Fractional Calculus in Nuclear Multistep Decay: Analytical Solutions, Existence and Uniqueness Analysis of the Actinium Series

Abstract

This paper provides a thorough examination of the Actinium radioactive decay series, which converts Uranium-235 into the stable Lead-207 isotope via a succession of alpha, beta, and gamma decays. For the first time, the series is modeled using fractional calculus, employing two innovative analytical methods: the Sumudu Residual Power Series Method (SRPSM) and the Temimi Ansari Method (TAM). The study discusses the well-posedness of the fractional-order model in the Caputo sense within a Banach space setting. These fractional models capture complex, non-ideal decay behaviors more accurately than traditional exponential models. Mathematica is used to do numerical computations for four different Actinium series scenarios. The results are tabulated and visually depicted to show how radionuclide concentrations change over time. The findings demonstrate that SRPSM and TAM effectively simplify the complex differential equations governing nuclear decay, offering enhanced precision and flexibility. This work provides a robust framework for modeling the Actinium series, with potential applications in nuclear physics, radiometric dating, and radiation safety studies.

1. Introduction

Physics, which studies natural phenomena, experienced significant advancements at the beginning of the 20th century with the emergence of modern physics, where time is considered the fourth dimension (in addition to the three geometric dimensions), previously regarded as independent and absolute. Moreover, modern physics has undergone a revolution in many subfields of physics where quantum mechanics was created [1,2,3].

One achievement of this revolution is the discovery of radioactivity, where Becquerel observed an unknown type of radiation in Uranium nuclei in 1896. After that, physicists detected the same behavior in other elements, whereas two new radioactive elements, known as Polonium and Radium, were discovered. This radiation was classified into three types: alpha, beta, or gamma decay.

Heavy nuclei are unstable and undergo multiple radioactive decays to become stable; these nuclei are classified by their atomic mass with regard to four radioactive series: the Thorium series, which starts with Thorium-232 (4n); the Neptunium series, which starts with Neptunium-237 (4n + 1), and Uranium series which starts with Uranium-238 (4n + 2); and the Actinium series which starts with Uranium-235 (4n + 3), where n is the atomic mass [4,5,6,7,8].

In the Actinium series, Uranium isotope-235 undergoes a sequence of radioactive decay steps to reach a stable Lead-207 isotope. Other works studied different cases of multi-step radioactive series [9,10,11].

There are several intermediate radioactive elements in the series, and each one decays further. In this series, a succession of alpha (Helium nuclei), beta (electrons or positrons), and gamma radiation decays is emitted. This series is crucial for understanding natural background radiation, nuclear power generation, and radiometric dating, among other uses. The Actinium decay series can be considered mathematically in fractional order as

$$\begin{array}{ccc}\hfill {\mathfrak{D}}_t^{ϵ}{N}_1\left(t\right)& =& -{\lambda }_1{N}_1\left(t\right),\hfill \\ \hfill {\mathfrak{D}}_t^{ϵ}{N}_2\left(t\right)& =& {\lambda }_1{N}_1\left(t\right)-{\lambda }_2{N}_2\left(t\right),\hfill \\ \hfill {\mathfrak{D}}_t^{ϵ}{N}_3\left(t\right)& =& {\lambda }_2{N}_2\left(t\right)-{\lambda }_3{N}_3\left(t\right),\hfill \\ \hfill ⋮& ⋮& ⋮\hfill \\ \hfill {\mathfrak{D}}_t^{ϵ}{N}_{j-1}\left(t\right)& =& {\lambda }_{j-2}{N}_{j-2}\left(t\right)-{\lambda }_{j-1}{N}_{j-1}\left(t\right),\hfill \\ \hfill {\mathfrak{D}}_t^{ϵ}{N}_j\left(t\right)& =& {\lambda }_{j-1}{N}_{j-1}\left(t\right),\hfill \end{array}$$

(1)

the decay constants are denoted by ${\lambda }_i$. The following boundary constraints express the physical conservation laws for the system: the total number of nuclei remains constant for all $t\ge 0$, and the overall fractional rate of change is zero. Mathematically, these conditions are written as

$$\begin{array}{ccc}\hfill N_1\left(t\right)+N_2\left(t\right)+N_3\left(t\right)+\cdots +N_j\left(t\right)& =& N_0,\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill {\mathfrak{D}}_t^{ϵ}{N}_1\left(t\right)+{\mathfrak{D}}_t^{ϵ}{N}_2\left(t\right)+{\mathfrak{D}}_t^{ϵ}{N}_3\left(t\right)+\cdots +{\mathfrak{D}}_t^{ϵ}{N}_j\left(t\right)& =& 0,\hfill \end{array}$$

(3)

where ${\mathfrak{D}}_t^{ϵ}$ denotes the Caputo fractional derivative of order $0<ϵ\le 1$. Equation (2) enforces total-mass conservation, while Equation (3) states that the sum of fractional rates of change of all nuclide populations is zero. In the fractional-order setting, these constraints incorporate memory effects and non-exponential decay behavior absent from classical models.

Notably, no prior research has been carried out on the fractional form of the Actinium series system as it is given in this work. While non-fractional models of other radioactive decay series, such as the Uranium and Thorium series, have been explored [12,13], and analytical solutions for simplified two-step decay processes have been developed, the application of fractional calculus to the Actinium series remains unprecedented [14,15]. Furthermore, other nuclear decay systems have been studied and solved analytically using classical differential equations [16], but these approaches do not account for the memory effects and non-standard decay rates that fractional calculus can capture. Recent advances in fractional-order analysis within control and fuzzy systems further underscore the breadth of applications of nonlocal dynamics. For example, Rguigui and Elghribi establish a separation principle for Caputo–Hadamard fractional-order fuzzy systems [17] and provide practical stabilization results for tempered fractional nonlinear fuzzy systems [18]. While these studies focus on control settings, their operator-level insights (Caputo–Hadamard and tempered derivatives) complement our Caputo-based treatment of multistep nuclear decay by reinforcing the value of fractional dynamics for systems with memory and non-exponential responses.

The Temimi–Ansari Method (TAM) [19,20,21] is an analytical technique employed in this study to solve the fractional differential equations governing the Actinium series. This approach, which is based on iterative approximations, provides a strong foundation for obtaining precise solutions and is especially useful for dealing with non-linear and fractional-order systems. To ensure that the solutions accurately represent the physical behavior of the radioactive isotopes involved, this method is used in this study to describe the intricate dynamics of the decay chain.

Similarly, the Sumudu Residual Power Series Method (SRPSM) [22,23,24] is utilized to address the fractional order system of equations. To provide very accurate approximations of solutions, SRPSM combines a power series expansion with the Sumudu transform, which is ideal for solving differential equations. The Actinium series is modeled in this work using SRPSM, which allows for the decay behavior over many isotopes to be analytically represented. Unlike classical methods that assume exponential decay, SRPSM accounts for the fractional derivatives’ ability to incorporate historical dependencies, providing a more comprehensive understanding of the decay process. When examining the long-term behavior of the Actinium series, where complicated decay patterns are produced by the interaction of many isotopes with different half-lives, this technique is very useful.

In this paper, the fractional order system of equations describing the radioactive Actinium series is solved analytically using both SRPSM and TAM. Using the characteristics of the Caputo fractional derivative to guarantee mathematical rigor, the study further investigates the existence and uniqueness of the solutions. To validate the theoretical framework, numerical implementations are conducted that offer insight into the practical applicability of the proposed models. Along with improving decay prediction accuracy, the fractional technique presents new opportunities for the investigation of additional radioactive decay series with comparable complexity.

The research is organized as follows: Section 2 introduces preliminary concepts and definitions related to fractional calculus; Section 3 investigates the existence and uniqueness of the model solutions; Section 4 elaborates on the theoretical foundations of the Sumudu Residual Power Series Method and the Temimi–Ansari method; Section 5 presents analytical solutions to the fractional nuclear radioactive series equations; and Section 6 discusses the numerical results, providing a detailed comparison with theoretical predictions to confirm the validity of the model.

2. Preliminaries

Definition 1.

For $N(x,t)$, the Caputo operator of order ϵ is defined as [25]

$${\mathfrak{D}}_t^{ϵ}N(x,t)=J_t^{n-ϵ}{\partial }_t^{n}N(x,t),n-1<ϵ\le n,x\in I.t>0.$$

(4)

The fractional order operator of the Caputo derivative is represented by ${\mathfrak{D}}_t^{ϵ}$, and $n\in \mathbb{N},I$ represents an interval.

Definition 2.

For each function $N\left(t\right)$ over a set $\mathbb{A}$, the Sumudu transform is defined as [25]

$$\Lambda \left(\zeta \right)=S\left[N\left(\mathrm{t}\right)\right]={\mathsf{\int }}_0^{\infty }N\left(\zeta \mathrm{t}\right)e^{-\mathrm{t}}d\mathrm{t},\zeta \in \left(-{ϵ}_1,{ϵ}_2\right),$$

(5)

where $N\left(\mathrm{t}\right)$ is continual function over Lemma 1 and

$$\mathbb{A}=\left\{N\left(t\right)\mid \mathit{there}\mathit{exists}\Delta ,{ϵ}_1,{ϵ}_2>0,\left|N\left(t\right)\right|<\Delta e^{{\displaystyle \frac{\left|t\right|}{{ϵ}_i}}}\right\} t\in {(-1)}^i\times [0,\infty ).$$

(6)

For the Caputo operator of ${\mathfrak{D}}_t^{ϵ}N\left(\mathrm{t}\right)$, the Sumudu transform is defined as

$$S\left[{\mathfrak{D}}_t^{ϵ}N\left(\mathrm{t}\right)\right]={\zeta }^{-ϵ}S\left[N\left(\mathrm{t}\right)\right]-\sum _{\lambda =0}^{n-1}{\zeta }^{(-ϵ+\lambda )}{N}^{\left(\lambda \right)}\left(0\right),n\ge 1,n-1<ϵ\le n,ϵ>0,n\in \mathbb{N}.$$

(7)

We outline certain properties of the fractional derivative ${\mathcal{D}}_t^{ϵ}$ in the following lemma, which is crucial for understanding the next portions of this work. Further characteristics are listed in the reference.

