{w(r^2-w^2{\alpha }^2)}{{(r^2+w^2{\alpha }^2)}^2}.$$

Proof: First, we show that:

$${\aleph }^{+}[\cos (\alpha \eta )]=\frac{r}{r^2+w^2{\alpha }^2}.$$

(3.2)

To do this, use the N-transform definition and then integrate by parts twice to get:

$$\begin{array}{c}{\aleph }^{+}[\cos (\alpha \eta )]=\frac{1}{w}{\int }_0^{\infty }{e}^{\frac{-r\eta }{w}}\cos (\alpha \eta )d\eta \hfill \\ =\frac{1}{w}([\frac{1}{\alpha }\sin (\alpha \eta )e^{\frac{-r\eta }{w}}{|}_0^{\infty }]+\frac{r}{w\alpha }{\int }_0^{\infty }{e}^{\frac{-r\eta }{w}}\sin (\alpha \eta )d\eta )\hfill \\ =\frac{r}{w^2\alpha }{\int }_0^{\infty }{e}^{\frac{-r\eta }{w}}\sin (\alpha \eta )d\eta \hfill \\ =\frac{r}{w^2\alpha }([\frac{-1}{\alpha }\cos (\alpha \eta )e^{\frac{-r\eta }{w}}{|}_0^{\infty }]+\frac{r}{w\alpha }{\int }_0^{\infty }{e}^{\frac{-r\eta }{w}}\cos (\alpha \eta )d\eta )\hfill \\ =\frac{r}{w^2\alpha }(\frac{1}{\alpha }+\frac{r}{\alpha w}{\aleph }^{+}[\cos (\alpha \eta )]).\hfill \end{array}$$

(3.3)

Which is equivalent to:

$$(1+\frac{r^2}{w^2{\alpha }^2}){\aleph }^{+}[\cos (\alpha \eta )]=\frac{-r}{w^2{\alpha }^2}.$$

Thus,

$${\aleph }^{+}[\cos (\alpha \eta )]=\frac{r}{r^2+w^2{\alpha }^2}.$$

Now, differentiate equation (3.3) we arrive at:

$$\frac{d}{\mathrm{dr}}{\aleph }^{+}[\cos (\alpha \eta )]=\frac{d}{\mathrm{dr}}(\frac{1}{w}{\int }_0^{\infty }{e}^{\frac{-r\eta }{w}}\cos (\alpha \eta )d\eta ).$$

Using equation (3.2), and Leibniz rule, we have:

$$\begin{array}{c}\frac{d}{\mathrm{dr}}\frac{r}{(r^2+w^2{\alpha }^2)}=\frac{1}{w}{\int }_0^{\infty }\frac{d}{\mathrm{dr}}(e^{\frac{-r\eta }{w}}\cos (\alpha \eta ))d\eta \hfill \\ \frac{(r^2+w^2{\alpha }^2)-2r^2}{{(r^2-w^2{\alpha }^2)}^2}=\frac{1}{w}{\int }_0^{\infty }\frac{-\eta }{w}{e}^{\frac{-r\eta }{w}}\cos (\alpha \eta )d\eta .\hfill \end{array}$$

which is equivalent to:

$$\frac{-(r^2+w^2{\alpha }^2)}{{(r^2-w^2{\alpha }^2)}^2}=\frac{-1}{w^2}{\int }_0^{\infty }\eta e^{\frac{-r\eta }{w}}\cos (\alpha \eta )d\eta .$$

(3.4)

Multiplying equation (3.4) by $w$ , we have:

$$\frac{w(r^2-w^2{\alpha }^2)}{{(r^2+w^2{\alpha }^2)}^2}=\frac{1}{w}{\int }_0^{\infty }\eta e^{\frac{-r\eta }{w}}\cos (\alpha \eta )d\eta .$$

(3.5)

However, from the definition of N-transformation, equation (3.5) becomes:

$${\aleph }^{+}[\eta \cos (\alpha \eta )]=\frac{w(r^2-w^2{\alpha }^2)}{{(r^2+w^2{\alpha }^2)}^2}.$$

Hence, the proof of Theorem 3.2 is now complete.

• Theorem 3.3 Let $\varphi \left(\eta \right)=\frac{1}{2{\alpha }^3}(\sinh (\alpha \eta )-\sin (\alpha \eta ))$ . Then its N-transformation is:

$${\aleph }^{+}\left[\frac{1}{2{\alpha }^3}(\sinh (\alpha \eta )-\sin (\alpha \eta ))\right]=\frac{w^3}{(r^4-w^4{\alpha }^4)}.$$

Proof: Implementing the linear property of N-transformation, we have:

$$\begin{array}{c}{\aleph }^{+}\left[\frac{1}{2{\alpha }^3}(\sinh (\alpha \eta )-\sin (\alpha \eta ))\right]=\frac{1}{2{\alpha }^3}\hfill \\ ({\aleph }^{+}[\sinh (\alpha \eta )]-{\aleph }^{+}[\sin (\alpha \eta )]).\hfill \end{array}$$

First, we show that:

$${\aleph }^{+}[\sin (\alpha \eta )]=\frac{w\alpha }{r^2+w^2{\alpha }^2}.$$

and,

$${\aleph }^{+}[\sinh (\alpha \eta )]=\frac{w\alpha }{r^2-w^2{\alpha }^2}.$$

Thus,

$$\begin{array}{c}{\aleph }^{+}\left[\frac{1}{2{\alpha }^3}(\sinh (\alpha \eta )-\sin (\alpha \eta ))\right]=\frac{1}{2{\alpha }^3}{\aleph }^{+}[\sinh (\alpha \eta )]\hfill \\ -{\aleph }^{+}[\sin (\alpha \eta )]=\frac{1}{2{\alpha }^3}(\frac{w\alpha }{r^2-w^2{\alpha }^2}-\frac{w\alpha }{r^2+w^2{\alpha }^2})\hfill \\ =\frac{w^3}{(r^4-w^4{\alpha }^4)}.\hfill \end{array}$$

Therefore,

$$\begin{array}{c}{\aleph }^{+}\left[\frac{1}{2{\alpha }^3}(\sinh (\alpha \eta )-\sin (\alpha \eta ))\right]=\frac{1}{2{\alpha }^3}\hfill \\ ({\aleph }^{+}[\sinh (\alpha \eta )]-{\aleph }^{+}[\sin (\alpha \eta )]).\hfill \end{array}$$

Hence, the proof of Theorem 3.3 is now complete.

• Theorem 3.4 Let $\varphi \left(\eta \right)=\frac{1}{2{\alpha }^2}(\cosh (\alpha \eta )-\cos (\alpha \eta ))$ . Then its N-transformation is:

$${\aleph }^{+}\left[\frac{1}{2{\alpha }^2}(\cosh (\alpha \eta )-\cos (\alpha \eta ))\right]=\frac{r{w}^2}{(r^4-w^4{\alpha }^4)}.$$

Proof: Employ the linear property of the N-transformation and theorem (3.1), we have:

$$\begin{array}{c}\multicolumn{1}{c}{{\aleph }^{+}\left[\varphi \left(\eta \right)\right]=\frac{1}{2{\alpha }^2}({\aleph }^{+}[\cosh (\alpha \eta )]-{\aleph }^{+}[\cos (\alpha \eta )])}\\ =\frac{1}{2{\alpha }^2}(\frac{r}{r^2-w^2{\alpha }^2}-\frac{r}{r^2+w^2{\alpha }^2})\hfill \\ =\frac{r{w}^2}{(r^4-w^4{\alpha }^4)}.\hfill \end{array}$$

Hence, the proof of Theorem 3.4 is now complete.

