New View on Nonlinear Picture Fuzzy Integral Equations

Abstract

In this article, we solve the second type of nonlinear Volterra picture fuzzy integral equation (NVPFIE) using an accelerated form of the Adomian decomposition method (ADM). Based on $(\alpha ,\delta ,\beta )$-cut, we convert the NVPFIE to the nonlinear Volterra integral equations in a crisp form. An accelerated version of the ADM is used to solve this transformed system, which is based on a new formula for the Adomian polynomial. The sufficient condition that guarantees a unique solution is obtained using this new Adomian polynomial, error estimates are given, and the convergence of the series solution is proven. Numerical cases are discussed to illustrate the effectiveness of this approach.

1. Introduction

In recent years, many scientists and academics have become interested in fuzzy integral equations of the second kind. These equations are frequently used in economic systems, approximation reasoning, fuzzy control, and fuzzy finance [1]. Integral equations and differential equations are also used to model dynamical systems in mathematical biology. In particular, indications concerning real phenomena described by PDEs should be considered [2,3], where analyses of dynamical systems modeling chemotaxis mechanisms are developed. In order to define fuzzy numbers and apply them to fuzzy control [4] and approximation reasoning problems [5], Zadeh [6] first established the idea of fuzzy sets. Dubois and Prade first introduced the concept of fuzzy functions [7]. Later, various approaches were suggested by Goetschel and Voxman [8], Kaleva [9], Nanda [10], and others. Various methods have been presented in recent years to resolve Volterra integral equations [11]. For nonlinear integral equations, Tricomi [12] was the first to present the successive approximations method. After Abbasbandy [13], who used it to solve fuzzy Volterra integral equations of the second kind, Liao [14] used the homotopy analysis method to address nonlinear problems. In order to directly solve a system of integral equations numerically, Babolian et al. [15] employed orthogonal triangular functions. Legendre wavelets have been used by Jafarian et al. [16] to resolve systems of linear integral equations. Integral equations were solved using the Taylor expansion method by Kanwal and Liu [17] and the variational iteration method by Xu [18]. Amawi and Qatanani have also looked into the analytical and numerical treatment of the fuzzy linear Fredholm integral equation [19,20]. Fuzzy Volterra integral equations have been solved by Hamaydi [21] using a variety of analytical and numerical methods.

Some technical, medical, and life devices and applications have data computation due to uncertainty in calculating measurements, which is the motivating factor behind combining fuzzy sets with integral equations. Furthermore, by converting these patterns as a result of not fully defining the variables that appear in these models, we turn to adding fuzzy properties to the variables that make the given function not a curve. The differentials and integrals have a lower and an upper limit, which affect the concepts of the integral equations by making the integral equation the result of an area with a lower and upper limit, and this is much better than the full definition of the variables. Working with a picture fuzzy model is much better than transferring models to a fuzzy form because picture fuzzy depends on three viewpoints or three characteristics that meet at the same time, namely the view of neutrality, the opposition, and the supporter. These make the accuracy of the calculation higher than the fuzzy. The process of calculating engineering and medical models depends on the opinion of an expert to transfer these models to a mathematical application. The ordinary integral equation is a very special case of the partial fractional integral equation when the opposing viewpoint is equal to zero and the pro and neutral are 100% in proportion. This does not achieve the process of converting any picture of equations. Using the picture fuzzy is more general than formulating it with the fuzzy integral equation because it is a special case of the picture fuzzy integral equation when the opposing viewpoint is equal to zero and the pro and neutral are equal. As a result, because the variables indicated by the equation are hazy, there is a high degree of idealism that does not occur in reality due to the blurring of the variables defined by the equation. Therefore, converting the integral equation to the picture fuzzy integral equation form is the best in terms of adding characteristics and the inaccuracy of the variables entering the equation. There must be an error in defining them or an attribute that affects the resulting algebraic operations within the integral equations. We have studied the integral equation and the fuzzy variables and unknowns inside it, as well as the algebraic equations inside the fuzzy equation. In addition, we constructed a new generalization by making the variables, unknowns, and algebraic operations inside the equation subject to the rules of picture fuzzy, as the equation can be studied from three points of view. We have studied the generalization of the classical fuzzy Adomian method, which is based on the fuzzy variables of the Volterra-type integral equation, and we have generalized a new concept of these variables and algebraic operations to be picture fuzzy. Suvankar found an $\alpha$-level in a differential equation that contains a linear and a nonlinear part in more than one equation, which was solved in an Adomian way by applying the same technique that was previously used to transform the differential equation with a linear and a nonlinear part into a fuzzy equation [22]. We have discovered a new generalization of fuzzy concepts using picture fuzzy, as the concept covers all possible characteristics by which variables can be described, using three of the membership, and we cover most of the possible ways to describe variables in picture fuzzy euclidean space. This is a generalization of the ordinary differential equation using fuzzy concepts, then solving it with fuzzy Adomian.

In this study, we propose a new version of ADM for solving NVPFIE based on the $(\alpha ,\delta ,\beta )$-$cut$:

$$\tilde{u}\left(y\right)=\tilde{g}\left(y\right)\oplus {\int }_0^{y}k(y,\tau )\odot f\left(\tilde{u}\left(\tau \right)\right)d\tau .$$

(1)

If $\tilde{u},\tilde{g}:A$ = $\left[a,b\right]\times \left[c,d\right]\to$$P{E}^1$ are picture fuzzy continuous functions, k:$A^2\to$$P{E}^1,f$$:P{E}^1$→$P{E}^1$ are continuous functions on $P{E}^1,$ where $P{E}^1$ is called the set of all Picture Fuzzy Numbers (PFNs).

This paper is structured as follows: In Section 2, some fundamental definitions for picture fuzzy integers, picture fuzzy functions, and picture fuzzy integrals are introduced. A parametric form of the NVPFIE is introduced in Section 3, and subsequently ADM is used to solve Equation (1). In Section 4, we discuss the existence, uniqueness, and convergence of the proposed method. Finally, a numerical case using MATHEMATICA software is introduced in Section 5. In Section 6, we discuss random simulated data.

2. Preliminaries

In this section, we briefly introduce a quick overview of some definitions and a lemma which will be used throughout the rest of the paper.

Definition 1

([23]). A Picture Fuzzy Set (PFS) A is an object of the form $A=\left\{\right(x,{\mu }_{{}_A}\left(x\right),{\eta }_{{}_A}\left(x\right)$, ${\upsilon }_{{}_A}\left(x\right):x\in X\}$ in a universe X, where ${\mu }_{{}_A}\left(x\right):X\to \left[0,1\right],{\eta }_{{}_A}\left(x\right):X\to \left[0,1\right]$ and ${\upsilon }_{{}_A}\left(x\right):X\to \left[0,1\right]$ are denoted by, respectively, the degree of positive membership, the degree of neutral membership, and the degree of negative membership of $x\in A$. ${\mu }_{{}_A}\left(x\right),{\eta }_{{}_A}\left(x\right),{\upsilon }_{{}_A}\left(x\right)$ must satisfy the condition ${\mu }_{{}_A}\left(x\right)+{\eta }_{{}_A}\left(x\right)+{\upsilon }_{{}_A}\left(x\right)\le 1\forall x\in X.$ Then, $\forall x\in X,$ the degree of refusal membership ${\rho }_{{}_A}\left(x\right)=$$\left(1-({\mu }_{{}_A}\left(x\right)+{\eta }_{{}_A}\left(x\right)+{\upsilon }_{{}_A}\left(x\right))\right)$.

Definition 2

([23]). A Picture Fuzzy Number (PFN) for a fixed $x\in A,({\mu }_{{}_A}\left(x\right),{\eta }_{{}_A}\left(x\right),{\upsilon }_{{}_A}\left(x\right),{\rho }_{{}_A}$ $\left(x\right))$ is called a PFN where ${\mu }_{{}_A}\left(x\right)\in \left[0,1\right],{\eta }_{{}_A}\left(x\right)\in \left[0,1\right],{\upsilon }_{{}_A}\left(x\right)\in \left[0,1\right],{\rho }_{{}_A}\left(x\right)\in \left[0,1\right]and{\mu }_{{}_A}\left(x\right)$ $+{\eta }_{{}_A}\left(x\right)+{\upsilon }_{{}_A}\left(x\right)+{\rho }_{{}_A}\left(x\right)=1.$ Simply, the PFN can be represented as $({\mu }_{{}_A}\left(x\right),{\eta }_{{}_A}\left(x\right),{\upsilon }_{{}_A}\left(x\right)).$

Definition 3

([24]). Let $(\alpha ,\delta ,\beta )$-$cut$ of PFN A of a universe set X be given as a crisp subset defined by $C_{\alpha ,\delta ,\beta }\left(A\right)=\left\{x:x\in X,suchthat{\mu }_{{}_A}\left(x\right)\ge \alpha ,{\eta }_{{}_A}\left(x\right)\le \delta ,{\upsilon }_{{}_A}\left(x\right)\le \beta \right\}$ such that $(\alpha ,\delta ,\beta )\in \left[0,1\right]$ with $\alpha +\delta +\beta \le 1.$ That is, positive membership function, ${\alpha }_{{}_{A+}}\left(A\right)=\left\{x:x\in X:{\mu }_{{}_A}\left(x\right)\ge \alpha \right\},$ Neutral membership function, ${\delta }_{{}_{A\pm }}\left(A\right)=\left\{x:x\in X:{\eta }_A\left(x\right)\le \delta \right\},$ Negative membership function, ${\beta }_{{}_{A-}}\left(A\right)=\left\{x:x\in X:{\upsilon }_{{}_A}\left(x\right)\le \beta \right\}.$