Lemma 1.

Let $N\left(\mathrm{t}\right)$ and $M\left(\mathrm{t}\right)$ be a piecewise continuous function of the exponential order on $[0,\infty )$, and $\Xi \left(\zeta \right)=S\left[N\right(\mathrm{t}\left)\right],{\rm Y}\left(\zeta \right)=S\left[M\right(\mathrm{t}\left)\right]$, where $u,v,w,a$ are constants [25].

Then (i) $S\left[uN\right(\mathrm{t})+vM(\mathrm{t}\left)\right]=uS\left[N\right(\mathrm{t}\left)\right]+vS\left[M\right(\mathrm{t}\left)\right]=u\Xi \left(\zeta \right)+v{\rm Y}\left(\zeta \right)$,

(ii) $S^{-1}[u\Xi \left(\zeta \right)+v{\rm Y}\left(\zeta \right)]=uN\left(\mathrm{t}\right)+vM\left(\mathrm{t}\right)$.

(iii) $S\left[e^{wt}N\left(\mathrm{t}\right)\right]={\displaystyle \frac{1}{(1-w\zeta )}}\Xi \left({\displaystyle \frac{\zeta }{1-w\zeta }}\right)$.

(iv) ${lim}_{\zeta \to 0}\Xi \left(\zeta \right)=N\left(0\right)$.

(v) $S\left[N\right(a\mathrm{t}\left)\right]=a\Xi \left(\zeta \right),a>0$, and ${lim}_{\mathrm{t}\to 0}N\left(\mathrm{t}\right)={lim}_{\zeta \to 0}\Xi \left(\zeta \right)$.

3. Existence and Uniqueness

In this section, the uniqueness and existence of the solution to the proposed model using the Caputo operator are explored. Consider the real-valued and continuous function $\mathfrak{Z}(\Gamma )$, which satisfies the sup-norm property, forming a Banach space over the interval $J=[0,b]$. Let $\Gamma =[0,\kappa ]$, $\kappa >0$ denotes the length of the time interval $[0,\kappa ]$ on which the solution is considered, and define $P=\mathfrak{Z}(\Gamma )\times \mathfrak{Z}(\Gamma )\times \mathfrak{Z}(\Gamma )\times \cdots \times \mathfrak{Z}(\Gamma )\times \mathfrak{Z}(\Gamma )$ with the norm given as $∥\left(N_1\left(t\right),N_2\left(t\right),N_3\left(t\right),\cdots ,N_{j-1}\left(t\right),N_j\left(t\right)\right)∥= \parallel N_1\left(t\right)\parallel +\parallel N_2\left(t\right)\parallel +\parallel N_3\left(t\right)\parallel +\cdots +∥N_{j-1}\left(t\right)∥+∥N_j\left(t\right)∥$, where the individual components are defined as $\parallel N_1\left(t\right)\parallel ={sup}_{t\in \Gamma }\left|N_1\left(t\right)\right|$, $\parallel N_2\left(t\right)\parallel ={sup}_{t\in \Gamma }\left|N_2\left(t\right)\right|$, $\parallel N_3\left(t\right)\parallel ={sup}_{t\in \Gamma }\left|N_3\left(t\right)\right|$, …, $∥N_{j-1}\left(t\right)∥={sup}_{t\in \Gamma }\left|N_{j-1}\left(t\right)\right|$, and $∥N_j\left(t\right)∥={sup}_{t\in \Gamma }\left|N_j\left(t\right)\right|$. When the Caputo derivative is applied to the model results in the following expression:

$$\left\{\begin{array}{c}{N}_1\left(t\right)-N_1\left(0\right)={\mathfrak{D}}_t^{ϵ}{N}_1\left(t\right)\left\{-{\lambda }_1{N}_1\left(t\right)\right\},\hfill \\ N_2\left(t\right)-N_2\left(0\right)={\mathfrak{D}}_t^{ϵ}{N}_2\left(t\right)\left\{{\lambda }_1{N}_1\left(t\right)-{\lambda }_2{N}_2\left(t\right)\right\},\hfill \\ N_3\left(t\right)-N_3\left(0\right)={\mathfrak{D}}_t^{ϵ}{N}_3\left(t\right)\left\{{\lambda }_2{N}_2\left(t\right)-{\lambda }_3{N}_3\left(t\right)\right\},\hfill \\ ⋮⋮⋮\hfill \\ N_{j-1}\left(t\right)-N_{j-1}\left(0\right)={\mathfrak{D}}_t^{ϵ}{N}_{j-1}\left(t\right)\left\{{\lambda }_{j-2}{N}_{j-2}\left(t\right)-{\lambda }_{j-1}{N}_{j-1}\left(t\right)\right\},\hfill \\ N_j\left(t\right)-N_j\left(0\right)={\mathfrak{D}}_t^{ϵ}{N}_j\left(t\right)\left\{{\lambda }_{j-1}{N}_{j-1}\left(t\right)\right\}.\hfill \end{array}\right.$$

(8)

After calculation

$$\begin{array}{cc}\hfill & N_1\left(t\right)-N_1\left(0\right)={\displaystyle \frac{1}{\Gamma \left(ϵ\right)}}{\int }_0^t{(t-\eta )}^{ϵ-1}{\mathcal{B}}_1(ϵ,\eta ,N_1\left(\eta \right))d\eta ,\hfill \\ \hfill & N_2\left(t\right)-N_2\left(0\right)={\displaystyle \frac{1}{\Gamma \left(ϵ\right)}}{\int }_0^t{(t-\eta )}^{ϵ-1}{\mathcal{B}}_2(ϵ,\eta ,N_2\left(\eta \right))d\eta ,\hfill \\ \hfill & N_3\left(t\right)-N_3\left(0\right)={\displaystyle \frac{1}{\Gamma \left(ϵ\right)}}{\int }_0^t{(t-\eta )}^{ϵ-1}{\mathcal{B}}_3(ϵ,\eta ,N_3\left(\eta \right))d\eta ,\hfill \\ \hfill & ⋮⋮⋮\hfill \\ \hfill & N_{j-1}\left(t\right)-N_{j-1}\left(0\right)={\displaystyle \frac{1}{\Gamma \left(ϵ\right)}}{\int }_0^t{(t-\eta )}^{ϵ-1}{\mathcal{B}}_{j-1}(ϵ,\eta ,N_{j-1}\left(\eta \right))d\eta ,\hfill \\ \hfill & N_j\left(t\right)-N_j\left(0\right)={\displaystyle \frac{1}{\Gamma \left(ϵ\right)}}{\int }_0^t{(t-\eta )}^{ϵ-1}{\mathcal{B}}_j(ϵ,\eta ,N_j\left(\eta \right))d\eta .\hfill \end{array}$$

(9)

where we use the normalization constant $\mathcal{H}\left(ϵ\right)=1/\Gamma \left(ϵ\right)$ for the Caputo kernel.

Here

$$\left\{\begin{array}{c}{N}_1\left(t\right)-N_1\left(0\right)={\mathfrak{D}}_t^{ϵ}{N}_1\left(t\right)\left\{-{\lambda }_1{N}_1\left(t\right)\right\},\hfill \\ N_2\left(t\right)-N_2\left(0\right)={\mathfrak{D}}_t^{ϵ}{N}_2\left(t\right)\left\{{\lambda }_1{N}_1\left(t\right)-{\lambda }_2{N}_2\left(t\right)\right\},\hfill \\ N_3\left(t\right)-N_3\left(0\right)={\mathfrak{D}}_t^{ϵ}{N}_3\left(t\right)\left\{{\lambda }_2{N}_2\left(t\right)-{\lambda }_3{N}_3\left(t\right)\right\},\hfill \\ ⋮⋮⋮\hfill \\ N_{j-1}\left(t\right)-N_{j-1}\left(0\right)={\mathfrak{D}}_t^{ϵ}{N}_{j-1}\left(t\right)\left\{{\lambda }_{j-2}{N}_{j-2}\left(t\right)-{\lambda }_{j-1}{N}_{j-1}\left(t\right)\right\},\hfill \\ N_j\left(t\right)-N_j\left(0\right)={\mathfrak{D}}_t^{ϵ}{N}_j\left(t\right)\left\{{\lambda }_{j-1}{N}_{j-1}\left(t\right)\right\}.\hfill \end{array}\right.$$

(10)

$$\left\{\begin{array}{c}{\mathcal{B}}_1(ϵ,t,N_1\left(t\right))=-{\lambda }_1{N}_1\left(t\right),\hfill \\ {\mathcal{B}}_2(ϵ,t,N_2\left(t\right))={\lambda }_1{N}_1\left(t\right)-{\lambda }_2{N}_2\left(t\right),\hfill \\ {\mathcal{B}}_3\left(ϵ,t,N_3\left(t\right)\right)={\lambda }_2{N}_2\left(t\right)-{\lambda }_3{N}_3\left(t\right),\hfill \\ ⋮⋮⋮\hfill \\ {\mathcal{B}}_{j-1}\left(ϵ,t,N_{j-1}\left(t\right)\right)={\lambda }_{j-2}{N}_{j-2}\left(t\right)-{\lambda }_{j-1}{N}_{j-1}\left(t\right),\hfill \\ {\mathcal{B}}_j\left(ϵ,t,N_j\left(t\right)\right)={\lambda }_{j-1}{N}_{j-1}\left(t\right).\hfill \end{array}\right.$$

