Theoretical analysis of J -transform decomposition method with applications of nonlinear ordinary differential equations

Abstract

One of the most noteworthy differential equations of the first order is the Riccati differential equation. It is applied in various branches of mathematics, including algebraic geometry, physics, and conformal mapping theory. The $\mathbb{J}$ -transform Adomian decomposition method is employed in the current study to find exact solutions for different kinds of nonlinear differential equations. We give thorough detailed proofs for new theorems related to the $\mathbb{J}$ -transform technique. The Adomian decomposition method and the $\mathbb{J}$ -transform method serve as the foundation for this technique. For certain differential equations, the theoretical analysis of the $\mathbb{J}$ -transform Adomian decomposition method is examined and is computed using readily computed terms. Our findings are contrasted with exact solutions found in the literature that were produced using alternative techniques. The significant features of the $\mathbb{J}$ -transform Adomian decomposition method are described in the article. It has been shown that the $\mathbb{J}$ -transform Adomian decomposition method is very efficient, useful, and adaptable to a broad variety of linear and nonlinear differential equations. Most of the symbolic and numerical calculations were performed using Mathematica.

Introduction

Solving differential and integral equations with initial value conditions requires the use of integral transformations, which are crucial. For the purpose of solving differential and integral equations, one of the most used methods is the integral transform method.^1–11 In the literature, the Laplace transform is also the most often employed method. ¹² Watugala ¹³ first suggested the Laplace-Carson transform in 1993. It is also known as the $p$ -multiplied variant of the standard Laplace transform. It is closely related to the Sumudu transform, which was employed to resolve controlled engineering applications. Similar to the Sumudu integral transform and the Laplace-Carson transform, the N-transformation, also known as the natural transform, was first introduced in 2008. The N-transform can solve the unsteady fluid flow through a flat wall problem by changing the variables to offer both Laplace and Sumudu integral transforms.^5–8

Most linear and nonlinear approaches that exist in a variety of scientific areas, such as population models, fluid mechanics, solid state physics, plasma physics, and chemical kinetics, can be modeled using differential equations. As a result, it is still very difficult to obtain exact or approximate solutions to linear and nonlinear differential equations in applied mathematics and physics, which calls for new methods.^14–20 To obtain exact or approximate analytical solutions of those existing phenomena, a number of powerful mathematical techniques have been proposed, including the adomian decomposition method (ADM),^14–18 natural ADM (NADM), ²¹ reduced differential transform method (RDTM),^22–25 operational method, ²⁶ collocation method, ²⁷ wavelet operational matrix method, ²⁸ and homotopy perturbation method (HPM). ²⁹

Ordinary differential equations (ODEs) of integer order can be used to model most nonlinear physical phenomena. ²¹ Dynamical study of ODEs is a challenging task that requires analytical and numerical analysis. Notably, the most effective and successful techniques for figuring out the approximate analytical and exact solutions of ODEs are integral transformations. Also, perturbations and long-lasting polynomials are not included in integral transforms, which is crucial and important.

The $\mathbb{J}$ -transform ADM ( $\mathbb{J}ADM$ ), as described by Maitama and Zhao, ³⁰ is a new technique for solving both linear and nonlinear differential equations. It has been inspired and driven by the continuing research in this field. Additionally, implementing the $\mathbb{J}$ -transform method ( $\mathbb{J}TM$ ), Shehu Maitama and Wei dong Zhao established exact solutions to partial differential equations (PDEs). It is important to note that the exact solutions reported in that study were incorrect, and this can be simply verified by incorporating the initial conditions to the exact solutions.

This work is presented as follows: The definitions and significant properties of the $\mathbb{J}$ -transform and the Adomian decomposition technique are provided in “The $\mathbb{J}$ -transform ADM” section along with a background on the theory of integral transform. The $\mathbb{J}$ -transform theories are examined in full with proofs in the “ $\mathbb{J}$ -transform derivation” section. Exact solutions to nonlinear ODEs are provided in the “Analysis of the $\mathbb{J}ADM$ for nonlinear ODEs” section. In the “Applications of $\mathbb{J}ADM$ ” section, we present the theoretical analysis of the JADM for nonlinear ODEs. The “Conclusion” section focuses on providing precise solutions to nonlinear ODEs. We present our work’s conclusion in the “Acknowledgments” section. Most of the calculations in this work were carried out using the symbolic manipulation program Mathematica 13.

The $\mathbb{J}$ -transform ADM

The concept of $\mathbb{J}TM$ was introduced by Maitama and Zhao. ³⁰ We provide some background information regarding the nature of the $\mathbb{J}TM$ in this section.

Definition 1

Suppose that $\theta (s)$ is a piece-wise continuous function over $\mathbb{R}$ and $K,p>0$ . Suppose that $\mathcal{A}=\{\theta (s):|\theta (s)|<K{e}^{cs}{\chi }_{(0,\mathrm{\infty })}(s)\}$ , where ${\chi }_{(0,\mathrm{\infty })}(s)$ is the characteristic function and its given by:

$${\chi }_{(0,\mathrm{\infty })}(s)=\{\begin{array}{cc}1& s\in (0,\mathrm{\infty })\\ 0& s\notin (0,\mathrm{\infty })\end{array}$$

So, $|\theta (s)|\le K{e}^{cs}$ for $s⟶\mathrm{\infty }$ , that is, given any $\theta (s)\in \mathcal{A}$ , where $r,v>0$ we have

$$\begin{array}{rl}|{\int }_0^{\mathrm{\infty }}{e}^{-rs}\theta (sv)ds|& \le K{\int }_0^{\mathrm{\infty }}{e}^{-rs}{e}^{c\left|sv\right|}ds\\ & =K{\int }_0^{\mathrm{\infty }}{e}^{(cv-r)s}ds\end{array}$$

If $cv-r<0$ , the above is convergent. Therefore, $\theta (s)$ is of exponentially order.

The $\mathbb{J}$ -transformation is then provided as follows:

$$\mathbb{J}(\theta (s))=\mathrm{\Theta }(r,v)=v{\int }_0^{\mathrm{\infty }}{e}^{\frac{-rs}{v}}\theta (s)ds,s,v>0$$

(1)

where $\mathbb{J}$ is the $\mathbb{J}$ -transformation of $\theta (s)$ and $r,v$ are the $\mathbb{J}$ -transformation variables.

So, equation (1) can be written as follows:

$$\mathbb{J}(\theta (s))=\mathrm{\Theta }(r,v)=v^2{\int }_0^{\mathrm{\infty }}{e}^{-rs}\theta (vs)ds,r,v\in (0,\mathrm{\infty })$$

(2)

It is crucial to understand the inverse property of the $\mathbb{J}$ -transform before we demonstrate its applications. First, we review the following important theorems.