Definition 4

([24]). A triangle Picture Fuzzy Set is represented by $A=\{[p_{{}_1},q_{{}_1},r_{{}_1}:{\gamma }_{{}_1}],[\stackrel{"}{p_{{}_1}},q_{{}_1},\stackrel{"}{r_{{}_1}}:{\vartheta }_{{}_1}],[{\acute{p}}_{{}_1},q_{{}_1},{\acute{r}}_{{}_1}:{\zeta }_{{}_1}]\}$ and defined by:

$$\begin{array}{ccc}\hfill {\mu }_{{}_A}\left(x\right)& =& \left\{\begin{array}{c}{\gamma }_1\frac{x-p_{{}_1}}{q_{{}_1}-p_{{}_1}}{,p}_{{}_1}\le x\le q_{{}_1}\\ {\gamma }_1\frac{r_{{}_1}-x}{r_{{}_1}-q_{{}_1}}{,q_{{}_1}\le x\le r}_{{}_1}\\ 0,otherwise\end{array}\right\}{,\eta }_{{}_A}\left(x\right)=\left\{\begin{array}{c}\frac{{(q}_{{}_1}{-x)+(x-\stackrel{"}{p_{{}_1}})\vartheta }_1}{q_1-\stackrel{"}{p_{{}_1}}},\stackrel{"}{p_{{}_1}}\le x\le q_{{}_1}\\ \frac{(x-q_{{}_1})+(\stackrel{"}{r_{{}_1}}-x){\vartheta }_1}{\stackrel{"}{r_{{}_1}}-q_1},q_{{}_1}\le x\le \stackrel{"}{r_{{}_1}}\\ 1,otherwise\end{array}\right\},\hfill \\ & & \\ \hfill {\upsilon }_{{}_A}\left(x\right)& =& \left\{\begin{array}{c}\frac{(q_{{}_1}-x)+(x-{\acute{p}}_{{}_1}){\zeta }_{{}_1}}{q_{{}_1}-{\acute{p}}_{{}_1}}{,\acute{p}}_{{}_1}\le x\le q_{{}_1}\\ \frac{(x-q_{{}_1})+({\acute{r}}_{{}_1}-x){\zeta }_1}{{\acute{r}}_{{}_1}-q_{{}_1}}{,q_{{}_1}\le x\le \acute{r}}_{{}_1}\\ 1,otherwise\end{array}\right\}.\hfill \end{array}$$

Definition 5

([25]). If $\stackrel{\sim }{A}$ and $\stackrel{\sim }{B}$ are two PFSs, then $\stackrel{\sim }{A}\oplus \stackrel{\sim }{B}$ can be formed by the ($\alpha ,\delta ,\beta )$-$cut$ of Ã$\oplus \stackrel{\sim }{B}$, where [$\stackrel{\sim }{A}$]${}_{(\alpha ,\delta ,\beta )}$ and [$\stackrel{\sim }{B}$]${}_{(\alpha ,\delta ,\beta )}$ are defined as:

$$\begin{array}{l}{\left[\tilde{A}\oplus \tilde{B}\right]}_{{}_{(}\alpha ,\delta ,\beta )}=\{{\left[a_{{}_1}+b_{{}_1},a_{{}_2}+b_{{}_2}\right]}_{\alpha },{\left[\stackrel{"}{a_{{}_1}}+\stackrel{"}{b_{{}_1}},\stackrel{"}{a_{{}_2}}+\stackrel{"}{b_{{}_2}}\right]}_{\delta },{\left[{\acute{a}}_{{}_1}+{\acute{b}}_{{}_1},{\acute{a}}_{{}_2}+{\acute{b}}_{{}_2}\right]}_{\beta }\}.\\ {\left[\tilde{A}\odot \tilde{B}\right]}_{{}_{(}\alpha ,\delta ,\beta )}=\{{\left[\begin{array}{c}min(a_{{}_1}{b}_{{}_1},a_{{}_1}{b}_{{}_2},a_{{}_2}{b}_{{}_1},a_{{}_2}{b}_{{}_2}),\\ max(a_{{}_1}{b}_{{}_1},a_{{}_1}{b}_{{}_2},a_{{}_2}{b}_{{}_1},a_{{}_2}{b}_{{}_2})\end{array}\right]}_{\alpha },{\left[\begin{array}{c}min(\stackrel{"}{a_{{}_1}}\stackrel{"}{b_{{}_1}},\stackrel{"}{a_{{}_1}}\stackrel{"}{b_{{}_2}},\stackrel{"}{a_{{}_2}}\stackrel{"}{b_{{}_1}},\stackrel{"}{a_{{}_2}}\stackrel{"}{b_{{}_2}}),\\ max(\stackrel{"}{a_{{}_1}}\stackrel{"}{b_{{}_1}},\stackrel{"}{a_{{}_1}}\stackrel{"}{b_{{}_2}},\stackrel{"}{a_{{}_2}}\stackrel{"}{b_{{}_1}},\stackrel{"}{a_{{}_2}}\stackrel{"}{b_{{}_2}})\end{array}\right]}_{\delta },\\ {\left[\begin{array}{c}min({\acute{a}}_{{}_1}{\acute{b}}_{{}_1},{\acute{a}}_{{}_1}{\acute{b}}_{{}_2},{\acute{a}}_{{}_2}{\acute{b}}_{{}_1},{\acute{a}}_{{}_2}{\acute{b}}_{{}_2}),\\ max({\acute{a}}_{{}_1}{\acute{b}}_{{}_1},{\acute{a}}_{{}_1}{\acute{b}}_{{}_2},{\acute{a}}_{{}_2}{\acute{b}}_{{}_1},{\acute{a}}_{{}_2}{\acute{b}}_{{}_2})\end{array}\right]}_{\beta }\}.\end{array}$$

Definition 6

([26]). The distance for two PFNs is defined using the Hausdroff metric as:

$$\begin{array}{ccc}\hfill D\left(\right(a,\mu ,\eta ,\upsilon ),(b,\mu ,\eta ,\upsilon \left)\right)& =& max({\alpha }^{*},{\delta }^{*},{\beta }^{*}),where\hfill \\ & & \left[\begin{array}{c}{\alpha }^{*}=\underset{\alpha \in \left[0,1\right]}{sup}max\left\{\left|a_{{}_{1-}}^{\alpha }-a_{{}_{2-}}^{\alpha }\right|,\left|b_{{}_{1+}}^{\alpha }-b_{{}_{2+}}^{\alpha }\right|\right\}\\ {\delta }^{*}=\underset{\delta \in \left[0,1\right]}{sup}max\left\{\left|a_{{}_{1-}}^{\delta }-a_{{}_2-}^{\delta }\right|,\left|b_{{}_{1+}}^{\delta }-b_{{}_{2+}}^{\delta }\right|\right\}\\ {\beta }^{*}=\underset{\beta \in \left[0,1\right]}{sup}max\left\{\left|a_{{}_{1-}}^{\beta }-a_{{}_{2-}}^{\beta }\right|,\left|b_{{}_{1+}}^{\beta }-b_{{}_{2+}}^{\beta }\right|\right\}\end{array}\right]\hfill \end{array}$$

where the $\left(\right(\alpha ,\delta ,\beta )$-$cut$) distance of ${\upsilon }_{{}_A}$ is

$$[{\upsilon }_{{}_{\mathrm{A}}},{\upsilon }_{{}_{\overline{A}}}]=\left\{\begin{array}{c}{[(a+\frac{(b-a)\alpha }{{\upsilon }_{{}_A}}),(c-\frac{(c-b)\alpha }{{\upsilon }_{{}_A}})]}_{\alpha }\\ \left[\right(b-\frac{\delta -{\upsilon }_{{}_1}}{1-{\upsilon }_{{}_1}}(b-a){),(b+\frac{\delta -{\upsilon }_{{}_1}}{1-{\upsilon }_{{}_1}}(c-b))]}_{\delta }\\ \left[\right(b-\frac{\beta -{\xi }_{{}_1}}{1-{\xi }_{{}_1}}(b-a){),(b+\frac{\beta -{\upsilon }_{{}_1}}{1-{\upsilon }_{{}_1}}(c-b))]}_{\beta }\end{array}\right\}$$

Lemma 1.

Let f$:A\to P{E}^1$. Then, f is (PFH)-integrable if ($\underline{f}$, $\overline{f})$ is Henstock integrable ∀$X\in \left[0,1\right].$

$$\begin{array}{ccc}& & {\left[\left(PFH\right){\int }_a^{b}f\left(y\right)dy\right]}^{r_1,r_2,r_3}\hfill \\ & =& \left\{\begin{array}{c}\left[\left(H\right){\int }_a^b\underline{f}(y{,r}_{{}_1})dy,(H){\int }_a^b\overline{f}(y{,r}_{{}_1})dy\right]\\ \\ \left[\left(H\right){\int }_a^b\underline{f}\left({y,r}_{{}_2}\right)dy,\left(H\right){\int }_a^b\overline{f}{(y,r}_{{}_2})dy\right]\\ \\ \left[\left(H\right){\int }_a^b\underline{f}\left({y,r}_{{}_3}\right)dy,\left(H\right){\int }_a^b\overline{f}(y{,r}_{{}_3})dy\right]\end{array}\right\}\hfill \end{array}$$

From the definition of $\alpha$-$cut$ of three membership, since every $r_{{}_{{}_1}}$, $r_{{}_{{}_2}}$, $r_{{}_3}$ level has a lower pound and an upper pound for the function, the integration has three levels $r_{{}_{{}_1}}$, $r_{{}_{{}_2}}$, $r_{{}_3}$. In addition, f is continuous, so $\underline{f}(.,.,r)$ and $\overline{f}(.,.,r)$ are continuous for X$\in \left[0,1\right]$. Thus, they are defined as Henstock integrable and hence f becomes (PFH)-integrable.