(11)

If $N_1\left(t\right),N_2\left(t\right),N_3\left(t\right),\cdots ,N_{j-1}\left(t\right)$, and $N_j\left(t\right)$ are bounded above, then the constants ${\mathcal{B}}_1,{\mathcal{B}}_2,{\mathcal{B}}_3,\cdots ,{\mathcal{B}}_{j-1}$, and ${\mathcal{B}}_j$ must satisfy the Lipschitz condition. Additionally, it is important to highlight that $N_1\left(t\right)$ and $N_1^{*}\left(t\right)$ represent a pair of related functions, leading to the conclusion:

$$\begin{array}{cc}\hfill ∥{\mathcal{B}}_1(ϵ,t,N_1\left(t\right))-{\mathcal{B}}_1\left(ϵ,t,N_1^{*}\left(t\right)\right)∥& =∥-{\lambda }_1\left(N_1\left(t\right)-N_1^{*}\left(t\right)\right)∥\hfill \\ \hfill & \le ∥{\lambda }_1∥∥N_1\left(t\right)-N_1^{*}\left(t\right)∥.\hfill \end{array}$$

(12)

So, Equation (12) becomes

$$∥{\mathcal{B}}_1(ϵ,t,N_1\left(t\right))-{\mathcal{B}}_1\left(ϵ,t,N_1^{*}\left(t\right)\right)∥\le {\lambda }_1∥N_1\left(t\right)-N_1^{*}\left(t\right)∥.$$

(13)

Similarly

$$\begin{array}{cc}\hfill & ∥{\mathcal{B}}_2(ϵ,t,N_2\left(t\right))-{\mathcal{B}}_2\left(ϵ,t,N_2^{*}\left(t\right)\right)∥\le {\lambda }_2∥N_2\left(t\right)-N_2^{*}\left(t\right)∥,\hfill \\ \hfill & ∥{\mathcal{B}}_3(ϵ,t,N_3\left(t\right))-{\mathcal{B}}_3\left(ϵ,t,N_3^{*}\left(t\right)\right)∥\le {\lambda }_3∥N_3\left(t\right)-N_3^{*}\left(t\right)∥,\hfill \\ \hfill & ⋮⋮⋮\hfill \\ \hfill & ∥{\mathcal{B}}_{j-1}\left(ϵ,t,N_{j-1}\left(t\right)\right)-{\mathcal{B}}_{j-1}\left(ϵ,t,N_{j-1}^{*}\left(t\right)\right)∥\le {\lambda }_{j-1}∥N_{j-1}\left(t\right)-N_{j-1}^{*}\left(t\right)∥,\hfill \\ \hfill & ∥{\mathcal{B}}_j\left(ϵ,t,N_j\left(t\right)\right)-{\mathcal{B}}_j\left(ϵ,t,N_j^{*}\left(t\right)\right)∥\le {\lambda }_j∥N_j\left(t\right)-N_j^{*}\left(t\right)∥.\hfill \end{array}$$

(14)

Fixed-point operator and contraction.

Let $P=\mathfrak{Z}{(\Gamma )}^j$ with $\Gamma =[0,\kappa ]$ and norm $\parallel N\parallel ={\sum }_{i=1}^j{\parallel N_i\parallel }_{\infty }$, $\parallel N_i{\parallel }_{\infty }={sup}_{t\in [0,\kappa ]}\left|N_i\left(t\right)\right|$. Assume for each i there is ${\lambda }_i>0$ with $|{\mathcal{B}}_i(ϵ,t,u)-{\mathcal{B}}_i(ϵ,t,v)|\le {\lambda }_i|u-v|$. Define $T:P\to P$ componentwise by the Caputo integral form

$${\left(TN\right)}_i\left(t\right)=N_i\left(0\right)+{\displaystyle \frac{1}{\Gamma \left(ϵ\right)}}{\int }_0^t{(t-\eta )}^{ϵ-1}{\mathcal{B}}_i\left(ϵ,\eta ,N_i\left(\eta \right)\right)d\eta ,i=1,\cdots ,j.$$

For $X,Y\in P$ and each i,

$${\parallel (TX-TY)}_i{\parallel }_{\infty }\le {\displaystyle \frac{{\lambda }_i}{\Gamma \left(ϵ\right)}}\underset{t\in [0,\kappa ]}{sup}{\int }_0^t{(t-\eta )}^{ϵ-1}\parallel X_i-Y_i{\parallel }_{\infty }d\eta ={\displaystyle \frac{{\lambda }_i{\kappa }^{ϵ}}{\Gamma (ϵ+1)}}{\parallel X_i-Y_i\parallel }_{\infty }.$$

Summing over i gives

$$\parallel TX-TY\parallel \le L\parallel X-Y\parallel ,L:=\underset{1\le i\le j}{max}{\displaystyle \frac{{\kappa }^{ϵ}{\lambda }_i}{\Gamma (ϵ+1)}}.$$

Write ${\mathcal{B}}_i(ϵ,t,N_i)={\mathcal{B}}_i(ϵ,t,0)+[{\mathcal{B}}_i(ϵ,t,N_i)-{\mathcal{B}}_i(ϵ,t,0)]$ and set $M_i:={sup}_{t\in [0,\kappa ]}\left|{\mathcal{B}}_i(ϵ,t,0)\right|$ (finite by continuity). Then,

$${\parallel \left(TN\right)}_i{\parallel }_{\infty }\le \left|N_i\left(0\right)\right|+{\displaystyle \frac{{\kappa }^{ϵ}}{\Gamma (ϵ+1)}}\left(M_i+{\lambda }_i{\parallel N_i\parallel }_{\infty }\right),$$

so

$$\parallel TN\parallel \le A+L\parallel N\parallel ,A:=\sum _{i=1}^j\left|N_i\left(0\right)\right|+{\displaystyle \frac{{\kappa }^{ϵ}}{\Gamma (ϵ+1)}}\sum _{i=1}^j{M}_i.$$

Choose $R\ge A/(1-L)$. Then, $\parallel N\parallel \le R\Rightarrow \parallel TN\parallel \le A+LR\le R$, i.e., $T\left(\overline{B_R}\right)\subset \overline{B_R}$.

This indicates that the Lipschitz condition is satisfied for all five functions. Proceeding recursively, the expressions in (9) result in:

$$\begin{array}{cc}\hfill N_{1_n}\left(t\right)=& \mathcal{H}\left(ϵ\right){\int }_0^t{(t-\eta )}^{ϵ-1}{\mathcal{B}}_1\left(ϵ,\eta ,N_{1_{n-1}}\left(\eta \right)\right)d\eta ,\hfill \\ \hfill N_{2_n}\left(t\right)=& \mathcal{H}\left(ϵ\right){\int }_0^t{(t-\eta )}^{ϵ-1}{\mathcal{B}}_2\left(ϵ,\eta ,N_{2_{n-1}}\left(\eta \right)\right)d\eta ,\hfill \\ \hfill N_{3_n}\left(t\right)=& \mathcal{H}\left(ϵ\right){\int }_0^t{(t-\eta )}^{ϵ-1}{\mathcal{B}}_3\left(ϵ,\eta ,N_{3_{n-1}}\left(\eta \right)\right)d\eta ,\hfill \\ \hfill & ⋮⋮⋮\hfill \\ \hfill N_{{j-1}_n}\left(t\right)=& \mathcal{H}\left(ϵ\right){\int }_0^t{(t-\eta )}^{ϵ-1}{\mathcal{B}}_{j-1}\left(ϵ,\eta ,N_{{j-1}_{n-1}}\left(\eta \right)\right)d\eta ,\hfill \\ \hfill N_{j_n}\left(t\right)=& \mathcal{H}\left(ϵ\right){\int }_0^t{(t-\eta )}^{ϵ-1}{\mathcal{B}}_j\left(ϵ,\eta ,N_{j_{n-1}}\left(\eta \right)\right)d\eta ,\hfill \end{array}$$

(15)

The successive differences between terms can be determined by using the initial conditions $N_1\left(0\right)=N_{1_0}$, $N_2\left(0\right)=N_{2_0}$, $N_3\left(0\right)=N_{3_0}$, ·, $N_{j-1}\left(0\right)=N_{{j-1}_0}$, and $N_j\left(0\right)=N_{j_0}$, along with the relevant associated information.