Theorem 1 (Spiegel ³¹ )

Assume that the function $\theta (w)$ is analytic on a region which includes $\gamma$ and its interior, and that $\gamma$ is a simple closed curve. Assume $\gamma$ in counterclockwise orientation. Then, for every $w_0$ within $\gamma$ we have

$$\theta (w_0)=\frac{1}{2\pi i}{\int }_{\gamma }\frac{\mathrm{\Theta }(w)}{w-w_0}dw$$

Theorem 2 (Spiegel ³¹ )

Provided that $\gamma$ is a simple closed, positively oriented contour and $\theta$ is analytic on $\mathbb{C}$ except for a few points $w_1,w_2,\dots ,w_n$ inside it, then

$${\oint }_{\gamma }\theta (w)dw=2\pi i\sum _{k=1}^{n}Re{s}_{\theta }\left(w_k\right)$$

Theorem 1 leads to Theorem 2, which is crucial in computing the real integral by using the proper contour in $\mathbb{C}$ , see (Spiegel ³¹ ).

Definition 2 (Inverse $\mathbb{J}$ -transform [bin Rasedee et al.] ²⁷ )

Suppose ${\mathbb{J}}^{-1}$ is referred to as the inverse $\mathbb{J}$ -transform of $\mathrm{\Theta }(r,v)$ , where $\mathrm{\Theta }(r,v)$ is the $\mathbb{J}$ -transform of the function $\theta (s)$ . Then,

$${\mathbb{J}}^{-1}[\mathrm{\Theta }(r,v)]=\theta (s),\mathrm{for}s⩾0$$

Equivalently, based on Theorems 1 and 2, the definition of the complex inverse $\mathbb{J}$ -transform is

$$\begin{array}{cc}\theta (s)& =\underset{\beta \to \mathrm{\infty }}{lim}\frac{1}{2\pi i}{\displaystyle {\int }_{\alpha -i\beta }^{\alpha +i\beta }\frac{1}{w^2}{e}^{\left(\frac{rs}{w}\right)}\mathrm{\Theta }(r,w)dr}\\ & ={\displaystyle \sum residuesof\frac{1}{w^2}{e}^{\left(\frac{rs}{w}\right)}\mathrm{\Theta }(r,w)atthepolesof\mathrm{\Theta }(r,w)}\end{array}$$

Significant properties: The following are a few fundamental properties of $\mathbb{J}$ -transforms; for a summary of these properties, see Maitama and Zhao ³⁰ :

$\mathbb{J}[C]=\frac{C{v}^2}{r}$ .

$\mathbb{J}[{\zeta }^m]=\frac{m!v^{m+2}}{r^{m+1}}$ , where $m\ge 0.$

$\mathbb{J}[e^{c\zeta }]=\frac{v^2}{(r-cv)}$ .

Suppose the function $\theta (\zeta )$ has a $\mathbb{J}$ -transformation if ${\theta }^{(k)}(\zeta )$ is its $k$ th derivative. Then,

$$\mathbb{J}\left[{\theta }^{\left(k\right)}\left(\zeta \right)\right]={\mathrm{\Theta }}_k(r,v)=\frac{r^k}{v^k}\mathrm{\Theta }(r,v)-{\displaystyle \sum _{j=0}^{k-1}\frac{r^{k-(j+1)}}{v^{k-(j+2)}}{\theta }^{\left(j\right)}\left(0\right)}$$

Adomian polynomials evaluations: Now let’s introduce the Adomian polynomials, which are a useful tool for efficiently breaking down a complex nonlinear component into smaller, more manageable components that can be integrable as a Taylor series. The unknown function $\varphi$ can be represented as follows, as demonstrated by Wazwaz ¹⁴ :

$$\varphi =\sum _{m=0}^{\mathrm{\infty }}{\varphi }_m$$

(3)

where the components ${\varphi }_m,m\ge 0$ must be found by building a recursive relation. The following formula can be used to define $F(\varphi )$ as an infinite series, also referred to as Adomian polynomials $A_m$ , when working with nonlinear terms

$$F(\varphi )=\sum _{m=0}^{\mathrm{\infty }}{A}_m({\varphi }_0,{\varphi }_1,\dots .,{\varphi }_m)$$

(4)

where the $A_m$ of the nonlinear term $F(\varphi )$ can be computed using the formula by Adomian ¹⁷ :

$$A_m=\frac{1}{m!}\frac{d^m}{d{\mu }^m}{\left[F\left(\sum _{j=0}^m{\mu }^j{\varphi }_j\right)\right]}_{\mu =0},m=0,1,2,\dots$$

(5)

The general formula for equation (5) can then be expressed as follows:

Let the nonlinear function be represented by $F(\varphi )$ . Using the Adomian polynomial definition and equation (5), the following can be obtained:

$$\begin{array}{c}{A}_0=F({\varphi }_0)\\ A_1={\varphi }_1{F}^{\prime }({\varphi }_0)\\ A_2={\varphi }_2{F}^{\prime }({\varphi }_0)+\frac{1}{2!}{\varphi }_1^2{F}^{″}({\varphi }_0)\end{array}$$

(6)

Finally, a similar process can be used to construct the other terms. Two important observations are provided by the polynomials previously given in equation (6). The variables $A_0$ and $A_1$ rely only on ${\varphi }_0$ , $A_1$ and ${\varphi }_1$ , $A_2$ and ${\varphi }_0$ , ${\varphi }_1$ and ${\varphi }_2$ , and so on.

Additionally, by substituting equation (6) for equation (4), we get

$$\begin{array}{rl}F(\varphi )& =A_0+A_1+A_2+\cdots \\ & =F({\varphi }_0)+({\varphi }_1+{\varphi }_2+{\varphi }_3+\cdots )F^{\prime }({\varphi }_0)\\ & +\frac{1}{2!}({\varphi }_1^2+2{\varphi }_1{\varphi }_2+2\varphi {\varphi }_1{\varphi }_3+{\varphi }_2^2+\cdots )F^{″}({\varphi }_0)\\ & +\frac{1}{3!}({\varphi }_1^3+3{\varphi }_1^2{\varphi }_2+3{\varphi }_1^2{\varphi }_3+6\varphi {\varphi }_1{\varphi }_2{\varphi }_3+\cdots )F^{‴}({\varphi }_0)+\cdots \\ & =F({\varphi }_0)+(\varphi -{\varphi }_0)F^{\prime }({\varphi }_0)+\frac{1}{2!}(\varphi -{\varphi }_0{)}^2{F}^{″}({\varphi }_0)+\cdots \end{array}$$

$\mathbb{J}$ -transform derivation

The new and comprehensive proofs of several properties pertaining to the $\mathbb{J}$ -transformation will be examined in this section. Additionally, we will use these properties to solve some ODEs under appropriate initial conditions.

Theorem 3

Let $\mathrm{\Theta }(r,w)=\frac{2w^4}{r(r^2+w^2)}$ . Then its inverse $\mathbb{J}$ -transform is given by

$$\theta (s)=2-4cos(s)$$

(7)

Proof

First, note that the singularities of $\mathrm{\Theta }(r,w)$ are $r=0$ , $r=iw$ , $r=-iw$ .