• Theorem 3.5 Let $\varphi \left(\eta \right)=(\cos (\alpha \eta )\cosh (\alpha \eta ))$ . Then its N-transformation is:

$${\aleph }^{+}\left[(\cosh (\alpha \eta )\cos (\alpha \eta ))\right]=\frac{r^3}{(r^4+4w^4{\alpha }^4)}.$$

Proof: Employ the N-transformation along with the property of $\cosh (\alpha \eta )$ , we can write:

$$\begin{array}{c}{\aleph }^{+}[\cos (\alpha \eta )\cosh (\alpha \eta )]=\frac{1}{w}{\int }_0^{\infty }{e}^{\frac{-r\eta }{w}}\cosh (\alpha \eta )\cos (\alpha \eta )d\eta \hfill \\ =\frac{1}{2w}{\int }_0^{\infty }{e}^{\frac{-r\eta }{w}}\cos (\alpha t)(e^{\alpha \eta }+e^{-\alpha \eta })d\eta \hfill \\ =\frac{1}{2w}({\int }_0^{\infty }{e}^{\frac{-(r-\alpha w)\eta }{w}}\cos (\alpha \eta )d\eta +{\int }_0^{\infty }{e}^{\frac{-(r+\alpha w)\eta }{w}}\cos (\alpha \eta )d\eta ).\hfill \end{array}$$

Let $r_1=r-\alpha w$ and $r_2=r+\alpha w.$

Then,

$$\begin{array}{c}{\aleph }^{+}[\cos (\alpha \eta )\cosh (\alpha \eta )]=\frac{1}{2w}\hfill \\ ({\int }_0^{\infty }{e}^{\frac{-r_1\eta }{w}}\cos (\alpha \eta )d\eta +{\int }_0^{\infty }{e}^{\frac{r_2\eta }{w}}\cos (\alpha \eta )d\eta )\hfill \\ =\frac{1}{2}(J(r_1,w)+J(r_2,w))\hfill \\ =\frac{1}{2}(\frac{r_1}{{r_1}^2-w^2{\alpha }^2}+\frac{r_2}{{r_2}^2+w^2{\alpha }^2})\hfill \\ =\frac{1}{2}(\frac{r-\alpha w}{{(r-\alpha w)}^2-w^2{\alpha }^2}+\frac{r+\alpha w}{{(r+\alpha w)}^2+w^2{\alpha }^2})\hfill \\ =\frac{r^3}{(r^4+4w^4{\alpha }^4)}.\hfill \end{array}$$

Therefore,

$${\aleph }^{+}\left[(\cosh (\alpha \eta )\cos (\alpha \eta ))\right]=\frac{r^3}{(r^4+4w^4{\alpha }^4)}.$$

Hence, the proof of Theorem 3.5 is now complete.

• Theorem 3.6 Let $\varphi \left(\eta \right)=\frac{1}{2\alpha }({\eta }^2\sin (\alpha \eta ))$ . Then its N-transformation is:

$${\aleph }^{+}\left[\frac{1}{2\alpha }({\eta }^2\sin (\alpha \eta ))\right]=\frac{w^3(3r^2-w^2{\alpha }^2)}{{(r^2+w^2{\alpha }^2)}^3}.$$

Proof: Employ the N-transformation to arrive at:

$${\aleph }^{+}\left[(\frac{1}{2\alpha }{\eta }^2\sin (\alpha \eta ))\right]=\frac{1}{2w\alpha }{\int }_0^{\infty }{e}^{\frac{-r\eta }{w}}{\eta }^2\sin (\alpha \eta )d\eta$$

We first notice that:

$${\aleph }^{+}[\sinh (\alpha \eta )]=\frac{1}{w}{\int }_0^{\infty }{e}^{\frac{-r\eta }{w}}\sin (\alpha \eta )d\eta$$

(3.6)

But using the N-transformation of $\sinh (\zeta )$ above, we can rewrite equation (3.6) as:

$$\frac{w\alpha }{r^2+w^2{\alpha }^2}=\frac{1}{w}{\int }_0^{\infty }{e}^{\frac{-r\eta }{w}}\sin (\alpha \eta )d\eta .$$

(3.7)

Differentiating equation (3.7) twice, we arrive at:

$$\begin{array}{c}\frac{-2w{r}^2\alpha {(r^2+w^2{\alpha }^2)}^2+4\mathrm{wr}\alpha (r^2+w^2{\alpha }^2)2r}{{(r^2+w^2{\alpha }^2)}^4}\hfill \\ =\frac{1}{w}{\int }_0^{\infty }\frac{{\eta }^2}{w^2}{e}^{\frac{-r\eta }{w}}\sin (\alpha \eta )d\eta \hfill \\ =\frac{1}{w^2}{\int }_0^{\infty }{e}^{\frac{-r\eta }{w}}{\eta }^2\sin (\alpha \eta )d\eta \hfill \\ =\frac{1}{w^2}{\aleph }^{+}[{\eta }^2\sin (\alpha \eta )].\hfill \end{array}$$

(3.8)

Multiplying by $w^2$ and dividing by $2\alpha$ , then equation (3.8) becomes:

$${\aleph }^{+}[\frac{1}{2\alpha }{\eta }^2\sin (\alpha \eta )]=\frac{w^3(3r^3\alpha -w^2{\alpha }^2)}{{(r^2+w^2{\alpha }^2)}^3}.$$

Hence, the proof of Theorem 3.6 is now complete.

Systems of nonlinear ordinary differential equations

This section describes the natural decomposition method (NDM) algorithm for general linear and nonlinear nonhomogeneous systems of ODEs.

Describing the methodology of the NDM

Consider the general linear and nonlinear non-homogeneous system of ODEs shown below:

$$\begin{array}{c}{L}_1\left(v\right)+J_1\left(v\right)+N_1\left(v\right)=g_1\left(\eta \right)\hfill \\ L_2\left(u\right)+J_2\left(u\right)+N_2\left(u\right)=g_2\left(\eta \right).\hfill \end{array}$$

(4.1)

Along with I.C:

$$v\left(0\right)=h_1\left(\eta \right),u\left(0\right)=h_2\left(\eta \right).$$

(4.2)

Where $L_1\left(v\right)$ and $L_2\left(u\right)$ are an operators of the highest derivatives, $J_1\left(v\right)$ and $J_2\left(u\right)$ are remainders of differential operators $g_1\left(\eta \right)$ and $g_1\left(\eta \right),$ are the non-homogeneous term and $N_1\left(v\right)$ and $N_2\left(u\right)$ are the nonlinear terms. Suppose $L_1\left(v\right)$ and $L_2\left(u\right)$ are a differential operators of the first order, then by taking the N-transformation of equation (4.1), we have:

$$\begin{array}{c}\frac{rV(r,w)}{w}-\frac{v\left(0\right)}{w}+{\aleph }^{+}[J_1\left(v\right)+N_1\left(v\right)]={\aleph }^{+}\left[g_1\left(\eta \right)\right]\hfill \\ \frac{rU(r,w)}{w}-\frac{u\left(0\right)}{w}+{\aleph }^{+}[J_2\left(u\right)+N_2\left(u\right)]={\aleph }^{+}\left[g_2\left(\eta \right)\right].\hfill \end{array}$$

(4.3)

Combine both equations (4.2) and (4.3) to arrive at:

$$\begin{array}{c}V(r,w)=\frac{v\left(0\right)}{w}+\frac{w}{r}{\aleph }^{+}\left[g_1\left(\eta \right)\right]-\frac{w}{r}{\aleph }^{+}[J_1\left(v\right)+N_1\left(v\right)]\hfill \\ U(r,w)=\frac{u\left(0\right)}{w}+\frac{w}{r}{\aleph }^{+}\left[g_2\left(\eta \right)\right]-\frac{w}{r}{\aleph }^{+}[J_2\left(u\right)+N_2\left(u\right)].\hfill \end{array}$$

(4.4)

Employing the inverse of the N-transformation for equation (4.4), one concludes:

$$\begin{array}{c}v\left(\eta \right)=G_1\left(\eta \right)-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[J_1\left(v\right)+N_1\left(v\right)]\right]\hfill \\ u\left(\eta \right)=G_2\left(\eta \right)-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[J_2\left(u\right)+N_2\left(u\right)]\right].\hfill \end{array}$$

(4.5)

Where $G_1\left(\eta \right)$ and $G_2\left(\eta \right)$ are source terms.