$$\begin{array}{cc}\hfill {\Xi }_{N_1,n}\left(t\right)=& N_{1_n}\left(t\right)-N_{1_{n-1}}\left(t\right)\hfill \\ \hfill =& \mathcal{H}\left(ϵ\right){\int }_0^t{(t-\eta )}^{ϵ-1}\left({\mathcal{B}}_1\left(ϵ,\eta ,N_{1_{n-1}}\left(\eta \right)\right)-{\mathcal{B}}_1\left(ϵ,\eta ,N_{1_{n-2}}\left(\eta \right)\right)\right)d\eta ,\hfill \\ \hfill {\Xi }_{N_2,n}\left(t\right)=& N_{2_n}\left(t\right)-N_{2_{n-1}}\left(t\right)\hfill \\ \hfill =& \mathcal{H}\left(ϵ\right){\int }_0^t{(t-\eta )}^{ϵ-1}\left({\mathcal{B}}_2\left(ϵ,\eta ,N_{2_{n-1}}\left(\eta \right)\right)-{\mathcal{B}}_2\left(ϵ,\eta ,N_{2_{n-2}}\left(\eta \right)\right)\right)d\eta ,\hfill \\ \hfill {\Xi }_{N_3,n}\left(t\right)=& N_{3_n}\left(t\right)-N_{3_{n-1}}\left(t\right)\hfill \\ \hfill =& \mathcal{H}\left(ϵ\right){\int }_0^t{(t-\eta )}^{ϵ-1}\left({\mathcal{B}}_3\left(ϵ,\eta ,N_{3_{n-1}}\left(\eta \right)\right)-{\mathcal{B}}_3\left(ϵ,\eta ,N_{3_{n-2}}\left(\eta \right)\right)\right)d\eta ,\hfill \\ \hfill & ⋮⋮⋮\hfill \\ \hfill {\Xi }_{N_{j-1},n}\left(t\right)=& N_{{j-1}_n}\left(t\right)-N_{{j-1}_{n-1}}\left(t\right)\hfill \\ \hfill =& \mathcal{H}\left(ϵ\right){\int }_0^t{(t-\eta )}^{ϵ-1}\left({\mathcal{B}}_{j-1}\left(ϵ,\eta ,N_{{j-1}_{n-1}}\left(\eta \right)\right)-{\mathcal{B}}_{j-1}\left(ϵ,\eta ,N_{{j-1}_{n-2}}\left(\eta \right)\right)\right)d\eta ,\hfill \\ \hfill {\Xi }_{N_j,n}\left(t\right)=& N_{j_n}\left(t\right)-N_{j_{n-1}}\left(t\right)\hfill \\ \hfill =& \mathcal{H}\left(ϵ\right){\int }_0^t{(t-\eta )}^{ϵ-1}\left({\mathcal{B}}_j\left(ϵ,\eta ,N_{j_{n-1}}\left(\eta \right)\right)-{\mathcal{B}}_j\left(ϵ,\eta ,N_{j_{n-2}}\left(\eta \right)\right)\right)d\eta .\hfill \end{array}$$

(16)

It is vital to observe that

$$\begin{array}{cc}\hfill N_{1_n}\left(t\right)=\sum _{m=0}^n{\Xi }_{N_1,m}\left(t\right),& N_{2_n}\left(t\right)=\sum _{m=0}^n{\Xi }_{N_2,m}\left(t\right),N_{3_n}\left(t\right)=\sum _{m=0}^n{\Xi }_{N_3,m}\left(t\right),\hfill \\ \hfill \cdots \cdots ,& N_{{j-1}_n}\left(t\right)=\sum _{m=0}^n{\Xi }_{N_{j-1},m}\left(t\right),N_{j_n}\left(t\right)=\sum _{m=0}^n{\Xi }_{N_j,m}\left(t\right).\hfill \end{array}$$

Furthermore, by referencing Equations (13) and (14) and taking into account that:

$$\begin{array}{cc}\hfill {\Xi }_{N_1,n-1}\left(t\right)=& {N_1}_{n-1}\left(t\right)-{N_1}_{n-2}\left(t\right),\hfill \\ \hfill {\Xi }_{N_2,n-1}\left(t\right)=& {N_2}_{n-1}\left(t\right)-{N_2}_{n-2}\left(t\right),\hfill \\ \hfill {\Xi }_{N_3,n-1}\left(t\right)=& {N_3}_{n-1}\left(t\right)-{N_3}_{n-2}\left(t\right),\hfill \\ \hfill ⋮& ⋮⋮\hfill \\ \hfill {\Xi }_{N_{j-1},n-1}\left(t\right)=& {N_{j-1}}_{n-1}\left(t\right)-{N_{j-1}}_{n-2}\left(t\right),\hfill \\ \hfill {\Xi }_{N_j,n-1}\left(t\right)=& {N_j}_{n-1}\left(t\right)-{N_j}_{n-2}\left(t\right),\hfill \end{array}$$

we reach

$$\begin{array}{c}∥{\Xi }_{N_1,n}\left(t\right)∥\le \mathcal{H}\left(ϵ\right){\lambda }_1{\int }_0^t{(t-\eta )}^{ϵ-1}∥{\Xi }_{N_1,n-1}\left(\eta \right)∥d\eta ,\hfill \\ ∥{\Xi }_{N_2,n}\left(t\right)∥\le \mathcal{H}\left(ϵ\right){\lambda }_2{\int }_0^t{(t-\eta )}^{ϵ-1}∥{\Xi }_{N_2,n-1}\left(\eta \right)∥d\eta ,\hfill \\ ∥{\Xi }_{N_3,n}\left(t\right)∥\le \mathcal{H}\left(ϵ\right){\lambda }_3{\int }_0^t{(t-\eta )}^{ϵ-1}∥{\Xi }_{N_3,n-1}\left(\eta \right)∥d\eta ,\hfill \\ ⋮⋮⋮\hfill \\ ∥{\Xi }_{N_{j-1},n}\left(t\right)∥\le \mathcal{H}\left(ϵ\right){\lambda }_{j-1}{\int }_0^t{(t-\eta )}^{ϵ-1}∥{\Xi }_{N_{j-1},n-1}\left(\eta \right)∥d\eta ,\hfill \\ ∥{\Xi }_{N_j,n}\left(t\right)∥\le \mathcal{H}\left(ϵ\right){\lambda }_j{\int }_0^t{(t-\eta )}^{ϵ-1}∥{\Xi }_{N_j,n-1}\left(\eta \right)∥d\eta .\hfill \end{array}$$

(17)

Theorem 1.

Let $0<ϵ\le 1$ and define the normalization constant for the Caputo fractional derivative as

$$\mathcal{H}\left(ϵ\right)={\displaystyle \frac{1}{\Gamma \left(ϵ\right)}}.$$

If the following condition is satisfied:

$${\displaystyle \frac{{\kappa }^{ϵ}{\lambda }_m}{\Gamma (ϵ+1)}}<1,m=1,2,3,\cdots ,(j-1),j,$$

(18)

then the proposed model admits a unique solution for $t\in [0,\kappa ]$.

Proof. 

From the Caputo form of the system, each component $N_i\left(t\right)$ satisfies

$$N_i\left(t\right)-N_i\left(0\right)={\displaystyle \frac{1}{\Gamma \left(ϵ\right)}}{\int }_0^t{(t-\eta )}^{ϵ-1}{\mathcal{B}}_i(ϵ,\eta ,N\left(\eta \right))d\eta .$$

It follows that $N_1\left(t\right)$, $N_2\left(t\right)$, …, $N_j\left(t\right)$ are bounded functions. Moreover, as demonstrated in Equations (13) and (14), each nonlinear term ${\mathcal{B}}_i$, $i=1,2,\cdots ,j$, satisfies the Lipschitz condition with constant ${\lambda }_i$.

Let ${\Xi }_{N_i,n}\left(t\right)=N_{i,n}\left(t\right)-N_{i,n-1}\left(t\right)$ denote the Picard iteration differences. Using the Lipschitz property and the above integral form, for $t\in [0,\kappa ]$ we have

$$\parallel {\Xi }_{N_i,n}\left(t\right)\parallel \le {\displaystyle \frac{{\lambda }_i}{\Gamma \left(ϵ\right)}}{\int }_0^t{(t-\eta )}^{ϵ-1}\parallel {\Xi }_{N_i,n-1}\left(\eta \right)\parallel d\eta \le {\displaystyle \frac{{\lambda }_i{\kappa }^{ϵ}}{\Gamma (ϵ+1)}}{\parallel {\Xi }_{N_i,n-1}\parallel }_{\infty }.$$

Hence,

$$\begin{array}{c}∥{\Xi }_{N_1,n}\left(t\right)∥\le ∥N_1\left(0\right)\left(t\right)∥{\left({\displaystyle \frac{{\kappa }^{ϵ}{\lambda }_1}{\Gamma (ϵ+1)}}\right)}^n,\hfill \\ ∥{\Xi }_{N_2,n}\left(t\right)∥\le ∥N_2\left(0\right)\left(t\right)∥{\left({\displaystyle \frac{{\kappa }^{ϵ}{\lambda }_2}{\Gamma (ϵ+1)}}\right)}^n,\hfill \\ ∥{\Xi }_{N_3,n}\left(t\right)∥\le ∥N_3\left(0\right)\left(t\right)∥{\left({\displaystyle \frac{{\kappa }^{ϵ}{\lambda }_3}{\Gamma (ϵ+1)}}\right)}^n,\hfill \\ ⋮\hfill \\ ∥{\Xi }_{N_{j-1},n}\left(t\right)∥\le ∥N_{j-1}\left(0\right)\left(t\right)∥{\left({\displaystyle \frac{{\kappa }^{ϵ}{\lambda }_{j-1}}{\Gamma (ϵ+1)}}\right)}^n,\hfill \\ ∥{\Xi }_{N_j,n}\left(t\right)∥\le ∥N_j\left(0\right)\left(t\right)∥{\left({\displaystyle \frac{{\kappa }^{ϵ}{\lambda }_j}{\Gamma (ϵ+1)}}\right)}^n.\hfill \end{array}$$

(19)

Since by hypothesis ${\displaystyle \frac{{\kappa }^{ϵ}{\lambda }_m}{\Gamma (ϵ+1)}}<1$ for all m, the sequences converge:

$$\parallel {\Xi }_{N_i,n}\left(t\right)\parallel \to 0\mathrm{as}n\to \infty ,i=1,\cdots ,j.$$