Using Definition 2, we obtain

$$\begin{array}{rl}\theta (s)& =\sum Res\left[\frac{e^{\frac{rs}{w}}}{w^2}\left(\frac{2w^4}{r(r^2+w^2)}\right)\right]\\ & =\underset{r\to 0}{lim}(r-0)e^{\frac{rs}{w}}\frac{2w^2}{r(r-iw)(r+iw)}+\underset{r\to iw}{lim}(r-iw)e^{\frac{rs}{w}}\frac{2w^2}{r(r-iw)(r+iw)}\\ & +\underset{r\to -iw}{lim}(r+iw)e^{\frac{rs}{w}}\frac{2w^2}{r(r-iw)(r+iw)}\\ & =\frac{2w^2}{w^2}+\frac{2w^2}{iw(2iw)}{e}^{is}+\frac{2w^2}{-iw(-2iw)}{e}^{-is}\\ & =2-2e^{is}-2e^{-is}\\ & =2-4\left(\frac{e^{is}+e^{-is}}{2}\right)\\ & =2-4\cosh (is)\end{array}$$

But,

$$\cosh (is)=\cos (s)$$

Thus,

$$\theta (s)=2-4\cos (s)$$

 □

Theorem 4

Let $\theta (s)=\frac{\sin (\alpha s)}{\alpha }\in \mathcal{A}$ , with $\alpha \ne 0$ . Then its $\mathbb{J}$ -transform is given by

$$\mathbb{J}\left[\frac{\sin (\alpha s)}{\alpha }\right]=\frac{w^3}{r^2+w^2{\alpha }^2}$$

(8)

Proof

By using integration by parts and the concept of the $\mathbb{J}$ -transform, we arrive at

$$\begin{array}{rl}\mathbb{J}\left[\frac{\sin (\alpha s)}{\alpha }\right]& =w{\int }_0^{\mathrm{\infty }}{e}^{\frac{-rs}{w}}\frac{\sin (\alpha s)}{\alpha }ds.\\ & =w[-e^{\frac{-rs}{w}}\frac{\cos (\alpha s)}{{\alpha }^2}{|}_0^{\mathrm{\infty }}-\frac{r}{w}{e}^{\frac{-rs}{w}}\frac{\sin (\alpha s)}{{\alpha }^3}{|}_0^{\mathrm{\infty }}]-w{\int }_0^{\mathrm{\infty }}\frac{r^2}{w^2}{e}^{\frac{-rs}{w}}\frac{\sin (\alpha s)}{{\alpha }^3}ds\end{array}$$

So,

$$\begin{array}{rl}w{\int }_0^{\mathrm{\infty }}{e}^{\frac{-rs}{w}}\frac{\sin (\alpha s)}{\alpha }ds& =\frac{w^2}{{\alpha }^2}-\frac{r^2}{w^2{\alpha }^2}w{\int }_0^{\mathrm{\infty }}{e}^{\frac{-rs}{w}}\frac{\sin (\alpha s)}{\alpha }ds\end{array}$$

Thus,

$$[1+\frac{r^2}{w^2{\alpha }^2}]w{\int }_0^{\mathrm{\infty }}{e}^{\frac{-rs}{w}}\frac{\sin (\alpha s)}{\alpha }ds=\frac{w}{{\alpha }^2}$$

and,

$$w{\int }_0^{\mathrm{\infty }}{e}^{\frac{-rs}{w}}\frac{\sin (\alpha s)}{\alpha }ds=\frac{u^3}{s^2+u^2{\alpha }^2}$$

Hence,

$$\mathbb{J}\left[\frac{\sin (\alpha s)}{\alpha }\right]=\frac{w^3}{r^2+w^2{\alpha }^2}$$

Theorem 5

Let $\theta (s)=\cos (\alpha s)\in \mathcal{A}$ . Then its $\mathbb{J}$ -transform is given by

$$\mathbb{J}[\cos (\alpha s)]=\frac{w^{2}r}{r^2+w^2{\alpha }^2}$$

(9)

Proof

By employing integration by parts and the concept of the $\mathbb{J}$ -transform, we arrive at

$$\begin{array}{rl}\mathbb{J}[\cos (\alpha s)]& =w{\int }_0^{\mathrm{\infty }}{e}^{\frac{-rs}{w}}\cos (\alpha s)ds\\ & =w[-e^{\frac{-rs}{w}}\frac{\sin (\alpha s)}{\alpha }{|}_0^{\mathrm{\infty }}-\frac{r}{w}{e}^{\frac{-rs}{w}}\frac{\cos (\alpha s)}{{\alpha }^2}{|}_0^{\mathrm{\infty }}]-w{\int }_0^{\mathrm{\infty }}\frac{r^2}{w^2}{e}^{\frac{-rs}{w}}\frac{\cos (\alpha s)}{{\alpha }^2}ds\end{array}$$

But,

$$\begin{array}{rl}w{\int }_0^{\mathrm{\infty }}{e}^{\frac{-rs}{w}}\cos (\alpha s)ds& =\frac{r}{{\alpha }^2}-\frac{r^2}{w^2{\alpha }^2}w{\int }_0^{\mathrm{\infty }}{e}^{\frac{-rs}{w}}\cos (\alpha s)ds\\ [1+\frac{r^2}{w^2{\alpha }^2}]w{\int }_0^{\mathrm{\infty }}{e}^{\frac{-rs}{w}}\cos (\alpha s)ds& =\frac{r}{{\alpha }^2}{\int }_0^{\mathrm{\infty }}{e}^{\frac{-rs}{w}}\cos (\alpha s)ds& =\frac{w^{2}r}{r^2+w^2{\alpha }^2}\end{array}$$

Hence,

$$\mathbb{J}[\cos (\alpha s)]=\frac{w^{2}r}{r^2+w^2{\alpha }^2}$$

Theorem 6

Let $\mathrm{erf}(s)=\frac{2}{\sqrt{\pi }}{\int }_0^s{e}^{-u}du$ be the error function. Its $\mathbb{J}$ -transform is subsequently given by

$$\mathbb{J}[\mathrm{erf}(s)]=\frac{w^2}{r}{e}^{\frac{r^2}{4w^2}}\mathrm{erf}\left(\frac{r}{2w}\right)$$

(10)

Proof

Using integration by parts and $\mathbb{J}$ -transform, we arrive at

$$\mathbb{J}[\mathrm{erf}(s)]=w^2{\int }_0^{\mathrm{\infty }}{e}^{-st}\mathrm{erf}(ws)ds$$

Let

$$\begin{array}{rl}v& =\mathrm{erf}(ws)\Rightarrow v=\frac{2}{\sqrt{\pi }}{\int }_0^{ws}{e}^{-u}du\Rightarrow dv=\frac{2}{\sqrt{\pi }}w{e}^{(ws{)}^2}\\ dR& =e^{-rs}\Rightarrow R=-\frac{1}{r}{e}^{-rs}\end{array}$$

Then,

$$\begin{array}{rl}\mathbb{J}[\mathrm{erf}(s)]& =w^2[-\frac{1}{r}{e}^{-rs}\mathrm{erf}(s){|}_0^{\mathrm{\infty }}+{\int }_0^{\mathrm{\infty }}\frac{e^{-rs}}{r}\frac{2w}{\sqrt{\pi }}{e}^{-w^2{s}^{2}ds}]\\ & =\frac{2w^3}{r\sqrt{\pi }}{\int }_0^{\mathrm{\infty }}{e}^{-(rs+w^2{s}^2)}ds\\ & =\frac{2w^3}{r\sqrt{\pi }}{\int }_0^{\mathrm{\infty }}{e}^{-w^2(s^2+\frac{r}{w^2}t)}ds\\ & =\frac{2w^3}{r\sqrt{\pi }}{\int }_0^{\mathrm{\infty }}{e}^{-w^2(s^2+\frac{r}{w^2}s+\frac{r^2}{4w^4})}{e}^{\frac{r^2}{4w^2}}ds\\ & =\frac{2w^3}{r\sqrt{\pi }}{e}^{\frac{r^2}{4w^2}}{\int }_0^{\mathrm{\infty }}{e}^{-{(ws+\frac{r}{2w})}^2}ds\end{array}$$