3. Parametric Form of NVPFIE

This section introduces the parametric formula for the NVPFIE (1), then applies ADM to solve it, considering $\left[u\right(y,\alpha )=(\underline{u}(y,\alpha ),\overline{u}(y,\alpha ))],$ $[u(y,\delta )=(\underline{u}(y,\delta ),\overline{u}(y,\delta ))],[u(y,\beta )=(\underline{u}(y,\beta ),\overline{u}(y,\beta ))]$ and $\left[g\right(y,\alpha )=(\underline{g}(y,\alpha ),\overline{g}(y,\alpha ))],$ $[g(y,\delta )=(\underline{g}(y,\delta ),\overline{g}(y,\delta ))],\left[g\right(y,\beta )=(\underline{g}(y,\beta ),\overline{g}(y,\beta )]$ to be the parametric forms of $\tilde{u}\left(y\right)$ and $\tilde{g}\left(y\right)$, respectively, in Equation (1) as follows:

Let $\tau \in A$. Then,

$$\left\{\begin{array}{c}\left[\begin{array}{c}H(\underline{u}(\tau ,\alpha ),\overline{u}(\tau ,\alpha ))=min\left\{f\left({\beta }_{{}_1}\right):\underline{u}(\tau ,\alpha )\le {\beta }_{{}_1}\le \overline{u}(\tau ,\alpha )\right\}\\ F(\underline{u}(\tau ,\alpha ),\overline{u}(\tau ,\alpha ))=max\left\{f\left({\beta }_{{}_1}\right):\underline{u}(\tau ,\alpha )\le {\beta }_{{}_1}\le \overline{u}(\tau ,\alpha )\right\}\end{array}\right],\hfill \\ \left[\begin{array}{c}H(\underline{u}(\tau ,\delta ),\overline{u}(\tau ,\delta ))=min\left\{f\left({\beta }_{{}_2}\right):\underline{u}(\tau ,\delta )\le {\beta }_{{}_2}\le \overline{u}(\tau ,\delta )\right\}\\ F(\underline{u}(\tau ,\delta ),\overline{u}(\tau ,\delta ))=max\left\{f\left({\beta }_{{}_2}\right):\underline{u}(\tau ,\delta ))\le {\beta }_{{}_2}\le \overline{u}(\tau ,\delta )\right\}\end{array}\right],\hfill \\ \left[\begin{array}{c}H(\underline{u}(\tau ,\beta ),\overline{u}(\tau ,\beta ))=min\left\{f\left({\beta }_{{}_3}\right):\underline{u}(\tau ,\beta )\le {\beta }_{{}_3}\le \overline{u}(\tau ,\beta )\right\}\\ F(\underline{u}(\tau ,\beta ),\overline{u}(\tau ,\beta ))=max\left\{f\left({\beta }_{{}_3}\right):\underline{u}(\tau ,\beta )\le {\beta }_{{}_3}\le \overline{u}(\tau ,\beta )\right\}\end{array}\right].\hfill \end{array}\right\}$$

$$\begin{array}{ccc}& & \left[\begin{array}{c}K(y,\tau )f\left(\overline{u}(\tau ,\alpha )\right)=\left\{\begin{array}{c}k(y,\tau )F(\underline{u}(\tau ,\alpha ),\overline{u}(\tau ,\alpha )),ifk(y,\tau )\ge 0,\\ \\ k(y,\tau )H\left(\underline{u}(\tau ,\alpha ),\overline{u}(\tau ,\alpha )\right),ifk(y,\tau )\prec 0.\end{array}\right\}\\ K(y,\tau )f\left(\underline{u}(\tau ,\alpha )\right)=\left\{\begin{array}{c}k(y,\tau )H\left(\underline{u}(\tau ,\alpha ),\overline{u}(\tau ,\alpha )\right),ifk(y,\tau )\ge 0,\\ \\ k(y,\tau )F\left(\underline{u}(\tau ,\alpha ),\overline{u}(\tau ,\alpha )\right),ifK(y,\tau )\prec 0.\end{array}\right\}\end{array}\right]\hfill \end{array}$$

$$\begin{array}{ccc}& & \left[\begin{array}{c}K(y,\tau )f\left(\overline{u}(\tau ,\delta )\right)=\left\{\begin{array}{c}k(y,\tau )F(\underline{u}(\tau ,\delta ),\overline{u}(\tau ,\delta )),ifk(y,\tau )\ge 0,\\ \\ k(y,\tau )H(\underline{u}(\tau ,\delta ),\overline{u}(\tau ,\delta )),ifk(y,\tau )\prec 0.\end{array}\right\}\\ K(y,\tau )f\left(\underline{u}(\tau ,\delta )\right)=\left\{\begin{array}{c}k(y,\tau )H\left(\underline{u}(\tau ,\delta ),\overline{u}(\tau ,\delta )\right),ifk(y,\tau )\ge 0,\\ \\ k(y,\tau )F\left(\underline{u}(\tau ,\delta ),\overline{u}(\tau ,\delta )\right),ifk(y,\tau )\prec 0.\end{array}\right\}\end{array}\right]\hfill \\ & & \\ & & \left[\begin{array}{c}K(y,\tau )f(\overline{u}(\tau ,\beta ))=\left\{\begin{array}{c}k(y,\tau )F(\underline{u}(\tau ,\beta ),\overline{u}(\tau ,\beta )),ifk(y,\tau )\ge 0,\\ \\ k(y,\tau )H(\underline{u}(\tau ,\beta ),\overline{u}(\tau ,\beta )),ifk(y,\tau )\prec 0.\end{array}\right\}\\ K(y,\tau )f(\underline{u}(\tau ,\beta ))=\left\{\begin{array}{c}k(y,\tau )H\left(\underline{u}(\tau ,\beta ),\overline{u}(\tau ,\beta )\right),ifk(y,\tau )\ge 0,\\ \\ k(y,\tau )F\left(\underline{u}(\tau ,\beta ),\overline{u}(\tau ,\beta )\right),ifk(y,\tau )\prec 0.\end{array}\right\}\end{array}\right]\hfill \end{array}$$

For each $0\le r\le 1$ and $K(t,\tau )\ge 0$, the functions $f({\beta }_{{}_1})$ are increasing for ${\beta }_1\in \left[\underline{u}(\tau ,r),\overline{u}(\tau ,r)\right]$, $f({\beta }_{{}_2})$ are increasing for ${\beta }_{{}_2}\in \left[\underline{u}(\tau ,\stackrel{"}{r}),\overline{u}(\tau ,\stackrel{"}{r})\right]$, and $f({\beta }_3)$ are increasing for ${\beta }_3\in \left[\underline{u}(\tau ,\acute{r}),\overline{u}(\tau ,\acute{r})\right].$ Then, parametric form that explains (1) becomes:

$$\begin{array}{c}\left\{\begin{array}{c}\underline{u}(y,\alpha )=\underline{g}(y,\alpha )+{\int }_0^{y}k(y,\tau )f\left(\underline{u}(\tau ,\alpha )\right)d\tau \\ \\ \overline{u}(y,\alpha )=\overline{g}(y,\alpha )+{\int }_0^{y}k(y,\tau )f\left(\overline{u}(\tau ,\alpha )\right)d\tau \end{array}\right\},\left\{\begin{array}{c}\underline{u}(y,\delta )=\underline{g}(y,\delta )+{\int }_0^{y}k(y,\tau )f\left(\underline{u}(\tau ,\delta )\right)d\tau \\ \\ \overline{u}(y,\delta )=\overline{g}(y,\delta )+{\int }_0^{y}k(y,\tau )f\left(\overline{u}(\tau ,\delta )\right)d\tau \end{array}\right\}\\ \\ \left\{\begin{array}{c}\underline{u}(y,\beta )=\underline{g}(y,\beta )+{\int }_0^{y}k(y,\tau )f\left(\underline{u}(\tau ,\beta )\right)d\tau \\ \\ \overline{u}(y,\beta )=\overline{g}(y,\beta )+{\int }_0^{y}k(y,\tau )f\left(\overline{u}(\tau ,\beta )\right)d\tau \end{array}\right\}\end{array}$$

Equation (1) may be seen to transform into a system of nonlinear picture Volterra equations in the crisp form.

4. Accelerated Solution of ADM for Solving the NVPFIE

Now, ADM [27] will be explained as a numerical algorithm to approximate the solution of this system of nonlinear integral equations in the crisp form. Then, an approximate solution to $u(x,\alpha ),u(x,\delta ),u(x,\beta )$ will be found.

The ADM assumes an infinite series solution for the unknown functions $(\underline{u}(y,\alpha ),\overline{u}(y,\alpha )),$ $(\underline{u}(y,\delta ),\overline{u}(y,\delta ))$, and $(\underline{u}(y,\beta ),\overline{u}(y,\beta ))$, given by:

$$\left\{\begin{array}{c}\left[\begin{array}{c}\underline{u}(y,\alpha )=\sum _{i=0}^{\infty }{\underline{u}}_i(y,\alpha )\\ \\ \overline{u}(y,\alpha )=\sum _{i=0}^{\infty }{\overline{u}}_i(y,\alpha )\end{array}\right],\left[\begin{array}{c}\underline{u}(y,\delta )=\sum _{i=0}^{\infty }{\underline{u}}_i(y,\delta )\\ \\ \overline{u}(y,\delta )=\sum _{i=0}^{\infty }{\overline{u}}_i(y,\delta )\end{array}\right]\\ \\ \left[\begin{array}{c}\underline{u}(y,\delta )=\sum _{i=0}^{\infty }{\underline{u}}_i(y,\delta )\\ \\ \overline{u}(y,\delta )=\sum _{i=0}^{\infty }{\overline{u}}_i(y,\delta )\end{array}\right].\end{array}\right\}$$

(2)