Furthermore, applying the triangle inequality, for any $s\in \mathbb{N}$ we obtain

$$\parallel N_{i,(n+s)}\left(t\right)-N_{i,\left(n\right)}\left(t\right)\parallel \le {\displaystyle \frac{X_i^{n+1}-X_i^{n+s+1}}{1-X_i}},X_i={\displaystyle \frac{{\kappa }^{ϵ}{\lambda }_i}{\Gamma (ϵ+1)}}<1.$$

(20)

Thus, $\left\{N_{i,n}\right\}$ is Cauchy in the Banach space $\mathfrak{Z}(\Gamma )$ for each i. By completeness, the limits exist and satisfy the original system (1). This proves the existence and uniqueness of solutions under condition (18). □

4. Principal Concepts of the Methods

4.1. The Temimi–Ansari Method

In order to clarify the main idea behind the proposed approach, we assume the following things regarding the general non-homogeneous fractional differential equation (FPDE) [19,20,21]:

$$\Xi \left(N\right(x,t\left)\right)=\Lambda \left(N\right(x,t\left)\right)+z(x,t),m-1<ϵ\le m,$$

(21)

with the following

$$\mathfrak{O}\left(N,{\displaystyle \frac{\partial N}{\partial x}}\right)=0,$$

(22)

where the fractional differential operator, denoted by $\Xi ={\mathfrak{D}}_t^{ϵ}={\displaystyle \frac{{\partial }^{ϵ}}{\partial t^{ϵ}}}$ is utilized and defined as a partial derivative of fractional order. The symbol $\Lambda$ represents standard differential operators, while $N(x,t)$ stands for the unknown function being solved. The function $z(x,t)$ is a known continuous function. The operator $\mathfrak{O}$ is used to express the boundary conditions. The main requirement in this context is the use of the fractional operator $\Xi$ as the general fractional differential operator. Linear expressions can also be formulated as needed to handle nonlinear terms. The initial step in this method involves extracting the initial condition by isolating the nonlinear component. This leads to the expression:

$${\mathfrak{D}}_t^{ϵ}{N}_0(x,t)=z(x,t),\mathfrak{O}\left(N_0,{\displaystyle \frac{\partial N_0}{\partial x}}\right)=0.$$

(23)

Following that, the next iteration is obtained by solving the subsequent equation

$${\mathfrak{D}}_t^{ϵ}{N}_1(x,t)=\Lambda \left(N_0(x,t)\right)+z(x,t),\mathfrak{O}\left(N_1,{\displaystyle \frac{\partial N_1}{\partial x}}\right)=0.$$

(24)

This provides a clear and straightforward iterative step to determine $N_{m+1}(x,t)$, making it well-suited for addressing both linear and nonlinear systems. The corresponding formulation takes the form:

$${\mathfrak{D}}_t^{ϵ}{N}_{m+1}(x,t)=\Lambda \left(N_m(x,t)\right)+z(x,t),\mathfrak{O}\left(N_{m+1},{\displaystyle \frac{\partial N_{m+1}}{\partial x}}\right)=0.$$

(25)

An important point to note is that in this technique, the function $N_{m+1}(x,t)$ is calculated independently in each step. The iterative process is straightforward to apply, and each new iteration enhances the result from the previous one. By continuing this procedure, it becomes possible to achieve a highly accurate approximation that closely matches the true solution. Therefore, the result for Equation (21) is derived accordingly as:

$$N(x,t)=\underset{m\to \infty }{lim}{N}_m(x,t)$$

4.2. The Sumudu Residual Power Series Method

The ST and a residual power series are used in the SRPSM to enable effective series solutions for fractional differential equations. The capacity of this approach to convert complicated fractional expressions into simpler algebraic forms makes it particularly helpful for generating analytical solutions to issues involving fractional operators. The following stages comprise the method for using SRPSM to solve the Actinium nuclear radioactive series of fractional order [22,23,24].

• First: Using the ST on Equation (21) as,

$$\mathcal{S}\left(\Xi \right(N(x,t)\left)\right)=\mathcal{S}\left(\Xi \right(N(x,t))+z(x,t\left)\right).$$

(26)

Then, using the ST’s differentiation property as: $\mathcal{S}\left[{\mathfrak{D}}_t^{\epsilon }N(x,t)\right]={\displaystyle \frac{1}{s^{\epsilon }}}\left\{\mathcal{S}\left(N(x,t)\right)-N_0\left(x\right)\right\}$, we obtain

$${\displaystyle \frac{1}{s^{\epsilon }}}\left\{\mathcal{S}\left(N(x,t)\right)-N_0\left(x\right)\right\}=\mathcal{S}\{\Xi \left(N(x,t)\right)+z(x,t)\},$$

(27)

i.e.,

$$\mathcal{S}\left(N(x,t)\right)=N_0\left(x\right)+s^{\epsilon }\mathcal{S}\{\Xi \left(N(x,t)\right)+z(x,t)\}.$$

(28)

• Second: Using the inverse function for ST in Equation (28) with the initial condition $N_0\left(x\right)$, we obtain

$$N(x,t)=N_0\left(x\right)+{\mathcal{S}}^{-1}\left[s^{\epsilon }\mathcal{S}\{\Xi \left(N(x,s)\right)+z(x,s)\}\right],$$

(29)

here.

• Third: Thus, the form of the $N(x,t)$ algorithm may be thought of as

$$N(x,t)=\stackrel{[}{n}=0]\infty \sum {\displaystyle \frac{f_n(x,t)t^{n\epsilon }}{\Gamma \left(1+n\epsilon \right)}}$$

• Finally: It is possible to express the Sumudu residual’s function as

$$\mathcal{S}Res\left(\Xi (x,s)\right)=\Xi (x,s)-{\displaystyle \frac{1}{s}}{N}_0\left(x\right)-{\displaystyle \frac{1}{s^{\epsilon }}}\{\Xi \left(N(x,s)\right)+z(x,s)\}.$$

(30)

5. Analytical Solution of the Fractional Nuclear Radioactive Series Equations

Here, we will construct the Analytical solution of the fractional nuclear radioactive series equations obtained using both the Sumudu-Residual Power Series Method and the Temimi–Ansari method as follows

5.1. Approximate Solution by SRPSM

The aim at this stage is to develop a series-based solution for fractional neutron diffusion equations by employing a combined analytical approach involving a specific integral transform (SRPSM). Applying this approach to Equation (1) leads us to observe the following outcome:

$$\begin{array}{ccc}\hfill \mathcal{S}\left[{\mathfrak{D}}_t^{ϵ}{N}_1\left(t\right)\right]& =& -{\lambda }_1\mathcal{S}\left[N_1\left(t\right)\right],\hfill \\ \hfill \mathcal{S}\left[{\mathfrak{D}}_t^{ϵ}{N}_2\left(t\right)\right]& =& {\lambda }_1\mathcal{S}\left[N_1\left(t\right)\right]-{\lambda }_2\mathcal{S}\left[N_2\left(t\right)\right],\hfill \\ \hfill \mathcal{S}\left[{\mathfrak{D}}_t^{ϵ}{N}_3\left(t\right)\right]& =& {\lambda }_2\mathcal{S}\left[N_2\left(t\right)\right]-{\lambda }_3\mathcal{S}\left[N_3\left(t\right)\right],\hfill \\ \hfill ⋮& ⋮& ⋮\hfill \\ \hfill \mathcal{S}\left[{\mathfrak{D}}_t^{ϵ}{N}_{j-1}\left(t\right)\right]& =& {\lambda }_{j-2}\mathcal{S}\left[N_{j-2}\left(t\right)\right]-{\lambda }_{j-1}\mathcal{S}\left[N_{j-1}\left(t\right)\right],\hfill \\ \hfill \mathcal{S}\left[{\mathfrak{D}}_t^{ϵ}{N}_j\left(t\right)\right]& =& {\lambda }_{j-1}\mathcal{S}\left[N_{j-1}\left(t\right)\right].\hfill \end{array}$$

(31)

Based on Lemma 1 and the initial conditions given in Equations (2) and (3), we can derive Equation (1) as follows:

$$\begin{array}{ccc}\hfill s^{ϵ}{\Xi }_1\left(s\right)-s^{ϵ-1}{\left(N_1\right)}_0\left(t\right)& =& -{\lambda }_1{\Xi }_1\left(s\right),\hfill \\ \hfill s^{ϵ}{\Xi }_2\left(s\right)-s^{ϵ-1}{\left(N_2\right)}_0\left(t\right)& =& {\lambda }_1{\Xi }_1\left(s\right)-{\lambda }_2{\Xi }_2\left(s\right),\hfill \\ \hfill ⋮& ⋮& ⋮\hfill \\ \hfill s^{ϵ}{\Xi }_{j-1}\left(s\right)-s^{ϵ-1}{\left(N_{j-1}\right)}_0\left(t\right)& =& {\lambda }_{j-2}{\Xi }_{j-2}\left(s\right)-{\lambda }_{j-1}{\Xi }_{j-1}\left(s\right),\hfill \\ \hfill s^{ϵ}{\Xi }_j\left(s\right)-s^{ϵ-1}{\left(N_j\right)}_0\left(t\right)& =& {\lambda }_{j-1}{\Xi }_{j-1}\left(s\right),\hfill \end{array}$$