Thus, using substitution, one can arrive at

$$\begin{array}{rl}\mathbb{J}[\mathrm{erf}(s)]& =\frac{2w^3}{r\sqrt{\pi }}{e}^{\frac{r^2}{4w^2}}\frac{1}{w}{\int }_{\frac{r}{2w}}^{\mathrm{\infty }}{e}^{-v^2}dv\\ & =\frac{2w^3}{r\sqrt{\pi }}{e}^{\frac{r^2}{4w^2}}[{\int }_0^{\mathrm{\infty }}{e}^{-v^2}dv-{\int }_0^{\frac{r}{2w}}{e}^{-v^2}dv]\\ & =\frac{2w^3}{r\sqrt{\pi }}{e}^{\frac{r^2}{4w^2}}[\frac{\sqrt{\pi }}{2}-\frac{\sqrt{\pi }}{2}\mathrm{erf}\left(\frac{r}{2w}\right)]\\ & =\frac{w^2}{r}{e}^{\frac{r^2}{4w^2}}[1-\mathrm{erf}\left(\frac{r}{2w}\right)]\end{array}$$

Hence,

$$\mathbb{J}[\mathrm{erf}(s)]=\frac{w^2}{r}{e}^{\frac{r^2}{4w^2}}\mathrm{erf}\left(\frac{r}{2w}\right)$$

Theorem 7

Let $\mathrm{erf}(s)=\frac{2}{\sqrt{\pi }}{\int }_s^{\mathrm{\infty }}{e}^{-u}du$ be the complementary error function. Its $\mathbb{J}$ -transform is subsequently given by

$$\mathbb{J}[\mathrm{erfc}(s)]=\frac{w}{r}[w-w{e}^{\frac{r^2}{4w^2}}\mathrm{erf}\left(\frac{r}{2w}\right)]$$

(11)

Proof

Based on the $\mathbb{J}$ -transform of the error function in Theorem 4 and the fact that $\mathrm{erfc}0.03in(s)=1-\mathrm{erf}(s)$ , we deduce that

$$\begin{array}{rl}\mathbb{J}[\mathrm{erfc}(s)]& =\mathbb{J}[1-\mathrm{erf}(s)]\\ & =\mathbb{J}[1]-\mathbb{J}[\mathrm{erf}(s)]]\\ & =\frac{w^2}{r}-\frac{w^2}{r}{e}^{\frac{r^2}{4w^2}}\mathrm{erfc}\left(\frac{r}{2w}\right)\end{array}$$

Hence,

$$\mathbb{J}[\mathrm{erfc}(s)]=\frac{w}{r}[w-w{e}^{\frac{r^2}{4w^2}}\mathrm{erfc}\left(\frac{r}{2w}\right)]$$

Theorem 8

Consider the sine integral function, $Si(s)={\int }_0^s\frac{\sin (w)}{w}dw$ , then its $\mathbb{J}$ -transform is given by

$$\mathbb{J}[Si(s)]=\frac{w^2}{r}\arctan \left(\frac{w}{r}\right)$$

(12)

Proof

First,

$$Si(s)={\int }_0^s\frac{sin(w)}{w}dw$$

(13)

Using substitution, equation (13) becomes

$$Si(s)={\int }_0^1\frac{sin(sv)}{sv}sdv={\int }_0^1\frac{sin(sv)}{v}dv$$

(14)

Applying $\mathbb{J}$ -transform on equation (14), we get

$$\begin{array}{rl}\mathbb{J}[Si(s)]& =\mathbb{J}\left[{\int }_0^1\frac{sin(sv)}{v}dv\right]\\ & =w{\int }_0^{\mathrm{\infty }}{e}^{-rs}\left[{\int }_0^1\frac{sin(svw)}{v}dv\right]ds\\ & ={\int }_0^1\frac{1}{v}[w{\int }_0^{\mathrm{\infty }}{e}^{-rs}\sin (svw)ds]dv\\ & ={\int }_0^1\frac{1}{v}\mathbb{J}[\sin (sv)]dv\\ & ={\int }_0^1\frac{1}{v}\frac{w^3}{r^2+v^2{w}^2}vdv\\ & =w^3{\int }_0^1\frac{1}{r^2+(vw{)}^2}dv\\ & =\frac{w^2}{r}{\tan }^{-1}\left(\frac{vw}{r}\right){|}_0^1\\ & =\frac{w^2}{r}{\tan }^{-1}\left(\frac{w}{r}\right)\end{array}$$

Hence,

$$\mathbb{J}[Si(s)]=\frac{w^2}{r}\arctan \left(\frac{w}{r}\right)$$

Analysis of the $\mathbb{J}ADM$ for nonlinear ODEs

We now shall discuss the theoretical analysis, such as the convergence theorem, the uniqueness theorem, and the error estimate for the proposed scheme. Given the form of a nonlinear ODE:

$${\theta }_s(s)=\theta (s)+F(\theta )$$

(15)

Accompanied by its IC

$$\theta (0)={\theta }_0$$

(16)

where $F(\theta )={\theta }_s^2+{\theta }^2$ is the nonlinear source and the remaining terms of the equation.

Employing the $\mathbb{J}$ -transformation along with Property 4 in equation (15), we get

$$\begin{array}{c}\mathrm{\Theta }(r,w)=\frac{w}{r}{\theta }_0+\frac{w}{r}\mathbb{J}[\theta (s)+F(\theta )]\end{array}$$

(17)

With the inverse $\mathbb{J}$ -transformation applied to equation (17), we obtain

$$\theta (s)=\beta (s)+{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}[\theta (s)+F(\theta )]\right]$$

Note $\beta (s)$ describes the conditions and the non-homogeneous source.

Suppose we have a solution as follows:

$$\theta (s)=\sum _{j=0}^{\mathrm{\infty }}{\theta }_j(s)$$

(18)

The nonlinear term $M(F(\theta (s)))=\sum _{i=0}^{\mathrm{\infty }}{C}_i$ , where the $C_i$ ’s are the Adomian polynomials.

Using equation (18), equation (17) can be rewritten as follows:

$$\sum _{j=0}^{\mathrm{\infty }}{\theta }_j(s)=\beta (s)+{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}[\sum _{j=0}^{\mathrm{\infty }}{C}_j+\sum _{j=0}^{\mathrm{\infty }}{\theta }_j]\right]$$

(19)

By looking at equation (19), one can observe that ${\theta }_0(s)=\beta (s)$ .