The nonlinear operators $\left[f(\underline{u}(\tau ,\alpha )\right),f(\overline{u}(\tau ,\alpha )\left)\right],\left[f(\underline{u}(\tau ,\delta )\right),f\left(\overline{u}(\tau ,\delta )\right)]$, and $[f\left(\underline{u}(\tau ,\beta )\right),f\left(\overline{u}(\tau ,\beta )\right)]$ are given by

$$\left[\begin{array}{c}\left\{\begin{array}{c}f\left(\underline{u}(\tau ,\alpha )\right)=\sum _{n=0}^{\infty }{\underline{A}}_n({\underline{u}}_{{}_0}(\tau ,\alpha ),{\underline{u}}_{{}_1}{(\tau ,\alpha ),.,\underline{u}}_{{}_n}(\tau ,\alpha ))\\ \\ f\left(\overline{u}(\tau ,\alpha )\right)=\sum _{n=0}^{\infty }{\overline{A}}_n({\overline{u}}_{{}_0}(\tau ,\alpha ),{\overline{u}}_{{}_1}(\tau ,\alpha ),..,{\overline{u}}_{{}_n}(\tau ,\alpha ))\end{array}\right\}\\ \left\{\begin{array}{c}f\left(\underline{u}(\tau ,\delta )\right)=\sum _{n=0}^{\infty }{\underline{A}}_n({\underline{u}}_{{}_0}(\tau ,\delta ),{\underline{u}}_{{}_1}(\tau ,\delta ),.,{\underline{u}}_{{}_n}(\tau ,\delta ))\\ \\ f\left(\overline{u}(\tau ,\delta )\right)=\sum _{n=0}^{\infty }{\overline{A}}_n({\overline{u}}_{{}_0}(\tau ,\delta ),{\overline{u}}_{{}_1}(\tau ,\delta ),..,{\overline{u}}_{{}_n}(\tau ,\delta ))\end{array}\right\}\\ \left\{\begin{array}{c}f\left(\underline{u}(\tau ,\beta )\right)=\sum _{n=0}^{\infty }{{\underline{A}}_n(\underline{u}}_{{}_0}{(\tau ,\beta ),\underline{u}}_{{}_1}{(\tau ,\beta ),.,\underline{u}}_{{}_n}(\tau ,\beta ))\\ \\ f\left(\overline{u}(\tau ,\beta )\right)=\sum _{n=0}^{\infty }{\overline{A}}_n({\overline{u}}_{{}_0}(\tau ,\beta ),{\overline{u}}_{{}_1}(\tau ,\beta ),..,{\overline{u}}_{{}_n}(\tau ,\beta ))\end{array}\right\}\end{array}\right]$$

(3)

where $A_n=({\underline{A}}_n,{\overline{A}}_n),n\ge 0,$ are the so-called Adomian polynomials, defined by:

$$\left\{\begin{array}{c}{\underline{A}}_n({\underline{u}}_{{}_0},{\underline{u}}_{{}_1},......,{\underline{u}}_{{}_n})=\frac{1}{n!}\frac{d^n}{d{\lambda }^n}{\left[f\left(\sum _{i=0}^{\infty }{\lambda }^i{\underline{u}}_i\right)\right]}_{\lambda =0}\\ \\ {\overline{A}}_n({\overline{u}}_{{}_0},{\overline{u}}_1,......,{\overline{u}}_{{}_n})=\frac{1}{n!}\frac{d^n}{d{\lambda }^n}{\left[f\left(\sum _{i=0}^{\infty }{\lambda }^i{\overline{u}}_i\right)\right]}_{\lambda =0}\end{array}\right\}$$

where $\lambda$ $⪰0$ is a crisp parameter.

Substituting (3) and (4) into (2), we obtain

$$\left[\begin{array}{c}\left\{\begin{array}{c}\sum _{n=0}^{\infty }{\underline{u}}_n(y,\alpha )=\underline{g}(y,\alpha )+{\int }_0^{y}k(y,\tau )\sum _{n=0}^{\infty }{\underline{A}}_n({\underline{u}}_{{}_0}(\tau ,\alpha ),....,{\underline{u}}_{{}_n}(\tau ,\alpha ))d\tau \\ \\ \\ \sum _{n=0}^{\infty }{\overline{u}}_{{}_n}(y,\alpha )=\overline{g}(y,\alpha )+{\int }_0^{y}k(y,\tau )\sum _{n=0}^{\infty }{\overline{A}}_n({\overline{u}}_{{}_0}(\tau ,\alpha ),..,{\overline{u}}_n(\tau ,\alpha ))d\tau \end{array}\right\}\\ \left\{\begin{array}{c}\sum _{n=0}^{\infty }{\underline{u}}_n(y,\delta )=\underline{g}(y,\delta )+{\int }_0^{y}k(y,\tau )\sum _{n=0}^{\infty }{\underline{A}}_n({\underline{u}}_{{}_0}(\tau ,\delta ),...,{\underline{u}}_{{}_n}(\tau ,\delta ))d\tau \\ \\ \\ \sum _{n=0}^{\infty }{\overline{u}}_n(y,\delta )=\overline{g}(y,\delta )+{\int }_0^{y}k(y,\tau )\sum _{n=0}^{\infty }{\overline{A}}_n({\overline{u}}_{{}_0}(\tau ,\delta ),....,{\overline{u}}_{{}_n}(\tau ,\delta ))d\tau \end{array}\right\}\\ \left\{\begin{array}{c}\sum _{n=0}^{\infty }{\underline{u}}_n(y,\beta )=\underline{g}(y,\beta )+{\int }_0^{y}k(y,\tau )\sum _{n=0}^{\infty }{\underline{A}}_n({\underline{u}}_{{}_0}(\tau ,\beta ),...,{\underline{u}}_{{}_n}(\tau ,\beta ))d\tau \\ \\ \\ \sum _{n=0}^{\infty }{\overline{u}}_n(y,\beta )=\overline{g}(y,\beta )+{\int }_0^{y}k(y,\tau )\sum _{n=0}^{\infty }{\overline{A}}_n({\overline{u}}_{{}_0}(\tau ,\beta ),...,{\overline{u}}_{{}_n}(\tau ,\beta ))d\tau \end{array}\right\}\end{array}\right]$$

(4)

The components [${\underline{u}}_n(y,\alpha ),{\overline{u}}_{{}_n}(y,\alpha )],[{\underline{u}}_n(y,\delta ),{\overline{u}}_n(y,\delta )]$, and $[{\underline{u}}_n(y,\beta ),{\overline{u}}_n(y,\beta )],n\ge 0$ are calculated using the following relations:

$$\begin{array}{ccc}& & \left\{\begin{array}{c}{\underline{u}}_{{}_0}(y,\alpha )=\underline{g}(y,\alpha )\\ {\underline{u}}_{{}_1}(y,\alpha )={\int }_0^{y}k(y,\tau ){\underline{A}}_0\left({\underline{u}}_{{}_0}(\tau ,\alpha )\right)d\tau \\ \\ \\ {\underline{u}}_{{}_{n+1}}(y,\alpha )={\int }_0^{y}k(y,\tau ){\underline{A}}_n({\underline{u}}_{{}_0}(\tau ,\alpha ),{\underline{u}}_1(\tau ,\alpha ),......,{\underline{u}}_n(\tau ,\alpha ))d\tau \end{array}\right\}\hfill \\ & & \\ & & \left\{\begin{array}{c}{\overline{u}}_{{}_0}(y,\alpha )=\overline{g}(y,\alpha )\\ {\overline{u}}_{{}_1}(y,\alpha )={\int }_0^{y}k(y,\tau ){\overline{A}}_0\left({\overline{u}}_{{}_0}(\tau ,\alpha )\right)d\tau \\ \\ \\ {\overline{u}}_{{}_{n+1}}(y,\alpha )={\int }_0^{y}k(y,\tau ){\overline{A}}_n({\overline{u}}_0(\tau ,\alpha ),{\overline{u}}_1(\tau ,\alpha ),......,{\overline{u}}_n(\tau ,\alpha ))d\tau \end{array}\right\}\hfill \end{array}$$

(5)

similarly,

$$\begin{array}{ccc}& & \left\{\begin{array}{c}{\underline{u}}_{{}_0}(y,\delta )=\underline{g}(y,\delta )\\ {\underline{u}}_{{}_1}(y,\delta )={\int }_0^{y}k(y,\tau ){\underline{A}}_0\left({\underline{u}}_{{}_0}(\tau ,\delta )\right)d\tau \\ \\ \\ {\underline{u}}_{{}_{n+1}}(y,\delta )={\int }_0^{y}k(y,\tau ){\underline{A}}_n({\underline{u}}_{{}_0}(\tau ,\delta ),{\underline{u}}_1(\tau ,\delta ),......,{\underline{u}}_n(\tau ,\delta ))d\tau \end{array}\right\}\hfill \\ & & \\ & & \left\{\begin{array}{c}{\overline{u}}_{{}_0}(y,\delta )=\overline{g}(y,\delta )\\ {\overline{u}}_{{}_1}(y,\delta )={\int }_0^{y}k(y,\tau ){\overline{A}}_0\left({\overline{u}}_{{}_0}(\tau ,\delta )\right)d\tau \\ \\ \\ {\overline{u}}_{{}_{n+1}}(y,\delta )={\int }_0^{y}k(y,\tau ){\overline{A}}_n({\overline{u}}_0(\tau ,\delta ),{\overline{u}}_1(\tau ,\delta ),......,{\overline{u}}_n(\tau ,\delta ))d\tau \end{array}\right\}\hfill \end{array}$$

(6)

and

$$\begin{array}{ccc}& & \left\{\begin{array}{c}{\underline{u}}_{{}_0}(y,\beta )=\underline{g}(y,\beta )\\ {\underline{u}}_{{}_1}(y,\beta )={\int }_0^{y}k(y,\tau ){\underline{A}}_0\left({\underline{u}}_{{}_0}(\tau ,{\acute{r}}_{{}_1})\right)d\tau \\ \\ \\ {\underline{u}}_{{}_{n+1}}(y,\beta )={\int }_0^{y}k(y,\tau ){\underline{A}}_n({\underline{u}}_{{}_0}(\tau ,{\acute{r}}_{{}_1}),{\underline{u}}_{{}_1}(\tau ,{\acute{r}}_{{}_1}),.,{\underline{u}}_n(\tau ,{\acute{r}}_{{}_1}))d\tau \end{array}\right\},\hfill \\ & & \left\{\begin{array}{c}{\overline{u}}_{{}_0}(y,\beta )=\overline{g}(y,\beta )\\ {\overline{u}}_{{}_1}(y,\beta )={\int }_0^{y}k(y,\tau ){\overline{A}}_0\left({\overline{u}}_{{}_0}(\tau ,\beta )\right)d\tau \\ \\ \\ {\overline{u}}_{{}_{n+1}}(y,\beta )={\int }_0^{y}k(y,\tau ){\overline{A}}_n({\overline{u}}_0(\tau ,\beta ),{\overline{u}}_1(\tau ,\beta ),..,{\overline{u}}_n(\tau ,\beta ))d\tau \end{array}\right\}\hfill \end{array}$$

(7)

Another formula for the ADM is as follows:

$$\left\{\begin{array}{c}{\underline{A}}_n=f\left({\underline{S}}_n\right)-\sum _{j=0}^{n-1}{\underline{A}}_j\\ \\ {\overline{A}}_n{=f(\overline{S}}_n)-\sum _{j=0}^{n-1}{\overline{A}}_j\end{array}\right\}$$