(32)

where ${\Xi }_i\left(s\right)=\mathcal{S}\left[N_i\left(t\right)\right],i=1,2,\cdots ,j$. Equation (32) may be rewritten as

$$\begin{array}{ccc}\hfill {\Xi }_1\left(s\right)-{\displaystyle \frac{1}{s}}{\left(N_1\right)}_0\left(t\right)+{\displaystyle \frac{{\lambda }_1}{s^{ϵ}}}{\Xi }_1\left(s\right)& =& 0,\hfill \\ \hfill {\Xi }_2\left(s\right)-{\displaystyle \frac{1}{s}}{\left(N_2\right)}_0\left(t\right)-{\displaystyle \frac{{\lambda }_1}{s^{ϵ}}}{\Xi }_1\left(s\right)+{\displaystyle \frac{{\lambda }_2}{s^{ϵ}}}{\Xi }_2\left(s\right)& =& 0,\hfill \\ \hfill ⋮& ⋮& ⋮\hfill \\ \hfill {\Xi }_{j-1}\left(s\right)-{\displaystyle \frac{1}{s}}{\left(N_{j-1}\right)}_0\left(t\right)-{\displaystyle \frac{{\lambda }_{j-2}}{s^{ϵ}}}{\Xi }_{j-2}\left(s\right)+{\displaystyle \frac{{\lambda }_{j-1}}{s^{ϵ}}}{\Xi }_{j-1}\left(s\right)& =& 0,\hfill \\ \hfill {\Xi }_j\left(s\right)-{\displaystyle \frac{1}{s}}{\left(N_n\right)}_0\left(t\right)-{\displaystyle \frac{{\lambda }_{j-1}}{s^{ϵ}}}{\Xi }_{j-1}\left(s\right)& =& 0.\hfill \end{array}$$

(33)

Model (33) represents a linear system of partial differential equations involving derivatives with respect to the variable x. Following the SRPS approach, the series solution for model (33) must follow the structure outlined below:

$${\Xi }_1\left(s\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(f_1\right)}_n\left(t\right)}{s^{nϵ+1}}},{\Xi }_2\left(s\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(f_2\right)}_n\left(t\right)}{s^{nϵ+1}}},\cdots ,{\Xi }_j\left(s\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(f_j\right)}_n\left(t\right)}{s^{nϵ+1}}}.$$

(34)

By applying the result from Lemma 1, the truncated series of order k^th for the transformed functions ${\Xi }_i\left(s\right)=\mathcal{S}\left[f_i\left(t\right)\right]$, where $i=1,2,3$, can be written in the following form:

$$\begin{array}{ccc}\hfill {\left({\Xi }_1\right)}_k\left(s\right)& =& {\displaystyle \frac{{\left(f_1\right)}_0\left(t\right)}{s}}+\sum _{n=1}^k{\displaystyle \frac{{\left(f_1\right)}_n\left(t\right)}{s^{nϵ+1}}},t\in I,s>\delta \ge 0,\hfill \\ \hfill {\left({\Xi }_2\right)}_k\left(s\right)& =& {\displaystyle \frac{{\left(f_2\right)}_0\left(t\right)}{s}}+\sum _{n=1}^k{\displaystyle \frac{{\left(f_2\right)}_n\left(t\right)}{s^{nϵ+1}}},t\in I,s>\delta \ge 0,\hfill \\ \hfill ⋮& ⋮& ⋮\hfill \\ \hfill {\left({\Xi }_j\right)}_k\left(s\right)& =& {\displaystyle \frac{{\left(f_j\right)}_0\left(t\right)}{s}}+\sum _{n=1}^k{\displaystyle \frac{{\left(f_j\right)}_n\left(t\right)}{s^{nϵ+1}}},t\in I,s>\delta \ge 0.\hfill \end{array}$$

(35)

The following step involves determining the unknown coefficients in model (35) through the construction of the Sumudu residual functions corresponding to model (33):

$$\begin{array}{ccc}\hfill \mathcal{S}Res\left({\Xi }_1\left(s\right)\right)& =& {\Xi }_1\left(s\right)-{\displaystyle \frac{1}{s}}{\left(f_1\right)}_0\left(t\right)+{\displaystyle \frac{{\lambda }_1}{s^{ϵ}}}{\Xi }_1\left(s\right),\hfill \\ \hfill \mathcal{S}Res\left({\Xi }_2\left(s\right)\right)& =& {\Xi }_2\left(s\right)-{\displaystyle \frac{1}{s}}{\left(f_2\right)}_0\left(t\right)-{\displaystyle \frac{{\lambda }_1}{s^{ϵ}}}{\Xi }_1\left(s\right)+{\displaystyle \frac{{\lambda }_2}{s^{ϵ}}}{\Xi }_2\left(s\right),\hfill \\ \hfill ⋮& ⋮& ⋮\hfill \\ \hfill \mathcal{S}Res\left({\Xi }_{j-1}\left(s\right)\right)& =& {\Xi }_{j-1}\left(s\right)-{\displaystyle \frac{1}{s}}{\left(f_{j-1}\right)}_0\left(t\right)-{\displaystyle \frac{{\lambda }_{j-2}}{s^{ϵ}}}{\Xi }_{j-2}\left(s\right)+{\displaystyle \frac{{\lambda }_{j-1}}{s^{ϵ}}}{\Xi }_{j-1}\left(s\right),\hfill \\ \hfill \mathcal{S}Res\left({\Xi }_j\left(s\right)\right)& =& {\Xi }_j\left(s\right)-{\displaystyle \frac{1}{s}}{\left(f_j\right)}_0\left(t\right)-{\displaystyle \frac{{\lambda }_{j-1}}{s^{ϵ}}}{\Xi }_{j-1}\left(s\right).\hfill \end{array}$$

(36)

Additionally, the k^th Sumudu-residual functions as

$$\begin{array}{ccc}\hfill \mathcal{S}Re{s}_k\left({\Xi }_1\left(s\right)\right)& =& {\left({\Xi }_1\right)}_k\left(s\right)-{\displaystyle \frac{1}{s}}{\left(f_1\right)}_0\left(t\right)+{\displaystyle \frac{{\lambda }_1}{s^{ϵ}}}{\left({\Xi }_1\right)}_k\left(s\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \mathcal{S}Re{s}_k\left({\Xi }_2\left(s\right)\right)& =& {\left({\Xi }_2\right)}_k\left(s\right)-{\displaystyle \frac{1}{s}}{\left(f_2\right)}_0\left(t\right)-{\displaystyle \frac{{\lambda }_1}{s^{ϵ}}}{\left({\Xi }_1\right)}_k\left(s\right)+{\displaystyle \frac{{\lambda }_2}{s^{ϵ}}}{\left({\Xi }_2\right)}_k\left(s\right),\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill ⋮& ⋮& ⋮\hfill \\ \hfill \mathcal{S}Re{s}_k\left({\Xi }_j\left(s\right)\right)& =& {\left({\Xi }_j\right)}_k\left(s\right)-{\displaystyle \frac{1}{s}}{\left(f_j\right)}_0\left(t\right)-{\displaystyle \frac{{\lambda }_{j-1}}{s^{ϵ}}}{\left({\Xi }_{j-1}\right)}_k\left(s\right).\hfill \end{array}$$

(38)

Since $\mathcal{S}Res\left({\Xi }_i\left(s\right)\right)=0,i=1,2,\cdots ,j$, we see that $s^{kϵ+1}\mathcal{S}Res\left({\Xi }_i\left(s\right)\right)=0$. Therefore,

$$\underset{s\to \infty }{lim}\left(s^{kϵ+1}\mathcal{S}Re{s}_k\left({\Xi }_i\left(s\right)\right)\right)=0\mathrm{for}k=0,1,2,\cdots .$$

(39)

By replacing with the functions obtained through the Sumudu-residual approach as ${\Xi }_i\left(s\right)={\displaystyle \frac{{\left(f_i\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_i\right)}_1\left(t\right)}{s^{ϵ+1}}},i=1,2,\cdots ,j$, we then obtain ${\left(f_1\right)}_1\left(t\right),$${\left(f_2\right)}_1\left(t\right),$…, ${\left(f_j\right)}_1\left(t\right)$ in (9) as

$$\begin{array}{ccc}\hfill \mathcal{S}Re{s}_1\left({\Xi }_1\left(s\right)\right)& =& {\displaystyle \frac{{\left(f_1\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\lambda }_1}{s^{ϵ}}}\left[{\displaystyle \frac{{\left(f_1\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_1\right)}_1\left(t\right)}{s^{ϵ+1}}}\right],\hfill \\ \hfill \mathcal{S}Re{s}_1\left({\Xi }_2\left(s\right)\right)& =& {\displaystyle \frac{{\left(f_2\right)}_1\left(t\right)}{s^{ϵ+1}}}-{\displaystyle \frac{{\lambda }_1}{s^{ϵ}}}\left[{\displaystyle \frac{{\left(f_1\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_1\right)}_1\left(t\right)}{s^{ϵ+1}}}\right]+{\displaystyle \frac{{\lambda }_2}{s^{ϵ}}}\left[{\displaystyle \frac{{\left(f_2\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_2\right)}_1\left(t\right)}{s^{ϵ+1}}}\right],\hfill \\ \hfill ⋮& ⋮& ⋮\hfill \\ \hfill \mathcal{S}Re{s}_1\left({\Xi }_{j-1}\left(s\right)\right)& =& {\displaystyle \frac{{\left(f_{j-1}\right)}_1\left(t\right)}{s^{ϵ+1}}}-{\displaystyle \frac{{\lambda }_{j-2}}{s^{ϵ}}}\left[{\displaystyle \frac{{\left(f_{j-2}\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_{j-2}\right)}_1\left(t\right)}{s^{ϵ+1}}}\right]+{\displaystyle \frac{{\lambda }_{j-1}}{s^{ϵ}}}\left[{\displaystyle \frac{{\left(f_{j-1}\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_{j-1}\right)}_1\left(t\right)}{s^{ϵ+1}}}\right],\hfill \\ \hfill \mathcal{S}Re{s}_1\left({\Xi }_j\left(s\right)\right)& =& {\displaystyle \frac{{\left(f_j\right)}_1\left(t\right)}{s^{ϵ+1}}}-{\displaystyle \frac{{\lambda }_{j-1}}{s^{ϵ}}}\left[{\displaystyle \frac{{\left(f_{j-1}\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_{j-1}\right)}_1\left(t\right)}{s^{ϵ+1}}}\right].\hfill \end{array}$$