So, one can achieve the recursive formula as

$${\theta }_{j+1}(s)={\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}[C_j+{\theta }_j]\right],j\ge 0$$

(20)

In this case, our intended solution is

$$\theta (s)=\sum _{j=0}^{\mathrm{\infty }}{\theta }_j(s).$$

(21)

Theorem 9 (Uniqueness Theorem)

Suppose $0<\delta <1$ and $\delta =(K_1+K_2)s$ . Then equation (15) will have a unique solution.

Proof

Given the norm $\Vert .\Vert$ and $\mathrm{\Xi }=(C[\mathrm{\Omega }],\Vert .\Vert )$ as the Banach space of all continuous functions on $\mathrm{\Omega }=[c,d]$ , define $\mathrm{{\rm Y}}:\mathrm{\Xi }\to \mathrm{\Xi }$ , where

$${\theta }_{j+1}(s)=\beta (s)+{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}[M({\theta }_j(s))+L({\theta }_j(s))]\right]$$

Given $L[\theta (s)]=\theta (s)$ , $M[\theta (s)]=F(\theta (s))$ with $|M(\theta )-M(\hat{\theta })|<K_1|\theta -\hat{\theta }|$ and $|L(\theta )-L(\hat{\theta })|<K_2|\theta -\hat{\theta }|$ , where $K_1,K_2$ are constants related to Lipschitz and $\theta ,\hat{\theta }$ are distinct solutions for equation (15). Then,

$$\begin{array}{rl}& \Vert \mathrm{{\rm Y}}(\theta )-\mathrm{{\rm Y}}(\hat{\theta })\Vert =\underset{s\in \mathrm{\Omega }}{max}\left|\begin{array}{c}{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}[L(\theta )+M(\theta )]\right]\\ -{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}[L(\hat{\theta })+M(\hat{\theta }))]\right]\end{array}\right|\\ & =\underset{s\in \mathrm{\Omega }}{max}\left|\begin{array}{c}{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}[L(\theta )-L(\hat{\theta })]\right]\\ & +{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}[M(\theta )-M(\hat{\theta })]\right]\end{array}\right|\\ & \le \underset{s\in \mathrm{\Omega }}{max}[K_1{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}\left[|\theta -\hat{\theta }|\right]\right]+K_2{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}\left[|\theta -\hat{\theta }|\right]\right]]\\ & \le \underset{s\in \mathrm{\Omega }}{max}(K_1+K_2)\left[{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}\left[|\theta -\hat{\theta }|\right]\right]\right]\\ & \le (K_1+K_2)\left[{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}\left[\Vert \theta (s)-\hat{\theta }(s)\Vert \right]\right]\right]\\ & =\Vert \theta -\hat{\theta }\Vert (K_1+K_2)s\end{array}$$

The Banach fixed-point theorem for contraction states that there is a unique solution to equation (15) since $0<\delta <1$ implies that $\mathrm{{\rm Y}}$ is contraction mapping. Hence, the proof of Theorem 9 is now complete.

Theorem 10 (Convergence Theorem)

The solution for equation (15) with $0<\delta <1$ and $|{\theta }_1|<\mathrm{\infty }$ will eventually converge when the JADM applied.

Proof

Assume the $k$ th partial sum is $q_k$ , that is, $q_k=\sum _{j=0}^k{\theta }_j(s)$ . We shall prove that $\{q_k\}$ is a Cauchy sequence in the Banach space $\mathrm{\Xi }$ . Assume the Adomian polynomials mentioned by El-Kalla, ³² which is new format of $M(q_k)={\tilde{C}}_k+\sum _{j=0}^{k-1}{\tilde{C}}_j$ . Let $q_i$ and $q_k$ be any distinct sums with $k\ge i$ . Thus,

$$\begin{array}{rl}& \Vert q_k-q_i\Vert =\underset{s\in \mathrm{\Omega }}{max}|q_k-q_i|\\ & =\underset{s\in \mathrm{\Omega }}{max}\left|{\displaystyle \sum _{j=i+1}^k{\hat{\theta }}_i(s)}\right|,k=1,2,...\\ & \le \underset{s\in \mathrm{\Omega }}{max}|{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}\left[L\left({\displaystyle \sum _{j=i+1}^k{\theta }_{j-1}(s)}\right)\right]\right]+{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}\left[{\displaystyle \sum _{j=i+1}^k{C}_{j-1}(s)}\right]\right]|\\ & =\underset{s\in \mathrm{\Omega }}{max}|{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}\left[L\left({\displaystyle \sum _{j=i}^{k-1}{\theta }_i(s)}\right)\right]\right]+{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}\left[{\displaystyle \sum _{j=i}^{k-1}{C}_j(s)}\right]\right]|\\ & \le \underset{s\in \mathrm{\Omega }}{max}|{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}[L(q_{k-1})-L(q_{i-1})]\right]+{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}[M(q_{k-1})-M(q_{i-1})]\right]|\\ & \le K_1\underset{s\in \mathrm{\Omega }}{max}{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}\left[|q_{k-1}-q_{i-1}|\right]\right]+K_2\underset{s\in \mathrm{\Omega }}{max}{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}\left[|q_{k-1}-q_{i-1}|\right]\right]\\ & =(K_1+K_2)s\Vert q_{k-1}-q_{i-1}\Vert \end{array}$$

Thus, $\Vert q_k-q_i\Vert \le \delta \Vert q_{k-1}-q_{i-1}\Vert$ . Choose $k=i+1$ , then

$$\Vert q_{i+1}-q_i\Vert \le \delta \Vert q_i-q_{i-1}\Vert \le {\delta }^2\Vert q_{i-1}-q_{i-2}\Vert \le \cdots \le {\delta }^i\Vert q_1-q_0\Vert$$

Similarly, using the triangle inequality

$$\begin{array}{c}\Vert q_k-q_i\Vert \le \Vert q_{i+1}-q_i\Vert +\Vert q_{i+2}-q_{i+1}\Vert +\cdots +\Vert q_k-q_{k-1}\Vert \\ \le [{\delta }^i+{\delta }^{i+1}+\cdots +{\delta }^{m-1}]\Vert q_1-q_0\Vert \\ \le {\delta }^i\left[\frac{1-{\delta }^{k-i}}{1-\delta }\right]\Vert {\theta }_1\Vert \end{array}$$

But, $0<\delta <1$ , then $1-{\delta }^{k-i}<1$ . Thus

$$\Vert q_k-q_i\Vert \le \frac{{\delta }^i}{1-\delta }\underset{s\in \mathrm{\Omega }}{max}\left|{\theta }_1\right|$$

(22)

Since $\theta (s)$ is bounded, then $|{\theta }_1|<\mathrm{\infty }$ . So, as $i\to \mathrm{\infty }$ , then $\Vert q_k-q_i\Vert \to 0$ . Thus, the sequence $\{q_k\}$ is a Cauchy in $\mathrm{\Xi }$ . Therefore, $\theta (s)=\sum _{j=0}^{\mathrm{\infty }}{\theta }_j(s)$ converges.

Hence, the proof of Theorem 10 is now complete.