(8)

The partial sums are [${\overline{S}}_n={\sum }_{i=0}^n{\overline{u}}_i(y,\alpha ),{\underline{S}}_n={\sum }_{i=0}^n{\underline{u}}_i(y,\alpha )],[{\overline{S}}_n={\sum }_{i=0}^n{\overline{u}}_i(y,\delta ),{\underline{S}}_n={\sum }_{i=0}^n{\underline{u}}_i(y,\delta )]$, and $[{\overline{S}}_n={\sum }_{i=0}^n{\overline{u}}_i(y,\beta ),{\underline{S}}_n={\sum }_{i=0}^n{\underline{u}}_i(y,\beta )]$. Applying the ADM (1) yields

$$\left\{\begin{array}{c}\overline{u}(y,\alpha )=\sum _{i=0}^{\infty }{\overline{u}}_i(y,\alpha )\\ \\ \underline{u}(y,\alpha )=\sum _{i=0}^{\infty }{\underline{u}}_i(y,\alpha )\end{array}\right\},\left\{\begin{array}{c}\overline{u}(y,\delta )=\sum _{i=0}^{\infty }{\overline{u}}_i(y,\delta )\\ \\ \underline{u}(y,\delta )=\sum _{i=0}^{\infty }{\underline{u}}_i(y,\delta )\end{array}\right\},\left\{\begin{array}{c}\overline{u}(y,\beta )=\sum _{i=0}^{\infty }{\overline{u}}_i(y,\beta )\\ \\ \underline{u}(y,\beta )=\sum _{i=0}^{\infty }{\underline{u}}_i(y,\beta )\end{array}\right\}$$

where

$$\left[\begin{array}{c}\left\{\begin{array}{c}{\overline{u}}_{{}_0}(y,\alpha )=\overline{g}(y,\alpha )\\ \\ {\overline{u}}_i(y,\alpha )={\int }_0^{y}k(y,\tau ){\overline{A}}_{i-1}d\tau \end{array}\right\},\left\{\begin{array}{c}{\underline{u}}_0(y,\alpha )=\underline{g}(y,\alpha )\\ \\ {\underline{u}}_0(y,\alpha )={\int }_0^{y}k(y,\tau ){\underline{A}}_{i-1}d\tau \end{array}\right\}\\ \\ \left\{\begin{array}{c}{\overline{u}}_{{}_0}(y,\delta )=\overline{g}(y,\delta )\\ \\ {\overline{u}}_i(y,\delta )={\int }_0^{y}k(y,\tau ){\overline{A}}_{i-1}d\tau \end{array}\right\},\left\{\begin{array}{c}{\underline{u}}_0(x,\delta )=\underline{g}(y,\delta )\\ \\ {\underline{u}}_0(y,\delta )={\int }_0^{y}k(y,\tau ){\underline{A}}_{i-1}d\tau \end{array}\right\}\\ \\ \left\{\begin{array}{c}{\overline{u}}_{{}_0}(y,\beta )=\overline{g}(y,\beta )\\ \\ {\overline{u}}_i(y,\beta )={\int }_0^{y}k(y,\tau ){\overline{A}}_{i-1}d\tau \end{array}\right\},\left\{\begin{array}{c}{\underline{u}}_0(y,\beta )=\underline{g}(y,\beta )\\ \\ {\underline{u}}_0(y,\beta )={\int }_0^{x}k(y,\tau ){\underline{A}}_{i-1}d\tau \end{array}\right\}\end{array}\right]$$

(9)

5. Convergence Analysis

In this section, the existence of the NVPFIE is proven and the uniqueness and convergence of the ADM is discussed.

5.1. Existence and Uniqueness Theorem

The major importance of integral equations concept of solution is that there is only one solution, the kernel is a specific function $\left|k(y,\tau )\right|\le M\forall y\in [0,T]$, and the function f in Equation (1) fulfils the Lipschitz condition with constant $L_1.$ Equation (1) has a unique solution at $\omega =L_{1}MT$.

5.2. Convergence Analysis

Theorem 1.

Converge the series solutions of [$\overline{u}(y,\alpha )={\sum }_{i=0}^{\infty }{\overline{u}}_i(y,\alpha )],[\overline{u}(y,\delta )={\sum }_{i=0}^{\infty }{\overline{u}}_i(y,\beta )]$, and $[\overline{u}(y,\beta )={\sum }_{i=0}^{\infty }{\overline{u}}_i(y,\beta )]$ in (1) using the ADM. When $\omega <1,[\overline{u}(y,\alpha ),\overline{u}(y,\delta )$ and $\overline{u}(y,\beta )]$ are bounded ∀[$(y,\alpha ),(y,\delta ),and(y,\beta )],respectively.$

Proof. 

Define the sequence of partial sums [${\overline{S}}_n,{\overline{S}}_m$] as arbitrary partial sums at $n\ge m$; then, one can find that $\left\{{\overline{S}}_n\right\}$ is a Cauchy sequence at $P{E}^1$. Denote $P{E}^1=(C(A\times [0,1],R),∥.∥)$ to be a Banach space of all functions continuous at A by norm $∥u∥={max}_{\forall y\in A}{sup}_{\alpha \in [0,1]}\left|u(y,\alpha )\right|,$

$∥u∥={max}_{\forall y\in A}{sup}_{\delta \in [0,1]}\left|u(y,\delta )\right|and∥u∥={max}_{\forall y\in A}{sup}_{\beta \in [0,1]}\left|u(y,\beta )\right|.$ Now,

$$\begin{array}{ccc}\hfill ∥{\overline{S}}_n-{\overline{S}}_m∥& =& \underset{\forall y\in A}{max}\underset{\alpha \in [0,1]}{sup}\left|{\overline{S}}_n(y,\alpha )-{\overline{S}}_m(y,\alpha )\right|\hfill \\ & =& \underset{\forall y\in A}{max}\underset{\alpha \in [0,1]}{sup}\left|\sum _{i=m+1}^n{\overline{u}}_i\left(y,\alpha \right)\right|\hfill \\ & =& \underset{\forall y\in A}{max}\underset{\alpha \in [0,1]}{sup}\left|\sum _{i=m+1}^n{\int }_0^{y}K(y,\tau ){\overline{A}}_{i-1}d\tau \right|\hfill \\ & =& \underset{\forall y\in A}{max}\underset{\alpha \in [0,1]}{sup}\left|{\int }_0^{y}K(y,\tau )\sum _{i=m}^{n-1}{\overline{A}}_{i}d\tau \right|\hfill \end{array}$$

From (10), we have

$$\sum _{i=m}^{n-1}{\overline{A}}_i=f\left({\overline{S}}_{n-1}\right)-f\left({\overline{S}}_{m-1}\right)$$

Therefore,

$$\begin{array}{ccc}\hfill ∥{\overline{S}}_n-{\overline{S}}_m∥& =& \underset{\forall y\in A}{max}\underset{\alpha \in [0,1]}{sup}\left|{\int }_0^{y}K(y,\tau )[f\left({\overline{S}}_{n-1}\right)-f\left({\overline{S}}_{m-1}\right)]d\tau \right|\hfill \\ & \le & \underset{\forall y\in A}{max}\underset{\alpha \in [0,1]}{sup}{\int }_0^y\left|K(y,\tau )\right|\left|f\left({\overline{S}}_{n-1}\right)-f\left({\overline{S}}_{m-1}\right)\right|d\tau \hfill \\ & \le & \omega ∥{\overline{S}}_{n-1}-{\overline{S}}_{m-1}∥\hfill \end{array}$$

Let $n=m+1$. Then,

$$∥{\overline{S}}_{m+1}-{\overline{S}}_m∥\le \omega ∥{\overline{S}}_m-{\overline{S}}_{m-1}∥\le {\omega }^2∥{\overline{S}}_{m-1}-{\overline{S}}_{m-2}∥\le \cdots \le {\omega }^m∥{\overline{S}}_1-{\overline{S}}_0∥$$

From the triangle inequality,

$$\begin{array}{ccc}\hfill ∥{\overline{S}}_n-{\overline{S}}_m∥& \le & ∥{\overline{S}}_{m+1}-{\overline{S}}_m∥+∥{\overline{S}}_{m+2}-{\overline{S}}_{m+1}∥+\cdots +∥{\overline{S}}_n-{\overline{S}}_{n-1}∥\hfill \\ \hfill & \le & \left[{\omega }^m+{\omega }^{m+1}+\cdots +{\omega }^{m-1}\right]∥{\overline{S}}_1-{\overline{S}}_0∥\hfill \\ & \le & {\omega }^m\left[1+\omega +\cdots +{\omega }^{n-m-1}\right]∥{\overline{S}}_1-{\overline{S}}_0∥\hfill \\ \hfill & \le & {\omega }^m\left[\frac{1-{\omega }^{n-m}}{1-\omega }\right]∥{\overline{u}}_{{}_1}\left(y,\alpha \right)∥\hfill \end{array}$$

Since$0<\omega <1$, we have $(1-{\omega }^{n-m})<1$. Consequently,

$$∥{\overline{S}}_n-{\overline{S}}_m∥\le \frac{{\omega }^m}{1-\omega }\underset{\forall y\in A}{max}\underset{\alpha \in [0,1]}{sup}\left|{\overline{u}}_{{}_1}\left(y,\alpha \right)\right|$$

(10)

but, if $\left|{\overline{u}}_{{}_1}\left(y,\alpha \right)\right|\le \infty$, then $∥{\overline{S}}_n-{\overline{S}}_m∥\to 0$ as $m\to \infty$ and hence $\left\{{\overline{S}}_n\right\}$ is called a Cauchy sequence in $P{E}^1$. Thus,

$$\overline{u}\left(y,\alpha \right)=\underset{n\to \infty }{lim}{\overline{u}}_n\left(y,\alpha \right)$$

(11)