(40)

Now, by solving $\underset{s\to \infty }{lim}{s}^{ϵ+1}\mathcal{S}Re{s}_1\left({\Xi }_i\left(s\right)\right)=0$, we obtain

$${\left(f_1\right)}_1\left(t\right)=-z{\lambda }_1,{\left(f_2\right)}_1\left(t\right)=z{\lambda }_1,{\left(f_3\right)}_1\left(t\right)=0,{\left(f_4\right)}_1\left(t\right)=0,\cdots ,{\left(f_n\right)}_1\left(t\right)=0.$$

(41)

Substituting the second Sumudu-residual functions as ${\Xi }_i\left(s\right)={\displaystyle \frac{{\left(f_i\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_i\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_i\right)}_2\left(t\right)}{s^{2ϵ+1}}},i=1,2,\cdots ,j$, we then obtain ${\left(f_1\right)}_2\left(t\right),$${\left(f_2\right)}_2\left(t\right)$$,\cdots ,$${\left(f_j\right)}_2\left(t\right)$ in (9) as

$$\begin{array}{ccc}\hfill \mathcal{S}Re{s}_2\left({\Xi }_1\left(s\right)\right)& =& {\displaystyle \frac{{\left(f_1\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_1\right)}_2\left(t\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{{\lambda }_1}{s^{ϵ}}}\left[{\displaystyle \frac{{\left(f_1\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_1\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_1\right)}_2\left(t\right)}{s^{2ϵ+1}}}\right],\hfill \\ \hfill \mathcal{S}Re{s}_2\left({\Xi }_2\left(s\right)\right)& =& {\displaystyle \frac{{\left(f_2\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_2\right)}_2\left(t\right)}{s^{2ϵ+1}}}-{\displaystyle \frac{{\lambda }_1}{s^{ϵ}}}\left[{\displaystyle \frac{{\left(f_1\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_1\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_1\right)}_2\left(t\right)}{s^{2ϵ+1}}}\right]+{\displaystyle \frac{{\lambda }_2}{s^{ϵ}}}\left[{\displaystyle \frac{{\left(f_2\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_2\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_2\right)}_2\left(t\right)}{s^{2ϵ+1}}}\right],\hfill \\ \hfill ⋮& ⋮& ⋮\hfill \\ \hfill \mathcal{S}Re{s}_2\left({\Xi }_{j-1}\left(s\right)\right)& =& {\displaystyle \frac{{\left(f_{j-1}\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_{j-1}\right)}_2\left(t\right)}{s^{2ϵ+1}}}-{\displaystyle \frac{{\lambda }_{j-2}}{s^{ϵ}}}\left[{\displaystyle \frac{{\left(f_{j-2}\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_{j-2}\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_{j-2}\right)}_2\left(t\right)}{s^{2ϵ+1}}}\right]+{\displaystyle \frac{{\lambda }_{j-1}}{s^{ϵ}}}\left[{\displaystyle \frac{{\left(f_{j-1}\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_{j-1}\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_{j-1}\right)}_2\left(t\right)}{s^{2ϵ+1}}}\right],\hfill \\ \hfill \mathcal{S}Re{s}_2\left({\Xi }_j\left(s\right)\right)& =& {\displaystyle \frac{{\left(f_j\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_j\right)}_2\left(t\right)}{s^{2ϵ+1}}}-{\displaystyle \frac{{\lambda }_{j-1}}{s^{ϵ}}}\left[{\displaystyle \frac{{\left(f_{j-1}\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_{j-1}\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_{j-1}\right)}_2\left(t\right)}{s^{2ϵ+1}}}\right].\hfill \end{array}$$

(42)

Now, by solving $\underset{s\to \infty }{lim}{s}^{2ϵ+1}\mathcal{S}Re{s}_2\left({\Xi }_i\left(s\right)\right)=0,$ we obtain

$${\left(f_1\right)}_2\left(t\right)=z{\lambda }_1^2,{\left(f_2\right)}_2\left(t\right)=-z{\lambda }_1\left({\lambda }_1+{\lambda }_2\right),{\left(f_3\right)}_2\left(t\right)=z{\lambda }_1{\lambda }_2,{\left(f_4\right)}_2\left(t\right)=0,\cdots ,{\left(f_j\right)}_2\left(t\right)=0.$$

(43)

Substituting the third Sumudu-residual functions as ${\Xi }_i\left(s\right)={\displaystyle \frac{{\left(f_i\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_i\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_i\right)}_2\left(t\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{{\left(f_i\right)}_3\left(t\right)}{s^{3ϵ+1}}},i=1,2,\cdots ,j$, we then obtain ${\left(f_1\right)}_3\left(t\right),$${\left(f_2\right)}_3\left(t\right)$$,\cdots ,$${\left(f_j\right)}_3\left(t\right)$ in (9) as

$$\begin{array}{ccc}\hfill \mathcal{S}Re{s}_3\left({\Xi }_1\left(s\right)\right)& =& {\displaystyle \frac{{\left(f_1\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_1\right)}_2\left(t\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{{\left(f_1\right)}_3\left(t\right)}{s^{3ϵ+1}}}+{\displaystyle \frac{{\lambda }_1}{s^{ϵ}}}\left[{\displaystyle \frac{{\left(f_1\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_1\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_1\right)}_2\left(t\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{{\left(f_1\right)}_3\left(t\right)}{s^{3ϵ+1}}}\right],\hfill \\ \hfill \mathcal{S}Re{s}_3\left({\Xi }_2\left(s\right)\right)& =& {\displaystyle \frac{{\left(f_2\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_2\right)}_2\left(t\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{{\left(f_2\right)}_3\left(t\right)}{s^{3ϵ+1}}}-{\displaystyle \frac{{\lambda }_1}{s^{ϵ}}}\left[{\displaystyle \frac{{\left(f_1\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_1\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_1\right)}_2\left(t\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{{\left(f_1\right)}_3\left(t\right)}{s^{3ϵ+1}}}\right]\hfill \\ & & +{\displaystyle \frac{{\lambda }_2}{s^{ϵ}}}\left[{\displaystyle \frac{{\left(f_2\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_2\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_2\right)}_2\left(t\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{{\left(f_2\right)}_3\left(t\right)}{s^{3ϵ+1}}}\right],\hfill \\ \hfill ⋮& ⋮& ⋮\hfill \\ \hfill \mathcal{S}Re{s}_3\left({\Xi }_{j-1}\left(s\right)\right)& =& {\displaystyle \frac{{\left(f_{j-1}\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_{j-1}\right)}_2\left(t\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{{\left(f_{j-1}\right)}_3\left(t\right)}{s^{3ϵ+1}}}-{\displaystyle \frac{{\lambda }_{j-2}}{s^{ϵ}}}\left[{\displaystyle \frac{{\left(f_{j-2}\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_{j-2}\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_{j-2}\right)}_2\left(t\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{{\left(f_{j-2}\right)}_3\left(t\right)}{s^{3ϵ+1}}}\right]\hfill \\ & & +{\displaystyle \frac{{\lambda }_{j-1}}{s^{ϵ}}}\left[{\displaystyle \frac{{\left(f_{j-1}\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_{j-1}\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_{j-1}\right)}_2\left(t\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{{\left(f_{j-1}\right)}_3\left(t\right)}{s^{3ϵ+1}}}\right],\hfill \\ \hfill \mathcal{S}Re{s}_3\left({\Xi }_j\left(s\right)\right)& =& {\displaystyle \frac{{\left(f_j\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_j\right)}_2\left(t\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{{\left(f_j\right)}_3\left(t\right)}{s^{3ϵ+1}}}-{\displaystyle \frac{{\lambda }_{j-1}}{s^{ϵ}}}\left[{\displaystyle \frac{{\left(f_{j-1}\right)}_0\left(t\right)}{s}}+{\displaystyle \frac{{\left(f_{j-1}\right)}_1\left(t\right)}{s^{ϵ+1}}}+{\displaystyle \frac{{\left(f_{j-1}\right)}_2\left(t\right)}{s^{2ϵ+1}}}+{\displaystyle \frac{{\left(f_{j-1}\right)}_3\left(t\right)}{s^{3ϵ+1}}}\right].\hfill \end{array}$$

(44)

Now, by solving ${\displaystyle \underset{s\to \infty }{lim}}{s}^{3ϵ+1}\mathcal{S}Re{s}_3\left({\Xi }_i\left(s\right)\right)=0$, we find that

$$\begin{array}{c}{\left(f_1\right)}_3\left(t\right)=-z{\lambda }_1^3,{\left(f_2\right)}_3\left(t\right)=z{\lambda }_1\left({\lambda }_1^2+{\lambda }_1{\lambda }_2+{\lambda }_2^2\right),{\left(f_3\right)}_3\left(t\right)=-z{\lambda }_1\left({\lambda }_1{\lambda }_2+{\lambda }_2^2+{\lambda }_2{\lambda }_3\right),\\ {\left(f_4\right)}_3\left(t\right)=z{\lambda }_1{\lambda }_2{\lambda }_3,{\left(f_5\right)}_3\left(t\right)=0,\cdots ,{\left(f_j\right)}_3\left(t\right)=0.\end{array}$$

(45)

In a similar manner, we substitute the k^th

Sumudu-residual function $\mathcal{S}Re{s}_k\left({\Xi }_i\left(s\right)\right)$ for the k^th truncated series ${\Xi }_i\left(s\right)$. We then multiply the resultant equations by $s^{kϵ+1}$, and when $s\to \infty$, we take the limit into consideration. The system (7) series solution is as previously mentioned.