Suppose we have an infinite series of solutions of the form:

$$v\left(\eta \right)=\sum _{j=0}^{\infty }{v}_j\left(\eta \right),u\left(\eta \right)=\sum _{j=0}^{\infty }{u}_j\left(\eta \right).$$

(4.6)

From equation (4.6), one can write equation (4.5) as:

$$\begin{array}{c}\sum _{n=0}^{\infty }{v}_j\left(\eta \right)=G_1\left(\eta \right)-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\sum _{j=0}^{\infty }{v}_j\left(\eta \right)+\sum _{j=0}^{\infty }{C}_j]\right]\hfill \\ \sum _{n=0}^{\infty }{u}_j\left(\eta \right)=G_2\left(\eta \right)-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\sum _{j=0}^{\infty }{u}_j\left(\eta \right)+\sum _{j=0}^{\infty }{D}_j]\right],\hfill \end{array}$$

(4.7)

where $C_j$ and $D_j$ are the nonlinear terms related to the Adomian polynomials.

Looking at equation (4.7), one can easily build a recursive formula as follows:

$$\begin{array}{c}{v}_0\left(\eta \right)=G_1\left(\eta \right)\hfill \\ u_0\left(\eta \right)=G_2\left(\eta \right)\hfill \\ v_1\left(\eta \right)=-\left[{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[v_0\left(\eta \right)+C_0]\right]\right]\hfill \\ v_2\left(\eta \right)=-\left[{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[v_1\left(\eta \right)+C_1]\right]\right]\hfill \\ v_3\left(\eta \right)=-\left[{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[v_2\left(\eta \right)+C_2\left(\eta \right)]\right]\right],\hfill \end{array}$$

and,

$$\begin{array}{c}{u}_1\left(\eta \right)=-\left[{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\left(u_0\left(\eta \right)\right)+D_0]\right]\right]\hfill \\ u_2\left(\eta \right)=-\left[{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\left(u_1\left(\eta \right)\right)+D_1]\right]\right]\hfill \\ u_3\left(\eta \right)=-\left[{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\left(u_2\left(\eta \right)\right)+D_2]\right]\right].\hfill \end{array}$$

On the long run, one can arrive at a general formula as:

$$\begin{array}{c}{v}_{j+1}\left(\eta \right)=-\left[{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[v_j\left(\eta \right)+C_j]\right]\right],j\ge 0\hfill \\ u_{j+1}\left(\eta \right)=-\left[{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[u_j\left(\eta \right)+D_j]\right]\right],j\ge 0.\hfill \end{array}$$

(4.8)

Finally, the intended solution can be written as:

$$v\left(\eta \right)=\sum _{j=0}^{\infty }{v}_j\left(\eta \right),u\left(\eta \right)=\sum _{j=0}^{\infty }{u}_j\left(\eta \right).$$

(4.9)

Worked examples

• Example 4.1 Given the first-order ordinary differential equations of the following nonlinear system:

$$\begin{array}{c}\frac{d\phi }{d\xi }=\frac{1}{2}\phi (\xi )\hfill \\ \frac{d\psi }{d\xi }=\psi (\xi )+{\phi }^2(\xi ).\hfill \end{array}$$

(4.10)

With I.C:

$$\phi (0)=1,\psi (0)=0.$$

(4.11)

Solution. Employ N-transformation to equation (4.10), to arrive at:

$$\begin{array}{c}\frac{r\Phi (r,w)}{w}-\frac{\phi (0)}{w}={\aleph }^{+}[\frac{1}{2}\phi (\xi )]\hfill \\ \frac{r\mathrm{\Psi }(r,w)}{w}-\frac{\psi (0)}{w}={\aleph }^{+}[\psi \left(\xi \right)+{\phi }^2\left(\xi \right)]\hfill \\ \frac{r\Phi (r,w)}{w}-\frac{1}{w}={\aleph }^{+}[\frac{1}{2}\phi (\xi )]\hfill \\ \frac{r\mathrm{\Psi }(r,w)}{w}-\frac{0}{w}={\aleph }^{+}[\psi \left(\xi \right)+{\phi }^2\left(\xi \right)].\hfill \end{array}$$

(4.12)

Substituting equation (4.11) into equation (12), to obtain:

$$\begin{array}{c}\mathrm{\Phi }(r,w)=\frac{1}{r}+\frac{w}{r}{\aleph }^{+}[\frac{1}{2}\phi (\xi )]\hfill \\ \mathrm{\Psi }(r,w)=\frac{w}{r}{\aleph }^{+}[\psi \left(\xi \right)+{\phi }^2\left(\xi \right)].\hfill \end{array}$$

(4.13)

Suppose our intended solutions are given in the form:

$$\phi (\xi )=\sum _{j=0}^{\infty }{\phi }_j(\xi ),\psi (\xi )=\sum _{j=0}^{\infty }{\psi }_j(\xi ).$$

(4.14)

Employ the inverse N-transformation of equation (4.13), one can arrive at:

$$\begin{array}{c}\sum _{j=0}^{\infty }{\phi }_j(\xi )=1+{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\frac{1}{2}\phi (\xi )]\right],j\ge 0.\hfill \\ \sum _{j=0}^{\infty }{\psi }_j(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\psi }_j(\xi )+\sum _{j=0}^{\infty }{C}_j]\right],j\ge 0.\hfill \end{array}$$

(4.15)

Where $C_j$ describes the Adomian polynomial for ${\phi }^2(\xi )$ .

Analyzing equation (4.15), one can generate the recursive relation as follows:

$$\begin{array}{c}{\phi }_0(\xi )=1\hfill \\ {\psi }_0(\xi )=0\hfill \\ {\varphi }_1(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\frac{1}{2}\phi (0)]\right]\hfill \\ {\psi }_1(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\psi }_0(\xi )+C_0]\right]\hfill \\ {\phi }_2(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\frac{1}{2}{\phi }_1(\xi )]\right]\hfill \\ {\psi }_2(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\psi }_1(\xi )+C_1]\right].\hfill \end{array}$$

(4.16)

Then, the general formula is given by:

$$\begin{array}{c}{\phi }_{j+1}(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\frac{1}{2}{\phi }_j(\xi )]\right]\hfill \\ {\psi }_{j+1}(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\psi }_j(\xi )+C_j]\right].\hfill \end{array}$$

(4.17)