Similarly, if $\left|\overline{u}(y,\delta )\right|<\infty ,\left|\overline{u}\left(y,\beta \right)\right|<\infty$. Then,

$$\begin{array}{ccc}\hfill \overline{u}\left(y,\delta \right)& =& \underset{n\to \infty }{lim}{\overline{u}}_n\left(y,\delta \right)\hfill \\ \hfill \overline{u}\left(y,\beta \right)& =& \underset{n\to \infty }{lim}{\overline{u}}_n\left(y,\beta \right)\hfill \end{array}$$

Thus, the series converges and this completes the proof. □

Similarly, let $\left\{{\underline{S}}_n\right\}$ be called a Cauchy sequence at $P{E}^1$; then, we obtain

$$\left[\begin{array}{c}\underline{u}\left(y,\alpha \right)=\underset{n\to \infty }{lim}{\underline{u}}_n\left(y,\alpha \right),therefore,\tilde{u}\left(y,\alpha \right)=\underset{n\to \infty }{lim}{\tilde{u}}_n\left(y,\alpha \right)\\ \underline{u}\left(y,\delta \right)=\underset{n\to \infty }{lim}{\underline{u}}_n\left(y,\delta \right),therefore,\tilde{u}\left(y,\delta \right)=\underset{n\to \infty }{lim}{\tilde{u}}_n\left(y,\delta \right)\\ \underline{u}\left(y,\beta \right)=\underset{n\to \infty }{lim}{\underline{u}}_n\left(y,\beta \right),therefore,\tilde{u}\left(y,\beta \right)=\underset{n\to \infty }{lim}{\tilde{u}}_n\left(y,\beta \right)\end{array}\right]$$

(12)

5.3. Error Estimate ([28,29,30])

To estimate the maximum absolute truncation error of the series solution for (1), where $\left|k(y,\tau )\right|\le M\forall y\in [0,T]$, and to satisfy the Lipschitz condition with constant $L_1$, let $L_2=\frac{{\omega }^{m+1}}{L_1(1-\omega )}$

$$\left\{\begin{array}{c}\underset{\forall y\in A}{max}\underset{\alpha \in [0,1]}{sup}\left|\overline{u}\left(y,\alpha \right)-{\displaystyle \sum _{i=0}^{\infty }}{\overline{u}}_i(y,\alpha )\right|{\le K}_1{L}_2{,K}_1=\underset{\forall y\in A}{max}\underset{\alpha \in [0,1]}{sup}\left|f(\overline{u}\left(y,\alpha \right))\right|\\ \underset{\forall y\in A}{max}\underset{\delta \in [0,1]}{sup}\left|\overline{u}\left(y,\delta \right)-{\displaystyle \sum _{i=0}^{\infty }}{\overline{u}}_i(y,\delta )\right|{\le K_{2}L}_2,K_2=\underset{\forall y\in A}{max}\underset{\delta \in [0,1]}{sup}\left|f(\overline{u}\left(y,\delta \right))\right|\\ \underset{\forall y\in A}{max}\underset{\beta \in [0,1]}{sup}\left|\overline{u}\left(y,\beta \right)-{\displaystyle \sum _{i=0}^{\infty }}{\overline{u}}_i(y,\beta )\right|{\le K_{3}L}_2{,K}_3=\underset{\forall y\in A}{max}\underset{\beta \in [0,1]}{sup}\left|f(\overline{u}\left(y,\beta \right))\right|\end{array}\right\}$$

(13)

$$\left\{\begin{array}{c}\underset{\forall y\in A}{max}\underset{\alpha \in [0,1]}{sup}\left|\underline{u}\left(y,\alpha \right)-{\displaystyle \sum _{i=0}^{\infty }}{\underline{u}}_i(y,\alpha )\right|\le K_1{L}_2,K_1=\underset{\forall y\in A}{max}\underset{\alpha \in [0,1]}{sup}\left|f(\underline{u}\left(y,\alpha \right))\right|\\ \underset{\forall y\in A}{max}\underset{\delta \in [0,1]}{sup}\left|\underline{u}\left(y,\delta \right)-{\displaystyle \sum _{i=0}^{\infty }}{\underline{u}}_i(y,\delta )\right|\le K_2{L}_2,K_2=\underset{\forall y\in A}{max}\underset{\delta \in [0,1]}{sup}\left|f(\underline{u}\left(y,\delta \right))\right|\\ \underset{\forall y\in A}{max}\underset{\beta \in [0,1]}{sup}\left|\underline{u}\left(y,\beta \right)-{\displaystyle \sum _{i=0}^{\infty }}{\underline{u}}_i\left(y,\beta \right)\right|\le K_3{L}_2,K_3=\underset{\forall y\in A}{max}\underset{\beta \in [0,1]}{sup}\left|f(\underline{u}\left(y,\beta \right))\right|\end{array}\right\}$$

(14)

6. Numerical Cases

Example 1.

Consider the following NVPFIE,

$$\tilde{u}\left(y\right)=\tilde{g}\left(y\right)\oplus {\int }_0^{y}k(y,\tau )\odot {\tilde{u}}^2\left(\tau \right))d\tau$$

where the triangle picture fuzzy = [1, 2, 3, $\mu ,\eta ,\upsilon ]$ with $(\alpha ,\delta ,\beta )$-$cut=[0.5,1,1.5,\alpha ,\beta ,\xi ]$, $k(y,\tau )=(y+\tau )$ and an exact solution at ($\alpha ,\upsilon )=(1,1),$($\beta ,{\upsilon }_{{}_1})=(0,0)$ and $({\delta }_{{}_1},{\xi }_{{}_1})=(0,0)$. Thus, we obtain (${\tilde{u}}_{{}_{exact}})=(2+\frac{3u_{-}^2{y}^2}{2}).$

The equation for $\alpha$ is

$$\left[\begin{array}{c}{\underline{u}}_{\alpha }=(1+\frac{\alpha }{\upsilon })\oplus {\int }_0^y(0.5+\frac{0.5\alpha }{\upsilon })\odot (y+\tau )\odot {\underline{u}}^{2}d\tau ,\\ {\overline{u}}_{\alpha }=(3-\frac{\alpha }{\upsilon })\oplus {\int }_0^y(1.5-\frac{0.5\alpha }{\upsilon })\odot (y+\tau )\odot {\overline{u}}^{2}d\tau ,\end{array}\right]$$

(15)

Applying the ADM to Equation (15)

$$\left\{\begin{array}{c}\left[\begin{array}{c}{\underline{u}}_{{}_0}=(1+\frac{\alpha }{\upsilon }).\\ {\underline{u}}_{{}_i}={\int }_0^y(0.5+\frac{0.5\alpha }{\upsilon })\odot (y+\tau )\odot A_{{}_{i-1}}d\tau ,i\ge 1,0\le r\le 1\\ {\underline{u}}_{{}_1}={\int }_0^y(0.5+\frac{0.5\alpha }{\upsilon })\odot (y+\tau )\odot {\underline{u}}_{{}_0}^{2}d\tau \\ {\underline{u}}_{{}_2}={\int }_0^y(0.5+\frac{0.5\alpha }{\upsilon })\odot (y+\tau )\odot {\underline{u}}_{{}_1}^{2}d\tau \\ {\underline{u}}_{{}_3}={\int }_0^y(0.5+\frac{0.5\alpha }{\upsilon })\odot (y+\tau )\odot {\underline{u}}_2^{2}d\tau \\ ...\end{array}\right]\\ \\ \left[\begin{array}{c}{\overline{u}}_{{}_0}=(3-\frac{\alpha }{\upsilon }).\\ {\overline{u}}_{{}_i}={\int }_0^y(1.5-\frac{0.5\alpha }{\upsilon })\odot (y+\tau )\odot B_{{}_{i-1}}d\tau ,i\ge 1.\\ {\overline{u}}_{{}_1}={\int }_0^y(1.5-\frac{0.5\alpha }{\upsilon })\odot (y+\tau )\odot {\overline{u}}_{{}_0}^{2}d\tau \\ {\overline{u}}_{{}_2}={\int }_0^y(1.5-\frac{0.5\alpha }{\upsilon })\odot (y+\tau )\odot {\overline{u}}_{{}_1}^{2}d\tau \\ {\overline{u}}_{{}_3}={\int }_0^y(1.5-\frac{0.5\alpha }{\upsilon })\odot (y+\tau )\odot {\overline{u}}_{{}_2}^{2}d\tau \\ ...\end{array}\right]\end{array}\right\}$$

where $A_i$ and $B_i$ are Adomian polynomials of the nonlinear terms ${\underline{u}}^2$ and ${\overline{u}}^2$, respectively. Thus, the solution will be

$$\left\{\begin{array}{c}{\underline{u}}_{\alpha }=\sum _{i=0}^q{\underline{u}}_i\\ \\ {\overline{u}}_{\alpha }=\sum _{i=0}^q{\overline{u}}_i\end{array}\right\}$$

Error analysis.

Absolute errors are computed as

$$\left\{\begin{array}{c}\underline{E}=\left|{\underline{u}}_{exact}-{\underline{u}}_{\alpha }\right|\\ \\ \overline{E}=\left|{\overline{u}}_{exact}-{\overline{u}}_{\alpha }\right|.\end{array}\right\}$$

Table 1. The exact and approximate solutions of the ADM at $u=0.5$ and $y=0.7$.

	Exact Solution			Approximate Solution		Error
$\mathit{i}$	${\tilde{\mathit{u}}}_{\mathbf{\alpha }\mathit{exact}}$	${\tilde{\mathit{u}}}_{\mathbf{\alpha }\mathit{exact}} \mathit{at} (\mathbf{\alpha }=\mathbf{0.4}\mathit{and}\mathbf{\upsilon }=\mathbf{0.8})$		${\tilde{\mathit{u}}}_{\mathbf{\alpha }} \mathit{at} (\mathbf{\alpha }=\mathbf{0.4}\mathit{and}\mathbf{\upsilon }=\mathbf{0.8})$
		${\underline{\mathit{u}}}_{\mathbf{\alpha }\mathit{exact}}$	${\overline{\mathit{u}}}_{\mathbf{\alpha }\mathit{exact}}$	${\underline{\mathit{u}}}_{\mathit{i}\mathbf{\alpha }}$	${\overline{\mathit{u}}}_{\mathit{i}\mathbf{\alpha }}$	${\underline{\mathit{E}}}_{\mathbf{\alpha }}$
0	2.18375	1.653125	10.22	1.5	2.5	$0.153125$
1	2.18375	1.653125	10.22	1.2403125	5.7421875	$0.4128125$
2	2.18375	1.653125	10.22	0.848029	30.29368401	$0.805096$
3	2.18375	1.653125	10.22	0.39643344	843.1435733	1.25669156
4	2.18375	1.653125	10.22	0.08663405	653,131.1845	1.56649095
5	2.18375	1.653125	10.22	4.137384182 $\times {10}^{-3}$	3.919206912 $\times {10}^{11}$	1.648987616
6	2.18375	1.653125	10.22	9.436268765 $\times {10}^{-6}$	$1.411216796\times {10}^{23}$	1.653115564

shows the exact and approximate solutions of the ADM at $u=0.5$ and $y=0.7$.