Theorem 11 (Error Estimate)

This series solution in equation (22) to equation (15) is expected to have a maximum absolute truncation error of

$$\underset{s\in \mathrm{\Omega }}{max}|\theta (s)-\sum _{k=0}^i{\theta }_k(s)|\le \frac{{\delta }^i}{1-\delta }\Vert {\theta }_1(s)\Vert$$

Proof

Using both Theorem 10 and equation (22) one can conclude: $\Vert q_k-q_i\Vert \le \frac{{\delta }^i}{1-\delta }$ ${max}_{s\in \mathrm{\Omega }}|{\theta }_1|$ . So as $k\to \mathrm{\infty }$ , we have $q_k\to \theta (s)$ . Then, $\Vert \theta (s)-q_i\Vert \le \frac{{\delta }^i}{1-\delta }{max}_{s\in \mathrm{\Omega }}|{\theta }_1(s)|$ .

Thus, the absolute truncation error reaches its maximum in $\mathrm{\Omega }$ as

$$\underset{s\in \mathrm{\Omega }}{max}|\theta (s)-\sum _{k=0}^i{\theta }_k(s)|\le \frac{{\delta }^i}{1-\delta }\underset{s\in \mathrm{\Omega }}{max}|{\theta }_1(s)|=\frac{{\delta }^i}{1-\delta }\Vert {\theta }_1(s)\Vert$$

Hence, the proof of Theorem 11 is now complete.

Remark: In the current work, the JADM for nonlinear ODE convergence analysis was effectively implemented. Because of its significance and the fact that it demonstrates how the approach converges and the answer is distinct, this is highly significant to the research community.

Applications of $\mathbb{J}ADM$

We now show that the $\mathbb{J}$ -transform may be used to solve a number of nonlinear ODEs. All of the nonlinear terms found in a nonlinear differential equation can be conveniently handled using the Adomian polynomials.

Suppose we have the nonlinear differential equation with the following I.V.P:

$$L(\theta )+R(\theta )+F(\theta )=g(s)$$

(23)

With its I.C.:

$$\theta (0)={\theta }_0$$

(24)

where the nonhomogeneous term is $g(s)$ , the nonlinear term is $F(\theta )$ , $L$ represents the operator of the largest derivative, and $R$ represents the remaining differential operator.

Applying $\mathbb{J}$ -transform and Property 4 to equation (23) we obtain

$$\frac{r^2\mathrm{\Theta }(r,w)}{w}-w\theta (0)+\mathbb{J}[R(\theta )]+\mathbb{J}[F(\theta )]=\mathbb{J}[g(s)]$$

(25)

Using equations (24) and (25), we arrive at

$$\mathrm{\Theta }(r,w)=\frac{{\theta }_0{w}^2}{r^2}+\frac{w}{r^2}\mathbb{J}[g(s)]-\frac{w}{r^2}\mathbb{J}[R(\theta )+F(\theta )]$$

(26)

Applying the inverse $\mathbb{J}$ -transform on equation (26) to get

$$\theta (s)=G(s)-{\mathbb{J}}^{-1}\left[\frac{w}{r^2}\mathbb{J}[R(\theta )+F(\theta )]\right]$$

(27)

where $G(s)$ represents both the initial condition and the nonhomogeneous part.

Assume that there exists an infinite series solution of the form

$$\theta (s)=\sum _{j=0}^{\mathrm{\infty }}{\theta }_j(s)$$

(28)

Rewrite equation (27) using equation (28) to arrive at:

$$\sum _{j=0}^{\mathrm{\infty }}{\theta }_j(s)=G(s)-{\mathbb{J}}^{-1}\left[\frac{w}{r^2}\mathbb{J}[R\sum _{j=0}^{\mathrm{\infty }}{\theta }_j(s)+\sum _{j=0}^{\mathrm{\infty }}{A}_j(s)]\right]$$

(29)

where the $A_j(s)$ are the Adomian polynomials that represent the nonlinear term $F(\theta (s))=\sum _{j=0}^{\mathrm{\infty }}{A}_j(s)$ .

Comparing both sides of equation (29), we obtain

$$\begin{array}{rl}{\theta }_0(s)& =G(s)\\ {\theta }_1(s)& =-{\mathbb{J}}^{-1}\left[\frac{w}{r^2}\mathbb{J}[R{\theta }_0(s)+A_0(s)]\right]\\ {\theta }_2(s)& =-{\mathbb{J}}^{-1}\left[\frac{w}{r^2}\mathbb{J}[R{\theta }_1(s)+A_1(s)]\right]\\ {\theta }_3(s)& =-{\mathbb{J}}^{-1}\left[\frac{w}{r^2}\mathbb{J}[R{\theta }_2(s)+A_2(s)]\right]\end{array}$$

Now, one can generate the following general relation:

$${\theta }_{j+1}(s)=-{\mathbb{J}}^{-1}\left[\frac{w}{r^2}\mathbb{J}[R{\theta }_j(s)+A_j(s)]\right],0.5cmj\ge 0$$

(30)

Hence, the exact solution is given by

$$\theta (s)=\sum _{j=0}^{\mathrm{\infty }}{\theta }_j(s)$$

(31)

Worked examples

In this section, we apply the JADM to some nonlinear ODE’s applications in order to demonstrate the effectiveness of the new developed scheme.

Example 1

Consider the nonlinear ODE:

$$\frac{d\theta }{ds}-s\theta (s)\frac{d\theta }{ds}={\theta }^2(s)$$

(32)

Accompanied by its condition

$$\theta (0)=1$$

(33)

Using $\mathbb{J}$ -transformation on equation (32), we get

$$\frac{r}{w}\mathrm{\Theta }(r,w)-w\theta (0)-\mathbb{J}[s\theta (s){\theta }^{\prime }(s)]=\mathbb{J}[{\theta }^2(s)]$$

(34)

Substitute equation (33) into equation (34) to produce

$$\mathrm{\Theta }(r,w)=\frac{w^2}{r}+\frac{w}{r}\left[\mathbb{J}[s\theta (s){\theta }^{\prime }(s)+{\theta }^2(s)]\right]$$

(35)

Employ the $\mathbb{J}$ -inverse transformation to equation (35) to get

$$\theta (s)=1+{\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[s\theta (s){\theta }^{\prime }(s)+{\theta }^2(s)]\right]\right]$$

(36)

Suppose the intended solution is of the form

$$\theta (s)=\sum _{j=0}^{\mathrm{\infty }}{\theta }_j(s)$$

(37)

Substituting equation (37) into equation (36) will results in

$$\sum _{j=0}^{\mathrm{\infty }}{\theta }_j(s)=1+{\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[\sum _{j=0}^{\mathrm{\infty }}s{A}_j(s)+\sum _{j=0}^{\mathrm{\infty }}{B}_j(s)]\right]\right]$$

(38)

where the Adomian polynomials $A_j(s)$ and $B_j(s)$ represent the nonlinear terms $\theta (s){\theta }^{\prime }(s)$ and ${\theta }^2(s)$ , respectively.