From equation (4.17), one can evaluate the rest of the terms as:

$$\begin{array}{c}\multicolumn{1}{c}{{\phi }_1(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\frac{1}{2}{\phi }_0(\xi )]\right]}\\ ={\aleph }^{-1}\left[\frac{w}{2r^2}\right]\hfill \\ =\frac{1}{2}\xi .\hfill \\ \multicolumn{1}{c}{{\psi }_1(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\psi }_0(\xi )+C_0]\right]}\\ \multicolumn{1}{c}{C_0(\xi )={{\varphi }_0}^2\left(\xi \right)=1}\\ \multicolumn{1}{c}{{\psi }_1(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[1\right]\right]}\\ \multicolumn{1}{c}{{\psi }_1(\xi )=\xi }\\ \multicolumn{1}{c}{{\psi }_2(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}{\phi }_1\right]}\\ ={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[\frac{1}{4}\xi \right]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{w^2}{4r^3}\right]\hfill \\ =\frac{1}{4}\frac{{\xi }^2}{2!}.\hfill \\ \multicolumn{1}{c}{{\psi }_2(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\psi }_1(\xi )+C_1]\right]}\\ \multicolumn{1}{c}{C_1(\xi )=2{\phi }_0(\xi ){\phi }_1(\xi )}\\ =\xi .\hfill \\ \multicolumn{1}{c}{{\psi }_2(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\xi +\xi ]\right]}\\ ={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[2\xi \right]\right]\hfill \\ =\frac{2{\xi }^2}{2!}.\hfill \\ \multicolumn{1}{c}{{\phi }_3(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\frac{1}{2}{\phi }_2(\xi )]\right]}\\ ={\aleph }^{-1}\left[r{\aleph }^{+}\left[\frac{1}{2}\frac{1}{8}{\xi }^2\right]\right]\hfill \\ =\frac{1}{8}\frac{{\xi }^3}{3!}.\hfill \end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{{\psi }_3(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\psi }_2(\xi )+C_2]\right]}\\ \multicolumn{1}{c}{C_2(\xi )=2{\phi }_0(\xi ){\phi }_2(\xi )+\phi {\left(1\right)}^2}\\ =2(1)\left(\frac{1}{8}{\xi }^2\right)+\left(\frac{1}{4}{\xi }^2\right)\hfill \\ =\frac{1}{2}{\xi }^2.\hfill \\ \multicolumn{1}{c}{{\phi }_3(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\xi }^2+\frac{1}{2}{\xi }^2]\right]}\\ ={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[\frac{3}{2}{\xi }^2\right]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{w}{r}\frac{3w^2}{r^3}\right]\hfill \\ =\frac{3{\xi }^3}{3!}.\hfill \\ \multicolumn{1}{c}{{\phi }_4(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\frac{1}{2}{\phi }_3(\xi )]\right]}\\ ={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[\frac{1}{2}\frac{1}{8}\frac{1}{6}{\xi }^3\right]\right]\hfill \\ =\frac{1}{16}\frac{1}{4!}{\xi }^4.\hfill \\ \multicolumn{1}{c}{{\psi }_4(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\psi }_3(\xi )+C_3]\right].}\end{array}$$

Note that,

$$\begin{array}{c}\multicolumn{1}{c}{C_3(\xi )=2{\phi }_0(\xi ){\phi }_3(\xi )+2{\phi }_1(\xi ){\phi }_2(\xi )}\\ =2\left(\frac{1}{8}\frac{1}{6}{\xi }^3\right)+\left(2\frac{1}{2}\xi \right)\left(\frac{1}{8}{\xi }^2\right)\hfill \\ =\frac{1}{6}{\xi }^3.\hfill \end{array}$$

Then,

$$\begin{array}{c}\multicolumn{1}{c}{{\psi }_4(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\frac{1}{2}{\xi }^3+\frac{1}{6}{\xi }^3]\right]}\\ ={\aleph }^{-1}\left[\frac{w}{r}\frac{4}{6}\frac{3!w^3}{r^4}\right]\hfill \\ ={\aleph }^{-1}\left[\frac{4w^4}{r^5}\right]\hfill \\ =\frac{4}{4!}{\xi }^4.\hfill \end{array}$$

We proceed to arrive at the exact solution:

$$\begin{array}{c}\multicolumn{1}{c}{\sum _{j=0}^{\infty }{\phi }_j(\xi )={\phi }_0(\xi )+{\phi }_1(\xi )+{\phi }_2(\xi )+...}\\ \multicolumn{1}{c}{=1+\frac{1}{2}\xi +\frac{1}{4}\frac{{\xi }^2}{2!}+\frac{1}{8}\frac{{\xi }^3}{3!}+\frac{1}{16}\frac{{\xi }^4}{4!}+...}\\ \multicolumn{1}{c}{\sum _{j=0}^{\infty }{\psi }_j(\xi )={\psi }_0(\xi )+{\psi }_1(\xi )+{\psi }_2(\xi )+...}\\ \multicolumn{1}{c}{=0+\xi +\frac{2{\xi }^2}{2!}+\frac{3{\xi }^3}{3!}+\frac{4{\xi }^4}{4!}+...}\end{array}$$

Hence, the intended solutions are given by:

$$\begin{array}{c}\varphi (\xi )=e^{\xi /2}\hfill \\ \psi (\xi )=\xi e^{\xi }.\hfill \end{array}$$

The precise answer is in perfect agreement with the outcome produced by (ADM).

• Example 4.2 Given a first-order linear system, ordinary differential equations of the form:

$$\begin{array}{c}\frac{d\varphi }{d\xi }=\phi (\xi )+\psi (\xi )\hfill \\ \frac{d\phi }{d\xi }=\varphi (\xi )+\psi (\xi )\hfill \\ \frac{d\psi }{d\xi }=\varphi (\xi )+\phi (\xi ).\hfill \end{array}$$

(4.18)

Along with the following conditions:

$$\varphi (0)=-1,\phi (0)=0,\psi (0)=1.$$

(4.19)

Solution. Employing the N-transformation to equation (4.18), we arrive at:

$$\begin{array}{c}\frac{r\Phi (r,w)}{w}-\frac{\varphi \left(0\right)}{w}={\aleph }^{+}[\phi (\xi )+\psi (\xi )]\hfill \\ \frac{r\mathrm{\Upsilon }(r,w)}{w}-\frac{\phi \left(0\right)}{w}={\aleph }^{+}[\varphi (\xi )+\psi (\xi )]\hfill \\ \frac{r\mathrm{\Upsilon }(r,w)}{w}-\frac{\phi \left(0\right)}{w}={\aleph }^{+}[\varphi (\xi )+\psi (\xi )]\hfill \end{array}$$

Substituting equation (4.19), we obtain:

$$\begin{array}{c}\Phi (r,w)=\frac{-1}{r}+\frac{w}{r}{\aleph }^{+}[\phi (\xi )+\psi (\xi )]\hfill \\ \mathrm{\Upsilon }(r,w)=\frac{w}{r}{\aleph }^{+}[\varphi (\xi )+\psi (\xi )]\hfill \\ \mathrm{\Psi }(r,w)=\frac{1}{r}+\frac{w}{r}{\aleph }^{+}[\varphi (\xi )+\phi (\xi )].\hfill \end{array}$$

(4.20)

Suppose our intended solutions are:

$$\varphi (\xi )=\sum _{j=0}^{\infty }{\varphi }_j(\xi ),\phi (\xi )=\sum _{j=0}^{\infty }{\phi }_j(\xi ),\psi (\xi )=\sum _{j=0}^{\infty }{\psi }_j(\xi ).$$

(4.21)

Taking the inverse N-transformation for equation (4.21), we arrive at:

$$\begin{array}{c}\sum _{j=0}^{\infty }{\varphi }_j(\xi )=-1+{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\phi (\xi )+\psi (\xi )]\right]\hfill \\ \sum _{j=0}^{\infty }{\phi }_j(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\varphi (\xi )+\psi (\xi )]\right]\hfill \\ \sum _{j=0}^{\infty }{\psi }_j(\xi )=1+{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\varphi (\xi )+\phi (\xi )]\right].\hfill \end{array}$$

(4.22)