The general form of the equation for $\beta$ is

$$\left[\begin{array}{c}{\underline{u}}_{\beta }=(2-\frac{\beta -{\upsilon }_1}{1-{\upsilon }_1}\left(1\right))\oplus {\int }_0^y(1-\frac{\beta -{\upsilon }_1}{1-{\upsilon }_1}\left(0.5\right))\odot (y+\tau )\odot {\underline{u}}^{2}d\tau ,\\ {\overline{u}}_{\beta }=(2+\frac{\beta -{\upsilon }_1}{1-{\upsilon }_1}\left(1\right))\oplus {\int }_0^y(1+\frac{\beta -{\upsilon }_1}{1-{\upsilon }_1}\left(0.5\right))\odot (y+\tau )\odot {\overline{u}}^{2}d\tau ,\end{array}\right]$$

(16)

Applying the ADM to Equation (16), we obtain

$$\left\{\begin{array}{c}\left[\begin{array}{c}{\underline{u}}_0=(2-\frac{\beta -{\upsilon }_1}{1-{\upsilon }_1}\left(1\right)).\\ {\underline{u}}_i={\int }_0^y(1-\frac{\beta -{\upsilon }_1}{1-{\upsilon }_1}\left(0.5\right))\odot (y+\tau )\odot A_{i-1}d\tau ,i\ge 1\\ {\underline{u}}_i={\int }_0^y(1-\frac{\beta -{\upsilon }_1}{1-{\upsilon }_1}\left(0.5\right))\odot (y+\tau )\odot {\underline{u}}_{{}_0}^{2}d\tau \\ {\underline{u}}_i={\int }_0^y(1-\frac{\beta -{\upsilon }_1}{1-{\upsilon }_1}\left(0.5\right))\odot (y+\tau )\odot {\underline{u}}_{{}_1}^{2}d\tau \\ {\underline{u}}_i={\int }_0^y(1-\frac{\beta -{\upsilon }_1}{1-{\upsilon }_1}\left(0.5\right))\odot (y+\tau )\odot {\underline{u}}_{{}_2}^{2}d\tau \\ ....\end{array}\right]\\ \\ \left[\begin{array}{c}{\overline{u}}_0=(2+\frac{\beta -{\upsilon }_1}{1-{\upsilon }_1}\left(1\right)).\\ {\overline{u}}_i={\int }_0^y(1+\frac{\beta -{\upsilon }_1}{1-{\upsilon }_1}\left(0.5\right))\odot (y+\tau )\odot B_{i-1}d\tau ,i\ge 1.\\ {\overline{u}}_i={\int }_0^y(1+\frac{\beta -{\upsilon }_1}{1-{\upsilon }_1}\left(0.5\right))\odot (y+\tau )\odot {\overline{u}}_{{}_0}^{2}d\tau \\ {\overline{u}}_i={\int }_0^y(1+\frac{\beta -{\upsilon }_1}{1-{\upsilon }_1}\left(0.5\right))\odot (y+\tau )\odot {\overline{u}}_{{}_1}^{2}d\tau \\ {\overline{u}}_i={\int }_0^y(1+\frac{\beta -{\upsilon }_1}{1-{\upsilon }_1}\left(0.5\right))\odot (y+\tau )\odot {\overline{u}}_{{}_2}^{2}d\tau \\ ....\end{array}\right]\end{array}\right\}$$

where $A_i$ and $B_i$ are Adomian polynomials of the nonlinear terms ${\underline{u}}^2$ and ${\overline{u}}^2$, respectively, and the solution will be

$$\left\{\begin{array}{c}{\underline{u}}_{\beta }=\sum _{i=0}^q{\underline{u}}_i\\ \\ {\overline{u}}_{\beta }=\sum _{i=0}^q{\overline{u}}_i\end{array}\right\}$$

Error estimate.

The absolute errors are computed as

$$\left\{\begin{array}{c}\underline{E}=\left|{\underline{u}}_{exact}-{\overline{u}}_{\beta }\right|\\ \\ \overline{E}=\left|{\overline{u}}_{exact}-{\overline{u}}_{\beta }\right|.\end{array}\right\}$$

Table 2. The exact and approximate solutions of the ADM at $u=0.5$ and $y=0.7$.

	Exact Solution			Approximate Solution		Error
$\mathit{i}$	${\tilde{\mathit{u}}}_{\mathbf{\beta }\mathit{exact}}$	${\tilde{\mathit{u}}}_{\mathbf{\beta }\mathit{exact}} \mathit{at} (\mathbf{\beta }=\mathbf{0.3}\mathit{and}{\mathbf{\upsilon }}_{\mathbf{1}}=\mathbf{0.6})$		${\tilde{\mathit{u}}}_{\mathbf{\beta }} \mathit{at} (\mathbf{\beta }=\mathbf{0.3}\mathit{and}{\mathbf{\upsilon }}_{\mathbf{1}}=\mathbf{0.6})$
		${\underline{\mathit{u}}}_{\mathbf{\beta }\mathit{exact}}$	${\overline{\mathit{u}}}_{\mathbf{\beta }\mathit{exact}}$	${\underline{\mathit{u}}}_{\mathit{i}\mathbf{\beta }}$	${\overline{\mathit{u}}}_{\mathit{i}\mathbf{\beta }}$	${\overline{\mathit{E}}}_{\mathbf{\beta }}$
0	2.18375	3.00265625	1.36484375	2.75	1.25	0.1148
1	2.18375	3.00265625	1.36484375	7.642851563	0.7177734375	0.6470
2	2.18375	3.00265625	1.36484375	59.03382004	0.2366694063	1.1281
3	2.18375	3.00265625	1.36484375	3522.019948	0.02573069987	1.339
4	2.18375	3.00265625	1.36484375	12,536,423.65	3.0413790 $\times {10}^{-4}$	1.364
5	2.18375	3.00265625	1.36484375	1.5883176 $\times {10}^{14}$	4.2492126$\times {10}^{-8}$	1.3648
6	2.18375	3.00265625	1.36484375	2.5495571 $\times {10}^{28}$	8.294386 $\times {10}^{-16}$	1.3648
7	2.18375	3.00265625	1.36484375	6.569306 $\times {10}^{56}$	3.1603554$\times {10}^{-31}$	1.3648

shows the exact and approximate solutions of the ADM at $u=0.5$ and $y=0.7$.

The general form of the equation for $\xi$ is

$$\left[\begin{array}{c}{\underline{u}}_{\xi }=(2-\frac{{\delta }_1-{\xi }_1}{1-{\xi }_1}\left(1\right))\oplus {\int }_0^y(1-\frac{{\delta }_1-{\xi }_1}{1-{\xi }_1}\left(0.5\right))\odot (y+\tau )\odot {\underline{u}}^{2}d\tau ,\\ {\overline{u}}_{\xi }=(2+\frac{{\delta }_1-{\xi }_1}{1-{\xi }_1}\left(1\right))\oplus {\int }_0^y(1+\frac{{\delta }_1-{\xi }_1}{1-{\xi }_1}\left(0.5\right))\odot (y+\tau )\odot {\overline{u}}^{2}d\tau ,\end{array}\right]$$

(17)

Applying the ADM to Equation (17), we obtain

$$\left\{\begin{array}{c}\begin{array}{c}{\underline{u}}_0=(2-\frac{{\delta }_1-{\xi }_1}{1-{\xi }_1}\left(1\right)).\\ {\underline{u}}_i={\int }_0^y(1-\frac{{\delta }_1-{\xi }_1}{1-{\xi }_1}\left(0.5\right))\odot (y+\tau )\odot A_{i-1}d\tau ,i\ge 1\\ {\underline{u}}_i={\int }_0^y(1-\frac{{\delta }_1-{\xi }_1}{1-{\xi }_1}\left(0.5\right))\odot (y+\tau )\odot {\underline{u}}_{{}_0}^{2}d\tau \\ {\underline{u}}_i={\int }_0^y(1-\frac{{\delta }_1-{\xi }_1}{1-{\xi }_1}\left(0.5\right))\odot (y+\tau )\odot {\underline{u}}_{{}_1}^{2}d\tau \\ {\underline{u}}_i={\int }_0^y(1-\frac{{\delta }_1-{\xi }_1}{1-{\xi }_1}\left(0.5\right))\odot (y+\tau )\odot {\underline{u}}_{{}_2}^{2}d\tau \\ ....\end{array}\\ \\ {\overline{u}}_0=(2+\frac{{\delta }_1-{\xi }_1}{1-{\xi }_1}\left(1\right)).\\ {\overline{u}}_i={\int }_0^y(1+\frac{{\delta }_1-{\xi }_1}{1-{\xi }_1}\left(0.5\right))\odot (y+\tau )\odot B_{i-1}d\tau ,i\ge 1.\\ {\overline{u}}_i={\int }_0^y(1+\frac{{\delta }_1-{\xi }_1}{1-{\xi }_1}\left(0.5\right))\odot (y+\tau )\odot {\overline{u}}_{{}_0}^{2}d\tau \\ {\overline{u}}_i={\int }_0^y(1+\frac{{\delta }_1-{\xi }_1}{1-{\xi }_1}\left(0.5\right))\odot (y+\tau )\odot {\overline{u}}_{{}_1}^{2}d\tau \\ {\overline{u}}_i={\int }_0^y(1+\frac{{\delta }_1-{\xi }_1}{1-{\xi }_1}\left(0.5\right))\odot (y+\tau )\odot {\overline{u}}_{{}_2}^{2}d\tau \\ .....\end{array}\right\}$$

where $A_i$ and $B_i$ are Adomian polynomials of the nonlinear terms ${\underline{u}}^2$ and ${\overline{u}}^2$, respectively, and the solution will be

$$\left\{\begin{array}{c}{\underline{u}}_{{\xi }_1}=\sum _{i=0}^q{\underline{u}}_i\\ \\ {\overline{u}}_{{\xi }_1}=\sum _{i=0\overline{u}i}^q\end{array}\right\}$$

Error analysis.