We continue in a similar manner to obtain

$$\begin{array}{rl}{\theta }_0(s)& =1\\ {\theta }_1(s)& ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[s{A}_0(s)+B_0(s)]\right]\right]\\ {\theta }_2(s)& ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[s{A}_1(s)+B_1(s)]\right]\right]\\ {\theta }_3(s)& ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[s{A}_2(s)+B_2(s)]\right]\right]\end{array}$$

Finally,

$${\theta }_{j+1}(s)={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[s{A}_j(s)+B_j(s)]\right]\right],j\ge 0$$

(39)

Using equation (39), one can arrive at

$$\begin{array}{rl}{\theta }_1(s)& ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[s{A}_0(s)+B_0(s)]\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[s{\theta }_0(s){\theta }_0^{\prime }(s)+{\theta }_0^2(s)]\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}\left[1\right]\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{w^3}{r^2}\right]\\ & =s\\ {\theta }_2(s)& ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[s{A}_1(s)+B_1(s)]\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[s{\theta }_1(s){\theta }_0^{\prime }+s{\theta }_0(s)v_1^{\prime }(s)+2{\theta }_0(s){\theta }_1(s)]\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}\left[3s\right]\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{3w^4}{r^3}\right]\\ & =\frac{3s^2}{2}\\ {\theta }_3(s)& ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[s{A}_2(s)+B_2(s)]\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[s{\theta }_2(s){\theta }_0^{\prime }(s)+s{\theta }_1(s){\theta }_1^{\prime }(s)+s{\theta }_2^{\prime }(s){\theta }_0(s)+2{\theta }_0(s){\theta }_2(s)+{\theta }_1^2(s)]\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}\left[8s^2\right]\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{16w^5}{r^4}\right]\\ & =\frac{8s^3}{3}\end{array}$$

Hence, the exact solution $\theta (s)$ is given by

$$\begin{array}{rl}\theta (s)& =\sum _{j=0}^{\mathrm{\infty }}{\theta }_j(s).\\ & ={\theta }_0(s)+{\theta }_1(s)+{\theta }_2(s)+{\theta }_3(s)+\cdots \\ & =1+s+\frac{3s^2}{2}+\frac{8s^3}{3}+\frac{125s^4}{24}+\cdots \end{array}$$

In this particular case, because $\theta (s)$ cannot be clearly represented in terms of $s$ , subsequently, the implicit solution to equation (32) is provided by

$$\theta (s)=e^{s\theta }$$

As a result, the exact solution is in excellent agreement with the result obtained by El-Sayed. ¹⁸

Example 2

Consider the following nonlinear Riccati differential equation:

$$\frac{d\theta }{ds}=1-s^2+{\theta }^2(s)$$

(40)

Accompanied by its condition

$$\theta (0)=0$$

(41)

Applying $\mathbb{J}$ -transformation to equation (40), we obtain

$$\frac{r\mathrm{\Theta }(r,w)}{w}-w\theta (0)=\frac{w^2}{r}-\frac{2w^4}{r^3}+\mathbb{J}[{\theta }^2(s)]$$

(42)

Substitute equation (41) into equation (42) to produce

$$\mathrm{\Theta }(r,w)=\frac{w^3}{r^2}-\frac{2w^5}{r^4}+\frac{w}{r}\mathbb{J}[{\theta }^2(s)]$$

(43)

Employ the $\mathbb{J}$ -inverse transformation to equation (43) to get

$$\theta (s)=s-\frac{s^3}{3}+{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}[{\theta }^2(s)]\right]$$

(44)

Suppose the intended solution is of the form

$$\theta (s)=\sum _{j=0}^{\mathrm{\infty }}{\theta }_j(s)$$

(45)

Substituting equation (45) into equation (44) will results in

$$\sum _{j=0}^{\mathrm{\infty }}{\theta }_j(s)=s-\frac{s^3}{3}+{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}[\sum _{j=0}^{\mathrm{\infty }}{A}_j(s)]\right]$$

(46)

where the nonlinear term ${\theta }^2(s)$ is represented by the Adomian polynomial $A_j$ .

We continue in a similar manner to obtain

$$\begin{array}{rl}{\theta }_0(s)& =s-\frac{s^3}{3}\\ {\theta }_1(s)& ={\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}[A_0(s)]\right]\\ {\theta }_2(s)& ={\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}[A_1(s)]\right]\\ {\theta }_3(s)& ={\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}[A_2(s)]\right]\end{array}$$

Finally,

$${\theta }_{j+1}(s)={\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}[A_j(s)]\right],j\ge 0$$

(47)

Using equation (47), we arrive at

$$\begin{array}{rl}{\theta }_1(s)& ={\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}[A_0(s)]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}[{\theta }_0^2(s)]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}\left[{(s-\frac{s^3}{3})}^2\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}\left[s^2\right]\right]-\frac{2}{3}{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}\left[s^4\right]\right]+\frac{1}{9}{\mathbb{J}}^{-1}\left[\frac{w}{r}\mathbb{J}\left[s^6\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{2w^5}{r^4}\right]-\frac{2}{3}{\mathbb{J}}^{-1}\left[\frac{4!w^7}{r^6}\right]+\frac{1}{9}{\mathbb{J}}^{-1}\left[\frac{6!w^9}{r^8}\right]\\ & =\frac{s^3}{3}-\frac{2s^5}{15}+\frac{s^7}{63}\end{array}$$

It is clear from ${\theta }_1(s)$ that the components ${\theta }_0(s)$ contain one noise term. After canceling the noise term from ${\theta }_0(s)$ , the exact solution can be obtained from the remaining non-canceled term of ${\theta }_0(s)$ . Substituting this into equation (40) makes it simple to verify.

Hence, the exact solution $\theta (s)$ is given by

$$\theta (s)=s$$

As a result, the exact solution is in excellent agreement with the result obtained by El-Sayed. ¹⁸

Example 3

Consider the nonlinear second-order differential below: Rawashdeh and Obeidat ²⁴ :

$$\frac{d^2\theta }{d{s}^2}+{\left(\frac{d\theta }{ds}\right)}^2+{\theta }^2(s)=1-\sin (s)$$

(48)

Accompanied by its conditions

$$\theta (0)=0,{\theta }^{\prime }(0)=1$$

(49)

Using $\mathbb{J}$ -transformation on equation (48), we obtain

$$\frac{r^2\mathrm{\Theta }(r,w)}{w^2}-r\theta (0)-w{\theta }^{\prime }(0)+\mathbb{J}\left[{\left(\frac{d\theta }{ds}\right)}^2\right]+\mathbb{J}[{\theta }^2(s)]=\frac{w^2}{r}-\frac{w^3}{r^2+w^2}$$

(50)

Substituting equation (49) into equation (50) to obtain

$$\mathrm{\Theta }(r,w)=\frac{w^3}{r^2}+\frac{w^4}{r^3}-\frac{w^5}{r^2(r^2+w^2)}-\frac{w^2}{r^2}\mathbb{J}[{\left(\frac{d\theta }{ds}\right)}^2+{\theta }^2(s)]$$

(51)

Employ the $\mathbb{J}$ -inverse transformation to equation (51) to get

$$\theta (s)=\frac{s^2}{2!}+\sin (s)-{\mathbb{J}}^{-1}\left[\frac{w^2}{r^2}\mathbb{J}[{\left(\frac{d\theta }{ds}\right)}^2+{\theta }^2(s)]\right]$$

(52)

Suppose the intended solution is of the form

$$\theta (s)=\sum _{j=0}^{\mathrm{\infty }}{\theta }_j(s)$$

(53)

Substituting equation (53) into equation (52) will results in

$$\sum _{j=0}^{\mathrm{\infty }}{\theta }_j(s)=\frac{s^2}{2!}+\sin (s)-{\mathbb{J}}^{-1}\left[\frac{w^2}{r^2}\mathbb{J}[\sum _{j=0}^{\mathrm{\infty }}{A}_j+\sum _{j=0}^{\mathrm{\infty }}{B}_j]\right]$$

(54)

The nonlinear terms’ Adomian polynomials $(\frac{d\theta }{ds}{)}^2$ and ${\theta }^2(s)$ are denoted by $A_j$ and $B_j$ , respectively.