Examining equation (4.22), one can come up with a formula like this:

$$\begin{array}{c}{\varphi }_0(\xi )=-1\hfill \\ {\phi }_0(\xi )=0\hfill \\ {\psi }_0(\xi )=1\hfill \\ {\varphi }_1\left(\xi \right)={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\phi }_0(\xi )+{\psi }_0(\xi )]\right]\hfill \\ {\phi }_1\left(\xi \right)={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\varphi }_0(\xi )+{\psi }_0(\xi )]\right]\hfill \\ {\psi }_1\left(\xi \right)={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\varphi }_0(\xi )+{\phi }_0(\xi )]\right].\hfill \end{array}$$

and,

$$\begin{array}{c}{\varphi }_2\left(\xi \right)={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\phi }_1(\xi )+{\psi }_1(\xi )]\right]\hfill \\ {\phi }_2\left(\xi \right)={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\varphi }_1(\xi )+{\psi }_1(\xi )]\right]\hfill \\ {\psi }_2\left(\xi \right)={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\varphi }_1(\xi )+{\phi }_1(\xi )]\right].\hfill \end{array}$$

(4.23)

Thus, the general formulas can be written as follows:

$$\begin{array}{c}{\varphi }_{j+1}(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\phi }_j(\xi )+{\psi }_j(\xi )]\right]\hfill \\ {\phi }_{j+1}(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\varphi }_j(\xi )+{\psi }_j(\xi )]\right]\hfill \\ {\psi }_{j+1}(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\varphi }_j(\xi )+{\phi }_j(\xi )]\right].\hfill \end{array}$$

(4.24)

Using equation (4.24), we can evaluate the rest of the terms as follows:

$$\begin{array}{c}\multicolumn{1}{c}{{\varphi }_1(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\phi }_0(\xi )+{\psi }_0(\xi )]\right]}\\ ={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[1\right]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{w}{r}\left[\frac{1}{r}\right]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{w}{r^2}\right]\hfill \\ =\xi .\hfill \\ \multicolumn{1}{c}{{\phi }_1(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\phi }_0(\xi )+{\psi }_0(\xi )]\right]}\\ ={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[-1+1]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{w}{r}\left[0\right]\right]\hfill \\ =0.\hfill \\ \multicolumn{1}{c}{{\psi }_1(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\varphi }_0(\xi )+{\phi }_0(\xi )]\right]}\\ ={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[-1+0]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{w}{r}\left[\frac{-1}{r}\right]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{-w}{r^2}\right]\hfill \\ =-\xi .\hfill \\ \multicolumn{1}{c}{{\varphi }_2(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\phi }_1(\xi )+{\psi }_1(\xi )]\right]}\\ ={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[-\xi ]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{-w}{r}\left[\frac{w}{r^2}\right]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{-w^2}{r^3}\right]\hfill \\ =\frac{-{\xi }^2}{2!}.\hfill \end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{{\phi }_2(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\varphi }_1(\xi )+{\psi }_1(\xi )]\right]}\\ ={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\xi -\xi ]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{w}{r}\left[0\right]\right]\hfill \\ =0.\hfill \\ \multicolumn{1}{c}{{\psi }_2(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\varphi }_1(\xi )+{\phi }_1(\xi )]\right]}\\ ={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[\xi \right]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{w}{r}\left[\frac{w}{r^2}\right]\right]\hfill \\ ={\aleph }^{-1}\left[\left[\frac{w^2}{r^3}\right]\right]\hfill \\ =\frac{{\xi }^2}{2!}.\hfill \\ \multicolumn{1}{c}{{\varphi }_3(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\phi }_2(\xi )+{\psi }_2(\xi )]\right]}\\ ={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[\frac{{\xi }^2}{2}\right]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{w}{r}\left[\frac{2!w^2}{2r^3}\right]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{w^3}{r^4}\right]\hfill \\ =\frac{{\xi }^3}{3!}.\hfill \\ \multicolumn{1}{c}{{\phi }_3(\zeta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\varphi }_2(\zeta )+{\psi }_2(\zeta )]\right]}\\ ={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\frac{-{\zeta }^2}{2!}+\frac{-{\zeta }^2}{2!}]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[0\right]\right]\hfill \\ =0.\hfill \end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{{\psi }_3(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\varphi }_2(\xi )+{\phi }_2(\xi )]\right]}\\ ={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[\frac{-{\xi }^2}{2}\right]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{-w}{r}{\aleph }^{+}\left[\frac{2!w^2}{2!r^3}\right]\right]\hfill \\ ={\aleph }^{-1}\left[\left[\frac{-w^3}{r^4}\right]\right]\hfill \\ =\frac{-{\xi }^3}{3!}.\hfill \\ \multicolumn{1}{c}{{\varphi }_4(\zeta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\phi }_3(\zeta )+{\psi }_3(\zeta )]\right]}\\ ={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[\frac{-{\xi }^3}{3!}\right]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{w}{r}\left[\frac{3!w^3}{3!r^4}\right]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{w^4}{r^5}\right]\hfill \\ =\frac{{\xi }^4}{4!}.\hfill \\ \multicolumn{1}{c}{{\phi }_4(\xi )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\phi }_3(\xi )+{\psi }_3(\xi )]\right]}\\ ={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\frac{{\xi }^3}{3!}-\frac{{\xi }^3}{3!}]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{w}{r}\left(0\right)\right]\hfill \\ =0.\hfill \end{array}$$

$$\begin{array}{c}{\psi }_4(t)={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\varphi }_3(\xi )+{\phi }_3(\xi )]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[\frac{{\xi }^3}{3!}\right]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{w}{r}\left[\frac{3!w^3}{3!r^4}\right]\right]\hfill \\ ={\aleph }^{-1}\left[\left[\frac{w^4}{r^5}\right]\right]\hfill \\ =\frac{{\xi }^4}{4!}.\hfill \end{array}$$

We proceed as usual to arrive at:

$$\begin{array}{c}\sum _{j=0}^{\infty }{\varphi }_j(\xi )={\varphi }_0(\xi )+{\varphi }_1(\xi )+{\varphi }_2(\xi )+{\varphi }_3(\xi )+{\varphi }_4(\xi )+...\hfill \\ =-1+\xi -\frac{{\xi }^2}{2!}+\frac{{\xi }^3}{3!}-\frac{{\xi }^4}{4!}+...\hfill \\ \sum _{j=0}^{\infty }{\phi }_j(\xi )={\phi }_0(\xi )+{\phi }_1(\xi )+{\phi }_2(\xi )+{\phi }_3(\xi )+{\phi }_4(\xi )+...\hfill \\ =0+0+0+0+0+...\hfill \\ \sum _{j=0}^{\infty }{\psi }_j(\xi )={\psi }_0(\xi )+{\psi }_1(\xi )+{\psi }_2(\xi )+{\psi }_3(\xi )+{\psi }_4(\xi )+...\hfill \\ =1-\xi +\frac{{\xi }^2}{2!}-\frac{{\xi }^3}{3!}+\frac{{\xi }^4}{4!}+...\hfill \end{array}$$

Hence, the intended solutions are:

$$\begin{array}{c}\varphi (\xi )=-e^{-\xi }.\hfill \\ \phi (\xi )=0.\hfill \\ \psi (\xi )=e^{-\xi }.\hfill \end{array}$$

The precise answer is in perfect agreement with the outcome produced by (ADM).