The absolute errors are computed as

$$\left\{\begin{array}{c}\underline{E}=\left|{\underline{u}}_{exact}-{\underline{u}}_{{\xi }_1}\right|\\ \\ \overline{E}=\left|{\overline{u}}_{exact}-{\overline{u}}_{{\xi }_1}\right|.\end{array}\right\}$$

Table 3. The exact and approximate solutions of the ADM at $u=0.5$ and $y=0.7$.

	Exact Solution			Approximate Solution		Error
$\mathit{i}$	${\tilde{\mathit{u}}}_{{\mathbf{\xi }}_{\mathit{1}}\mathit{exact}}$	${\tilde{\mathit{u}}}_{{\mathbf{\xi }}_{\mathit{1}}\mathit{exact}} \mathit{at} ({\mathbf{\delta }}_{{}_{\mathit{1}}}=\mathbf{0.2}\mathit{and}{\mathbf{\xi }}_{\mathbf{1}}=\mathbf{0.6})$		${\tilde{\mathit{u}}}_{{\mathbf{\xi }}_{\mathbf{1}}} \mathit{at} ({\mathbf{\delta }}_{\mathbf{1}}=\mathbf{0.2}\mathit{and}{\mathbf{\xi }}_{\mathbf{1}}=\mathbf{0.6})$
		${\underline{\mathit{u}}}_{{\mathbf{\xi }}_{\mathbf{1}}\mathit{exact}}$	${\overline{\mathit{u}}}_{{\mathbf{\xi }}_{\mathbf{1}}\mathit{exact}}$	${\underline{\mathit{u}}}_{\mathit{i}{\mathbf{\xi }}_{\mathbf{1}}}$	${\overline{\mathit{u}}}_{\mathit{i}{\mathbf{\xi }}_{\mathit{1}}}$	${\overline{\mathit{E}}}_{{\mathbf{\xi }}_{\mathbf{1}}}$
0	2.18375	3.275625	1.091875	3	1	0.0918
1	2.18375	3.275625	1.091875	9.9225	0.3675	0.7253
2	2.18375	3.275625	1.091875	108.5477469	0.04963317188	1.0422
3	2.18375	3.275625	1.091875	12,990.33122	9.053185183 $\times {10}^{-4}$	1.0909
4	2.18375	3.275625	1.091875	186,045,447.6	3.012035952 $\times {10}^{-7}$	1.0918
5	2.18375	3.275625	1.091875	3.81607317 $\times {10}^{16}$	3.334092511$\times {10}^{-14}$	1.0918
6	2.18375	3.275625	1.091875	1.605506192 $\times {10}^{33}$	4.085193531$\times {10}^{-28}$	1.0918
7	2.18375	3.275625	1.091875	2.841859271 $\times {10}^{66}$	6.133136272$\times {10}^{-56}$	1.0918

shows the exact and approximate solutions of the ADM at $u=0.5$ and $y=0.7$.

7. Simulation

In this section, we use random simulated data to evaluate the NVPFIE in

Table 4. Approximate Solution with [1, 2, 3, $\mu ,\eta ,\upsilon \left]and\right(\alpha ,\delta ,\beta )$-$cut=[0.5,1,1.5,\alpha ,\beta ,\xi ] at (y=0.7)$.

Approximate Solution
	with [1, 2, 3,$\mathbf{\mu },\mathbf{\eta },\mathbf{\upsilon }\left]\mathit{and}\right(\mathbf{\alpha },\mathbf{\delta },\mathbf{\beta })$-$\mathit{cut}=[\mathbf{0.5},\mathbf{1},\mathbf{1.5},\mathbf{\alpha },\mathbf{\beta },\mathbf{\xi }] \mathit{at} (\mathit{y}=\mathbf{0.7})$
$\mathit{i}$	${\tilde{\mathit{u}}}_{\mathbf{\alpha }} \mathit{at} (\mathbf{\alpha }=\mathbf{0.1}\mathit{and}\mathbf{\upsilon }=\mathbf{0.5})$		${\tilde{\mathit{u}}}_{\mathbf{\beta }} \mathit{at} (\mathbf{\beta }=\mathbf{0.5}\mathit{and}{\mathbf{\upsilon }}_{\mathbf{1}}=\mathbf{0.4})$		${\tilde{\mathit{u}}}_{{\mathbf{\xi }}_{\mathbf{1}}} \mathit{at} ({\mathbf{\delta }}_{\mathbf{1}}=\mathbf{0.3}\mathit{and}{\mathbf{\xi }}_{\mathbf{1}}=\mathbf{0.2})$
	${\underline{\mathit{u}}}_{\mathit{i}\mathbf{\alpha }}$	${\overline{\mathit{u}}}_{\mathit{i}\mathbf{\alpha }}$	${\underline{\mathit{u}}}_{\mathit{i}\mathbf{\beta }}$	${\overline{\mathit{u}}}_{\mathit{i}\mathbf{\beta }}$	${\underline{\mathit{u}}}_{\mathit{i}{\mathbf{\xi }}_{\mathbf{1}}}$	${\overline{\mathit{u}}}_{\mathit{i}{\mathbf{\xi }}_{\mathbf{1}}}$
0	1.2	2.8	1.833333	2.16667	1.875	2.125
1	0.63504	8.06736	2.264540376	3.73796289	2.422485352	3.526420898
2	0.1778446285	66.969884	3.455086425	11.12549688	4.043718684	9.711461012
3	0.01394826194	445.001793	8.042972958	98.55718212	11.26731627	73.65214843
4	8.57983189 $\times {10}^{-5}$	21915890.55	43.58449269	7734.388825	87.47814906	4236.303995
5	3.24635602 $\times {10}^{-9}$	4.942351 $\times {10}^{14}$	1279.860892	47,632,288.51	5273.000179	14,014,916.43

and

Table 5. Approximate Solution with [1, 2, 3, $\mu ,\eta ,\upsilon \left]and\right(\alpha ,\delta ,\beta )$-$cut=[0.5,1,1.5,\alpha ,\beta ,\xi ] at (y=0.5)$.

Approximate Solution
	with [1, 2, 3,$\mathbf{\mu },\mathbf{\eta },\mathbf{\upsilon }\left]\mathit{and}\right(\mathbf{\alpha },\mathbf{\delta },\mathbf{\beta })$-$\mathit{cut}=[\mathbf{0.5},\mathbf{1},\mathbf{1.5},\mathbf{\alpha },\mathbf{\beta },\mathbf{\xi }] \mathit{at} (\mathit{y}=\mathbf{0.5})$
$\mathit{i}$	${\tilde{\mathit{u}}}_{\mathbf{\alpha }} \mathit{at} (\mathbf{\alpha }=\mathbf{0.6}\mathit{and}\mathbf{\upsilon }=\mathbf{0.3})$		${\tilde{\mathit{u}}}_{\mathbf{\beta }} \mathit{at} (\mathbf{\beta }=\mathbf{0.2}\mathit{and}{\mathbf{\upsilon }}_{\mathbf{1}}=\mathbf{0.7})$		${\tilde{\mathit{u}}}_{{\mathbf{\xi }}_{\mathbf{1}}} \mathit{at} ({\mathbf{\delta }}_{\mathbf{1}}=\mathbf{0.4}\mathit{and}{\mathbf{\xi }}_{\mathbf{1}}=\mathbf{0.1})$
	${\underline{\mathit{u}}}_{\mathit{i}\mathbf{\alpha }}$	${\overline{\mathit{u}}}_{\mathit{i}\mathbf{\alpha }}$	${\underline{\mathit{u}}}_{\mathit{i}\mathbf{\beta }}$	${\overline{\mathit{u}}}_{\mathit{i}\mathbf{\beta }}$	${\underline{\mathit{u}}}_{\mathit{i}{\mathbf{\xi }}_{\mathbf{1}}}$	${\overline{\mathit{u}}}_{\mathit{i}{\mathbf{\xi }}_{\mathbf{1}}}$
0	3	1	3.666667	0.33333	1.666667	2.33333
1	5.0625	0.1875	9.243057236	6.944305556 $\times {10}^{-3}$	0.8680559028	2.38193763
2	14.41625977	6.591796875 $\times {10}^{-3}$	58.73594861	3.013961229 $\times {10}^{-6}$	0.2354753282	2.482211776
3	116.9035569	8.147209883 $\times {10}^{-6}$	2371.814266	5.67747643 $\times {10}^{-13}$	0.01732769694	2.695601693
4	7687.373411	1.244569291 $\times {10}^{-11}$	3,867,533.251	2.01460866 $\times {10}^{-26}$	9.3827837 $\times {10}^{-5}$	3.178992464
5	33,241,336.85	2.90428635 $\times {10}^{-23}$	1.028349 $\times {10}^{13}$	2.53665504 $\times {10}^{14}$	2.7511447 $\times {10}^{-9}$	4.421371974

.

8. Conclusions

A semi-analytical solution of the NVPFIE is obtained using an accelerated version of the ADM based on $(\alpha ,\delta ,\beta )$-$cut$; the solution is presented as an infinite series which converges rapidly to the accurate solution and the sufficient condition that guarantees a unique solution is identified. A convergence analysis is reliable enough to determine the maximum absolute truncation error of the Adomian series solution. Working with a picture fuzzy model is much more effective because picture fuzzy depends on three viewpoints or three characteristics that meet at the same time, namely the view of neutrality, opposition, and supporter, which makes the accuracy of the calculation higher than that of fuzzy. In the future, this work can be applied to engineering applications such as water desalination and the spread of viruses.