Finally,

$${\theta }_{j+1}(s)=-{\mathbb{J}}^{-1}\left[\frac{w^2}{r^2}\mathbb{J}[A_j+B_j]\right],j\ge 0$$

(55)

Using equation (55), we arrive at

$$\begin{array}{rl}{\theta }_0(s)& =\frac{s^2}{2!}+\sin (s)\\ {\theta }_1(s)& =-{\mathbb{J}}^{-1}\left[\frac{w^2}{r^2}\mathbb{J}[A_0+B_0]\right]\\ {\theta }_2(s)& =-{\mathbb{J}}^{-1}\left[\frac{w^2}{r^2}\mathbb{J}[A_1+B_1]\right]\\ {\theta }_3(s)& =-{\mathbb{J}}^{-1}\left[\frac{w^2}{r^2}\mathbb{J}[A_2+B_2]\right]\end{array}$$

Thus,

$$\begin{array}{rl}{\theta }_1(s)& =-{\mathbb{J}}^{-1}\left[\frac{w^2}{r^2}\mathbb{J}[A_0+B_0]\right]\\ & =-{\mathbb{J}}^{-1}\left[\frac{w^2}{r^2}\mathbb{J}[{\left({\theta }_0^{\prime }\right)}^2+{\theta }_0^2]\right]\\ & =-{\mathbb{J}}^{-1}\left[\frac{w^2}{r^2}\mathbb{J}\left[1\right]\right]+\cdots \\ & =-{\mathbb{J}}^{-1}\left[\frac{w^4}{r^3}\right]+\cdots \\ & =-\frac{s^2}{2!}\end{array}$$

Therefore, it can be observed that the non-canceled component of ${\theta }_0(s)$ still solves the provided differential equation by canceling the noise terms that emerge between ${\theta }_0(s)$ and ${\theta }_1(s)$ . This leads to an exact solution of the form:

$$\theta (s)=\sin (s)$$

As a result, the exact solution is in excellent agreement with the result obtained by El-Sayed. ¹⁸

Example 4

Consider the first-order nonlinear ODE of the form:

$$\frac{d\theta }{ds}-1={\theta }^2(s)$$

(56)

Accompanied by its condition:

$$\theta (0)=0$$

(57)

Applying $\mathbb{J}$ -transformation on equation (56), to obtain:

$$\frac{r}{w}\mathrm{\Theta }(r,w)-w\theta (0)-\frac{w^2}{r}=\mathbb{J}[{\theta }^2(s)]$$

(58)

Substitute equation (57) into equation (58) to produce

$$\mathrm{\Theta }(r,w)=\frac{w^3}{r^2}+\frac{w}{r}\left[\mathbb{J}[{\theta }^2(s)]\right]$$

(59)

Applying $\mathbb{J}$ -inverse transformation to equation (59), to get

$$\theta (s)=s+{\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[{\theta }^2(s)]\right]\right]$$

(60)

Suppose the intended solution is of the form

$$\theta (s)=\sum _{j=0}^{\mathrm{\infty }}{\theta }_j(s)$$

(61)

Substituting equation (61) into equation (60) will results in

$$\sum _{j=0}^{\mathrm{\infty }}{\theta }_j(s)=s+{\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[\sum _{j=0}^{\mathrm{\infty }}{A}_j(s)]\right]\right]$$

(62)

where the nonlinear term ${\theta }^2(s)$ is represented by the Adomian polynomial $A_j(s)$ .

We continue in a similar manner to obtain

$$\begin{array}{rl}{\theta }_0(s)& =s\\ {\theta }_1(s)& ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[A_0(s)]\right]\right]\\ {\theta }_2(s)& ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[A_1(s)]\right]\right]\\ {\theta }_3(s)& ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[A_2(s)]\right]\right]\end{array}$$

Finally,

$${\theta }_{j+1}(s)={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[A_j(s)]\right]\right],j\ge 0$$

(63)

Using equation (63), we can arrive at

$$\begin{array}{rl}{\theta }_1(s)& ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[A_0(s)]\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[{\theta }_0^2(s)]\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}\left[s^2\right]\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{2w^5}{r^4}\right]\\ & =\frac{1}{3}{s}^3\end{array}$$

Similarly,

$$\begin{array}{rl}{\theta }_2(s)& ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[A_1(s)]\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[2{\theta }_0(s){\theta }_1(s)]\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}\left[\frac{2s^4}{3}\right]\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{16w^7}{s^6}\right]\\ & =\frac{2s^5}{15}\\ {\theta }_3(s)& ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[A_2(s)]\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}[2{\theta }_0(s){\theta }_2(s)+{\theta }_1^2(t)]\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{w}{r}\left[\mathbb{J}\left[\frac{17s^6}{45}\right]\right]\right]\\ & ={\mathbb{J}}^{-1}\left[\frac{272w^9}{r^8}\right]\\ & =\frac{17s^7}{315}\end{array}$$

Hence, the exact solution $\theta (s)$ is given by

$$\begin{array}{rl}\theta (s)& =\sum _{j=0}^{\mathrm{\infty }}{\theta }_j(s)\\ & ={\theta }_0(s)+{\theta }_1(s)+{\theta }_2(s)+{\theta }_3(s)+\cdots \\ & =s+\frac{1}{3}{s}^3+\frac{2s^5}{15}+\frac{17s^7}{315}+\cdots \end{array}$$

Thus, the exact solution of equation (56) is given by

$$\theta (s)=\tan (s)$$

As a result, the exact solution is in excellent agreement with the result obtained by El-Sayed. ¹⁸

Remark. For some of Adomian’s terms calculations, we used Mathematica 13 and a CPU time range of 10–15 min to do these calculations.

Conclusion

One of the most intriguing differential equations of first order is the nonlinear Riccati differential equation, which we solve in this article using a unique strategy. It has applications in algebraic geometry, physics, and the theory of conformal mapping, among other branches of mathematics. Using the JADM, a higher-dimensional heat flow problem has been effectively resolved. We also provided exact solutions for three additional ODEs that are nonlinear. The results show that the convergence rate of the JADM is faster than other methods reported in the literature. The relevance of JADM was proved by its application in applied science and engineering fields. Furthermore, the effectiveness and applicability of the proposed technique were demonstrated when we applied it to multiple examples. The JADM may also be used to accurately solve other nonlinear ODEs and PDEs, including systems of ODEs and PDEs, which are frequently encountered in science and engineering, according to the study mentioned above. Thus, a deeper grasp of the real-world applications represented by these modeling issues will become evident from the NADM solutions to several situations.