• Example 4.3 Given the partial differential equations of the first-order nonlinear system:

$$\begin{array}{c}{\varphi }_{\eta }-{\phi }_{\tau }{\psi }_{\xi }=5\hfill \\ {\phi }_{\eta }-{\psi }_{\tau }{\varphi }_{\xi }=5\hfill \\ {\psi }_{\eta }-{\varphi }_{\tau }{\phi }_{\xi }=1.\hfill \end{array}$$

(4.25)

With conditions:

$$\varphi (\tau ,\xi ,0)=\tau -2\xi ,\phi (\tau ,\xi ,0)=-\tau +2\xi ,\psi (\tau ,\xi ,0,)=\tau +2\xi .$$

(4.26)

Solution. Employing the N-transformation to equation (4.25), we arrive at:

$$\begin{array}{c}\frac{r\mathrm{\Phi }(\tau ,\xi ,r,w)}{w}-\frac{\tau -2\xi }{w}-{\aleph }^{+}\left[{\phi }_{\tau }{\psi }_{\xi }\right]={\aleph }^{+}\left[\frac{5}{r}\right]\hfill \\ \frac{r\mathrm{\Upsilon }(\tau ,\xi ,r,w)}{w}-\frac{-\tau +2\xi }{w}-{\aleph }^{+}\left[{\psi }_{\tau }{\varphi }_{\xi }\right]={\aleph }^{+}\left[\frac{5}{r}\right]\hfill \\ \frac{r\mathrm{\Psi }(\tau ,\xi ,r,w)}{w}-\frac{\tau +2\xi }{w}-{\aleph }^{+}\left[{\varphi }_{\tau }{\phi }_{\xi }\right]={\aleph }^{+}\left[\frac{1}{r}\right].\hfill \end{array}$$

(4.27)

Thus,

$$\begin{array}{c}\mathrm{\Phi }(\tau ,\xi ,r,w)=\frac{\tau -2\xi }{r}+\frac{5w}{r^2}+\frac{w}{r}{\aleph }^{+}\left[{\phi }_{\tau }{\psi }_{\xi }\right]\hfill \\ \mathrm{\Upsilon }(\tau ,\xi ,r,w)=\frac{-\tau +2\xi }{r}+\frac{5w}{r^2}+\frac{w}{r}{\aleph }^{+}\left[{\psi }_{\tau }{\varphi }_{\xi }\right]\hfill \\ \mathrm{\Psi }(\tau ,\xi ,r,w)=\frac{\tau +2\xi }{w}+\frac{w}{r^2}+\frac{w}{r}{\aleph }^{+}\left[{\varphi }_{\tau }{\phi }_{\xi }\right].\hfill \end{array}$$

(4.28)

Taking the inverse N-transformation, we arrive at:

$$\begin{array}{c}\varphi (\tau ,\xi ,\eta )={\aleph }^{-1}[\frac{\tau -2\xi }{r}+\frac{5w}{r^2}+\frac{w}{r}{\aleph }^{+}\left[{\phi }_{\tau }{\psi }_{\xi }\right]]\hfill \\ \phi (\tau ,\xi ,\eta )={\aleph }^{-1}[\frac{-\tau +2\xi }{r}+\frac{5w}{r^2}+\frac{w}{r}{\aleph }^{+}\left[{\psi }_{\tau }{\varphi }_{\xi }\right]]\hfill \\ \psi (\tau ,\xi ,\eta )={\aleph }^{-1}[\frac{\tau +2\xi }{w}+\frac{w}{r^2}+\frac{w}{r}{\aleph }^{+}\left[{\varphi }_{\tau }{\phi }_{\xi }\right]].\hfill \end{array}$$

(4.29)

Suppose we have the following solutions:

$$\begin{array}{c}\multicolumn{1}{c}{\varphi (\tau ,\xi ,\eta )=\sum _{j=0}^{\infty }{\varphi }_j(\tau ,\xi ,\eta ),\phi (\tau ,\xi ,\zeta )}\\ =\sum _{j=0}^{\infty }{\phi }_j(\tau ,\xi ,\eta ),\psi (\tau ,\xi ,\eta )=\sum _{j=0}^{\infty }{\psi }_j(\tau ,\xi ,\eta ).\hfill \end{array}$$

(4.30)

$$\begin{array}{c}\sum _{j=0}^{\infty }{\varphi }_j(\tau ,\xi ,\eta )=\tau -2\xi +5\eta +{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[\sum _{j=0}^{\infty }{A}_j\right]\right]\hfill \\ \sum _{j=0}^{\infty }{\phi }_j(\tau ,\xi ,\eta )=-\tau +2\xi +5\eta +{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[\sum _{j=0}^{\infty }{B}_j\right]\right]\hfill \\ \sum _{j=0}^{\infty }{\psi }_j(\tau ,\xi ,\eta )=\tau +2\xi +\eta +{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[\sum _{j=0}^{\infty }{C}_j\right]\right].\hfill \end{array}$$

Where $A_j$ , $B_j$ , and $C_j$ represent the Adomian polynomials for ${\phi }_{\tau }{\psi }_{\xi }$ , ${\psi }_{\tau }{\varphi }_{\xi }$ , and ${\psi }_{\tau }{\phi }_{\xi }$ , respectively.

Looking at the above equations, we can generate the recursive relation as follows:

$$\begin{array}{c}{\varphi }_0(\tau ,\xi ,\eta )=\tau -2\xi +5\eta \hfill \\ {\phi }_0(\tau ,\xi ,\eta )=-\tau +2\xi +5\eta \hfill \\ {\psi }_0(\tau ,\xi ,\eta )=\tau +2\xi +\eta ,\hfill \\ {\varphi }_1(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[A_0\right]\right]\hfill \\ {\phi }_1(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[B_0\right]\right]\hfill \\ {\psi }_1(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[C_0\right]\right],\hfill \end{array}$$

and,

$$\begin{array}{c}{\varphi }_2(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[A_1\right]\right]\hfill \\ {\phi }_2(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[B_1\right]\right]\hfill \\ {\psi }_2(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[C_1\right]\right].\hfill \end{array}$$

Then, our general formula is given by:

$$\begin{array}{c}{\varphi }_{j+1}(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[A_j\right]\right]\hfill \\ {\phi }_{j+1}(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[B_j\right]\right]\hfill \\ {\psi }_{j+1}(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[C_j\right]\right].\hfill \end{array}$$

(4.31)

Using equation (4.31), we can evaluate the rest of the iterations as follows:

$$\begin{array}{c}\multicolumn{1}{c}{{\varphi }_1(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[A_0\right]\right]}\\ \multicolumn{1}{c}{A_0={\phi }_{0\tau }{\psi }_{0\xi }}\\ =-2.\hfill \\ \multicolumn{1}{c}{{\varphi }_1(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[-2]\right]}\\ ={\aleph }^{-1}\left[\frac{w}{r}\left[\frac{-2}{r}\right]\right]\hfill \\ =-2\eta .\hfill \\ \multicolumn{1}{c}{{\phi }_1(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[B_0\right]\right]}\\ \multicolumn{1}{c}{B_0={\psi }_{0\tau }{\varphi }_{0\xi }}\\ =-2.\hfill \\ \multicolumn{1}{c}{{\phi }_1(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[-2]\right]}\\ ={\aleph }^{-1}\left[\frac{w}{r}\left[\frac{-2}{r}\right]\right]\hfill \\ =-2\eta .\hfill \\ \multicolumn{1}{c}{{\psi }_1(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[C_0\right]\right]}\\ \multicolumn{1}{c}{C_0={\varphi }_{0\tau }{\phi }_{0\tau }}\\ \multicolumn{1}{c}{{\psi }_1(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[2\right]\right]}\\ ={\aleph }^{-1}\left[\frac{w}{r}\left[\frac{2}{r}\right]\right]\hfill \\ =-2\eta .\hfill \\ \multicolumn{1}{c}{{\varphi }_2(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[A_1\right]\right]}\\ \multicolumn{1}{c}{A_1=\left({\phi }_{1\tau }{\psi }_{0\xi }\right)+\left({\phi }_{0\tau }\tau {\psi }_{1\xi }\right)}\\ =0.\hfill \\ \multicolumn{1}{c}{{\varphi }_2(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[0\right]\right]}\\ =0.\hfill \end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{{\phi }_2(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[B_1\right]\right]}\\ \multicolumn{1}{c}{B_1=\left({\psi }_{1\tau }{\varphi }_{0\xi }\right)+\left({\psi }_{0\tau }{\varphi }_{1\xi }\right)}\\ =0.\hfill \\ \multicolumn{1}{c}{{\phi }_2(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[0\right]\right]}\\ =0.\hfill \\ \multicolumn{1}{c}{{\psi }_2(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[C_1\right]\right]}\\ \multicolumn{1}{c}{C_1=\left({\varphi }_{1\tau }{\phi }_{0\tau }\right)+\left({\varphi }_{0\tau }{\phi }_{1\tau }\right)}\\ \multicolumn{1}{c}{{\psi }_2(\tau ,\xi ,\eta )={\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[0\right]\right]}\\ \multicolumn{1}{c}{=0.}\end{array}$$

We proceed to arrive at the intended solutions to the form:

$$\begin{array}{c}\varphi (\tau ,\xi ,\eta )=\sum _{j=0}^{\infty }{\varphi }_j(\tau ,\xi ,\eta ),\phi (\tau ,\xi ,\eta )=\sum _{j=0}^{\infty }{\phi }_j(\tau ,\xi ,\eta ),\hfill \\ \psi (\tau ,\xi ,\eta )=\sum _{j=0}^{\infty }{\psi }_j(\tau ,\xi ,\eta ).\hfill \\ \sum _{j=0}^{\infty }{\varphi }_j(\tau ,\xi ,\eta )={\varphi }_0(\tau ,\xi ,\eta )+{\varphi }_1(\tau ,\xi ,\eta )+{\varphi }_2(\tau ,\xi ,\eta )+...\hfill \\ =(\tau -2\xi +5\eta )+(-2\eta )+0+...\hfill \\ \sum _{j=0}^{\infty }{\phi }_j(x,y,\eta )={\phi }_0(\tau ,\xi ,\eta )+{\phi }_1(\tau ,\xi ,\eta )+{\phi }_2(\tau ,\xi ,\eta )+...\hfill \\ =(-\tau +2\xi +5\eta )+(-2\eta )+0+...\hfill \\ \sum _{j=0}^{\infty }{\psi }_j(\tau ,\xi ,\eta )={\psi }_0(\tau ,\xi ,\eta )+{\psi }_1(\tau ,\xi ,\eta )+{\psi }_2(\tau ,\xi ,\eta )+...\hfill \\ =(\tau +2\xi +\eta )+\left(2\zeta \right)+0+...\hfill \end{array}$$

Hence, our intended solutions are as follows:

$$\begin{array}{c}\varphi (\tau ,\xi ,\eta )=\tau -2\xi +3\eta \hfill \\ \phi (\tau ,\xi ,\eta )=2\xi -\tau +3\eta \hfill \\ \psi (\tau ,\xi ,\eta )=\tau +2\xi +3\eta .\hfill \end{array}$$

The precise answer is in perfect agreement with the outcome produced by (ADM).

• Example 4.4 Given the Wu-Zhang nonlinear system of partial differential equations of the form:

$$\begin{array}{c}{\psi }_{\eta }=-\psi {\psi }_{\tau }-{\varphi }_{\tau }\hfill \\ {\varphi }_{\eta }=-\varphi {\psi }_{\tau }-\psi {\varphi }_{\tau }-\frac{1}{3}{\psi }_{\tau \tau \tau }.\hfill \end{array}$$

(4.32)

Along with conditions:

$$\psi (\tau ,0)=\frac{2}{\sqrt{3}}(1+\tanh (\tau )),\varphi (\tau ,0)=\frac{2}{3}{\mathrm{s}\mathrm{ech}}^2\left(\tau \right).$$

(4.33)

Solution. Employing the N-transformation of equation (4.32), we obtain:

$$\begin{array}{c}\frac{r\mathrm{\Psi }(\tau ,r,w)}{w}-\frac{2}{w\sqrt{3}}(1+\tanh (\tau ))=-{\aleph }^{+}[\psi {\psi }_{\tau }+{\varphi }_{\tau }]\hfill \\ \frac{r\mathrm{\Phi }(\tau ,r,w)}{w}-\frac{2}{3w}\mathrm{s}\mathrm{ech}{}^2(\tau )=-{\aleph }^{+}[\varphi {\psi }_{\tau }+\psi {\varphi }_{\tau }+\frac{1}{3}{\psi }_{\tau \tau \tau }].\hfill \end{array}$$

(4.34)

Thus,

$$\begin{array}{c}\mathrm{\Psi }(\tau ,r,w)=\frac{2}{\sqrt{3}s}(1+\tanh (\tau ))-\frac{w}{r}{\aleph }^{+}[\psi {\psi }_{\tau }+{\varphi }_{\tau }]\hfill \\ \mathrm{\Phi }(\tau ,r,w)=\frac{2}{3r}\mathrm{s}\mathrm{ech}{}^2(\tau )-\frac{w}{r}{\aleph }^{+}[\varphi {\psi }_{\tau }+\psi {\varphi }_{\tau }+\frac{1}{3}{\psi }_{\tau \tau \tau }].\hfill \end{array}$$

(4.35)

Take the inverse N-transformation of equation (4.35) to arrive at:

$$\begin{array}{c}\psi (\tau ,\eta )={\aleph }^{-1}[\frac{2}{\sqrt{3}s}(1+\tanh (\tau ))-\frac{w}{r}{\aleph }^{+}[\psi {\psi }_{\tau }+{\varphi }_{\tau }]]\hfill \\ {\varphi }_j(\tau ,\eta )={\aleph }^{-1}[\frac{2}{3r}\mathrm{s}\mathrm{ech}{}^2(\tau )-\frac{w}{r}{\aleph }^{+}[\varphi {\psi }_{\tau }+\psi {\varphi }_{\tau }+\frac{1}{3}{\psi }_{\tau \tau \tau }]].\hfill \end{array}$$

(4.36)

Suppose we have solutions given in the following forms:

$$\psi (\tau ,\eta )=\sum _{j=0}^{\infty }{\psi }_j(\tau ,\eta ),\varphi (\tau ,\eta )=\sum _{j=0}^{\infty }{\varphi }_j(\tau ,\eta ).$$

(4.37)

Substitute equation (4.37) into equation (4.36) to arrive at:

$$\begin{array}{c}\sum _{j=0}^{\infty }{\psi }_j(\tau ,\eta )=\frac{2}{\sqrt{3}}(1+\tanh (\tau ))-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\sum _{j=0}^{\infty }{A}_j+{{\varphi }_j}_{\tau }]\right]\hfill \\ \sum _{j=0}^{\infty }{\varphi }_j(\tau ,\eta )=\frac{2}{3}\mathrm{s}\mathrm{ech}{}^2(\tau )-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\sum _{j=0}^{\infty }{B}_j+\sum _{j=0}^{\infty }{C}_j+\frac{1}{3}{{\psi }_j}_{\tau \tau \tau }]\right].\hfill \end{array}$$

Here $A_j$ , $B_j$ , and $C_j$ describe the Adomian polynomials for $\psi \psi {}_{\tau }$ , $\varphi {\psi }_{\tau }$ , and $\psi {\varphi }_{\tau }$ , respectively.

Comparing both sides of the above formulas, one can conclude that:

$$\begin{array}{c}{\psi }_0(\tau ,\eta )=\frac{2}{\sqrt{3}}(1+\tanh (\tau ))\hfill \\ {\varphi }_0(\tau ,\eta )=\frac{2}{3}\mathrm{s}\mathrm{ech}{}^2(\tau ).\hfill \\ {\psi }_1(\tau ,\eta )=-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[A_0+{\varphi }_{0\tau }]\right]\hfill \\ {\varphi }_1(\tau ,\eta )=-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[B_0+C_0+\frac{1}{3}{\psi }_{0\tau \tau \tau }]\right]\hfill \\ {\psi }_2(\tau ,\eta )=-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[A_1+{\varphi }_{1\tau }]\right]\hfill \\ {\varphi }_2(\tau ,\eta )=-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[B_1+C_1+\frac{1}{3}{\psi }_{1\tau \tau \tau }]\right].\hfill \end{array}$$

Then the general formula is of the form:

$$\begin{array}{c}{\psi }_{j+1}(\tau ,\eta )=-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[A_j+{\varphi }_{j\tau }]\right]\hfill \\ {\varphi }_{j+1}(\tau ,\eta )=-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[B_j+C_j+\frac{1}{3}{\psi }_{j\tau \tau \tau }]\right].\hfill \end{array}$$

(4.38)

From equation (4.38), one can evaluate the rest of the terms as follows:

$$\begin{array}{c}{\psi }_1(\tau ,\eta )=-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[A_0+{\varphi }_{0\tau }]\right]\hfill \\ =-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\psi }_0{\psi }_{0\tau }+{\psi }_{0\tau }]\right]\hfill \\ =-{\aleph }^{-1}[\frac{w}{r}\frac{4}{3}\mathrm{s}\mathrm{ech}{}^2(\tau ){\aleph }^{+}\left[1\right]]\hfill \\ =-\frac{4}{3}\mathrm{s}\mathrm{ech}{}^2(x)\eta \hfill \\ {\varphi }_1(\tau ,\eta )=-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[B_0+C_0+\frac{1}{3}{\psi }_{0\tau \tau \tau }]\right]\hfill \\ =-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[{\varphi }_0{\psi }_{0\tau }+{\psi }_0{\varphi }_{0\tau }+\frac{1}{3}{\psi }_{0\tau \tau \tau }]\right]\hfill \\ =-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[\frac{-8}{3\sqrt{3}}\mathrm{s}\mathrm{ech}{}^2(\tau )\tanh (\tau )]\right]\hfill \\ =\frac{8}{3\sqrt{3}}\mathrm{s}\mathrm{ech}{}^2(\tau )\tanh (\tau )\eta \hfill \\ {\psi }_2(\tau ,\eta )=-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[A_1+{\varphi }_{1\tau }]\right]\hfill \\ =-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[({\psi }_1{\psi }_{0\tau }+{\psi }_0{\psi }_{1\tau })+{\varphi }_{1\tau }]\right]\hfill \\ =-{\aleph }^{-1}[\frac{w}{r}\frac{8}{3\sqrt{3}}\mathrm{s}\mathrm{ech}{}^2(\tau )(1+\tanh (\tau ))(-1+3\tanh (\tau )){\aleph }^{+}[-\eta ]]\hfill \\ =\frac{8}{3\sqrt{3}}\mathrm{s}\mathrm{ech}{}^2(x)(1+\tanh (x))(-1+3\tanh (x))\frac{{\eta }^2}{2!}\hfill \\ {\varphi }_2(\tau ,\eta )=-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[B_1+C_1+\frac{1}{3}{\psi }_{1\tau \tau \tau }]\right]\hfill \\ =-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}[({\psi }_0{\psi }_{1\tau }+{\varphi }_1{\psi }_{0\tau })+({\psi }_0{\varphi }_{1\tau }+{\psi }_1{\varphi }_{0\tau })+\frac{1}{3}{\psi }_{1\tau \tau \tau }]\right]\hfill \\ =-{\aleph }^{-1}\left[\frac{w}{r}{\aleph }^{+}\left[\frac{-16}{9}(-2+\cosh (2\tau )\mathrm{s}\mathrm{ech}{}^4(\tau ))\eta \right]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{w}{r}\frac{16}{9}(-2+\cosh (2\tau )\mathrm{s}\mathrm{ech}{}^4(\tau )){\aleph }^{+}\left[\frac{w}{r^2}\right]\right]\hfill \\ ={\aleph }^{-1}\left[\frac{16}{9}(-2+\cosh (2\tau )\mathrm{s}\mathrm{ech}{}^4(\tau ))\left[\frac{w^2}{r^3}\right]\right]\hfill \\ =\frac{16}{9}(-2+\cosh (2\tau )\mathrm{s}\mathrm{ech}{}^4(\tau ))\frac{{\eta }^2}{2!}.\hfill \end{array}$$

We proceed to arrive at the intended solutions presented in the form:

$$\begin{array}{c}\psi (\tau ,\eta )=\sum _{j=0}^{\infty }{\psi }_j(\tau ,\eta ),\varphi (\tau ,\eta )=\sum _{j=0}^{\infty }{\varphi }_j(\tau ,\eta ).\hfill \\ \sum _{j=0}^{\infty }{\psi }_j(\tau ,\eta )={\psi }_0(\tau ,\eta )+{\psi }_1(\tau ,\eta )+{\psi }_2(\tau ,\eta )+...\hfill \\ =\frac{2}{\sqrt{3}}(1+\tanh (\tau ))-\frac{4}{3}\mathrm{s}\mathrm{ech}{}^2(\tau )\eta \hfill \end{array}$$

$$\begin{array}{c}+\frac{8}{3\sqrt{3}}\mathrm{s}\mathrm{ech}{}^2(\tau )(1+\tanh (\tau ))(-1+3\tanh (\tau ))\frac{{\eta }^2}{2!}+...\hfill \\ \sum _{j=0}^{\infty }{\varphi }_j(\tau ,\eta )={\varphi }_0(\tau ,\eta )+{\varphi }_1(\tau ,\eta )+{\varphi }_2(\tau ,\eta )+...\hfill \\ =\frac{2}{3}\mathrm{s}\mathrm{ech}{}^2(\tau )+\frac{8}{3\sqrt{3}}\mathrm{s}\mathrm{ech}{}^2(\tau )\tanh (\tau )\eta \hfill \\ +\frac{16}{9}(-2+\cosh (2\tau )\mathrm{s}\mathrm{ech}{}^4(\tau ))\frac{{\eta }^2}{2!}+...\hfill \end{array}$$

Thus, the intended solutions are given by:

$$\begin{array}{c}\psi (\tau ,\eta )=\frac{2}{\sqrt{3}}(1+\tanh (\tau -\frac{2}{\sqrt{3}}\eta ))\hfill \\ \varphi (\tau ,\eta )=\frac{2}{3}\left(\mathrm{s}\mathrm{ech}{}^2(\tau +\frac{2}{\sqrt{3}}\eta )\right).\hfill \end{array}$$

The precise answer is in perfect agreement with the outcome produced by (ADM).

Conclusion

In the present research work, a novel approach for solving Wu-Zhang system of nonlinear ordinary and partial differential equations under proper initial conditions was developed by combining the natural transform method and the Adomian decomposition method. The outcomes demonstrate that the NDM’s rate of convergence is quicker than that of other approaches described in the literature. The NDM’s applicability in the engineering and applied scientific areas demonstrated its significance. Many researchers used various techniques on the Wu-Zhang system to find solitary wave solutions since the solitary wave is a localized wave that propagates without any change of shape or velocity. This kind of wave is stable against mutual collisions. Additionally, when we used the present scheme on a few numerical examples, its efficacy and applicability were proven. Based on the study provided above, the NDM can also be used to precisely solve additional non-linear ODEs and PDEs, including systems of ODEs and PDEs, which are often encountered in science and engineering.