Following the same procedure, the k^th residual function derived through the Sumudu method is used in place of the corresponding truncated series ${\Xi }_i\left(s\right)$. The resulting expressions are then scaled by $s^{kϵ+1}$, and the behavior as s approaches infinity is taken into account. Based on the earlier discussion, the series solution is obtained as follows:

$$\begin{array}{c}{\Xi }_i\left(s\right)={\displaystyle \frac{f_0\left(x\right)}{s}}+{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{{\left(f_i\right)}_n\left(t\right)}{s^{nϵ+1}}},\end{array}$$

(46)

Here, we may obtain the series solution for the system given in (1) by applying the inverse of ST in (24) as:

$$\begin{array}{c}{N}_i\left(t\right)={\left(f_i\right)}_0\left(t\right)+{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{{\left(f_i\right)}_n\left(t\right)t^{nϵ}}{\Gamma (nϵ+1)}},\end{array}$$

(47)

then

$$\begin{array}{ccc}\hfill {\left(N_1\right)}_n\left(t\right)& =& z-{\displaystyle \frac{z{\lambda }_1{t}^{ϵ}}{\Gamma (ϵ+1)}}+{\displaystyle \frac{z{\lambda }_1^2{t}^{2ϵ}}{2!\Gamma (2ϵ+1)}}-{\displaystyle \frac{z{\lambda }_1^3{t}^{3ϵ}}{3!\Gamma (3ϵ+1)}}+{\displaystyle \frac{z{\lambda }_1^4{t}^{4ϵ}}{4!\Gamma (4ϵ+1)}}-{\displaystyle \frac{z{\lambda }_1^5{t}^{5ϵ}}{5!\Gamma (5ϵ+1)}}+\cdots ,\hfill \\ \hfill {\left(N_2\right)}_n\left(t\right)& =& {\displaystyle \frac{z{\lambda }_1{t}^{ϵ}}{\Gamma (ϵ+1)}}-{\displaystyle \frac{z{\lambda }_1{t}^{2ϵ}}{2!\Gamma (2ϵ+1)}}\left({\lambda }_1+{\lambda }_2\right)+{\displaystyle \frac{z{\lambda }_1{t}^{3ϵ}}{3!\Gamma (3ϵ+1)}}\left({\lambda }_1^2+{\lambda }_1{\lambda }_2+{\lambda }_2^2\right)\hfill \\ & & -{\displaystyle \frac{z{\lambda }_1{t}^{4ϵ}}{4!\Gamma (4ϵ+1)}}\left({\lambda }_1^3+{\lambda }_1^2{\lambda }_2+{\lambda }_1{\lambda }_2^2+{\lambda }_2^3\right)+{\displaystyle \frac{z{\lambda }_1{t}^{5ϵ}}{5!\Gamma (5ϵ+1)}}\left({\lambda }_1^4+{\lambda }_1^3{\lambda }_2+{\lambda }_1^2{\lambda }_2^2+{\lambda }_1{\lambda }_2^3+{\lambda }_2^4\right)+\cdots ,\hfill \\ \hfill {\left(N_3\right)}_n\left(t\right)& =& {\displaystyle \frac{z{\lambda }_1{\lambda }_2{t}^{2ϵ}}{2!\Gamma (2ϵ+1)}}-{\displaystyle \frac{z{\lambda }_1{t}^{3ϵ}}{3!\Gamma (3ϵ+1)}}\left({\lambda }_1{\lambda }_2+{\lambda }_2^2+{\lambda }_2{\lambda }_3\right)+{\displaystyle \frac{z{\lambda }_1{t}^{4ϵ}}{4!\Gamma (4ϵ+1)}}\left({\lambda }_1^2{\lambda }_2+{\lambda }_1{\lambda }_2^2+{\lambda }_1{\lambda }_2{\lambda }_3+{\lambda }_2^3+{\lambda }_2{\lambda }_3^2+{\lambda }_2^2{\lambda }_3\right)+\cdots ,\hfill \\ \hfill ⋮& ⋮& ⋮\hfill \\ \hfill {\left(N_j\right)}_n\left(t\right)& =& {\displaystyle \frac{z{t}^{(j-1)ϵ}}{(j-1)!\Gamma \left[(j-1)ϵ+1\right]}}\stackrel{[}{m}=1]j-1\prod {\lambda }_m.\hfill \end{array}$$

(48)

5.2. Approximate Solution by TAM

Applying the fractional TAM method, one initially reformulates model (1) in the following manner:

$$\begin{array}{ccc}\hfill {\Xi }_1\left(N_1\left(t\right)\right)={\mathfrak{D}}_t^{ϵ}{N}_1\left(t\right),& {\Lambda }_1\left(N_1\left(t\right)\right)=-{\lambda }_1{N}_1\left(t\right),& p_1\left(t\right)=0\hfill \\ \hfill {\Xi }_2\left(N_2\left(t\right)\right)={\mathfrak{D}}_t^{ϵ}{N}_2\left(t\right),& {\Lambda }_2\left(N_2\left(t\right)\right)={\lambda }_1{N}_1\left(t\right)-{\lambda }_2{N}_2\left(t\right),& p_2\left(t\right)=0,\hfill \\ \hfill {\Xi }_3\left(N_3\left(t\right)\right)={\mathfrak{D}}_t^{ϵ}{N}_3\left(t\right),& {\Lambda }_3\left(N_3\left(t\right)\right)={\lambda }_2{N}_2\left(t\right)-{\lambda }_3{N}_3\left(t\right),& p_3\left(t\right)=0,\hfill \\ \hfill ⋮& ⋮& ⋮\hfill \\ \hfill {\Xi }_j\left(N_j\left(t\right)\right)={\mathfrak{D}}_t^{ϵ}{N}_j\left(t\right),& {\Lambda }_2\left(N_{j-1}\left(t\right)\right)={\lambda }_{j-1}{N}_{j-1}\left(t\right),& p_j\left(t\right)=0.\hfill \end{array}$$

(49)

The first problem has to be handled as

$$\left\{\begin{array}{cc}{\Xi }_1\left({\left(N_1\right)}_0\left(t\right)\right)=0,& {\left(N_1\right)}_0\left(0\right)=z,\\ {\Xi }_2\left({\left(N_2\right)}_0\left(t\right)\right)=0,& {\left(N_2\right)}_0\left(0\right)=0,\\ ⋮& ⋮\\ {\Xi }_j\left({\left(N_j\right)}_0\left(t\right)\right)=0,& {\left(N_j\right)}_0\left(0\right)=0.\end{array}\right.$$

(50)

The preceding equation can be addressed through a straightforward procedure, outlined below:

$$\left\{\begin{array}{cc}{I}^{ϵ}\left({\mathfrak{D}}_t^{ϵ}{\left(N_1\right)}_0\left(t\right)\right)=0,& {\left(N_1\right)}_0\left(0\right)=z,\\ I^{ϵ}\left({\mathfrak{D}}_t^{ϵ}{\left(N_2\right)}_0\left(t\right)\right)=0,& {\left(N_2\right)}_0\left(0\right)=0,\\ ⋮& ⋮\\ I^{ϵ}\left({\mathfrak{D}}_t^{ϵ}{\left(N_j\right)}_0\left(t\right)\right)=0,& {\left(N_j\right)}_0\left(0\right)=0.\end{array}\right.$$

(51)

As a result of the fundamental features of definition (2), we obtain the primary iteration as

$$\left\{\begin{array}{c}{\left(N_1\right)}_0\left(t\right)=z\\ {\left(N_2\right)}_0\left(t\right)=0,\\ ⋮\\ {\left(N_j\right)}_0\left(t\right)=0.\end{array}\right.$$

The next iteration can be computed as

$$\left\{\begin{array}{cc}{\Xi }_1\left({\left(N_1\right)}_1\left(t\right)\right)={\Lambda }_1\left({\left(N_1\right)}_0\left(t\right)\right)+p_1\left(t\right),& {\left(N_1\right)}_0\left(0\right)=z,\\ {\Xi }_2\left({\left(N_2\right)}_1\left(t\right)\right)={\Lambda }_2\left({\left(N_2\right)}_0\left(t\right)\right)+p_2\left(t\right),& {\left(N_2\right)}_0\left(0\right)=0,\\ ⋮& ⋮\\ {\Xi }_j\left({\left(N_j\right)}_1\left(t\right)\right)={\Lambda }_j\left({\left(N_j\right)}_0\left(t\right)\right)+p_j\left(t\right),& {\left(N_j\right)}_0\left(0\right)=0.\end{array}\right.$$

(52)

Next, by integrating both sides of the above equation and using the fundamental characteristics of definition (1), we obtain

$$\left\{\begin{array}{cc}{I}^{ϵ}\left({\mathfrak{D}}_t^{ϵ}{\left(N_1\right)}_1\left(t\right)\right)=I^{ϵ}\left(-{\lambda }_1{\left(N_1\right)}_0\left(t\right)\right),& {\left(N_1\right)}_0\left(0\right)=z,\\ I^{ϵ}\left({\mathfrak{D}}_t^{ϵ}{\left(N_2\right)}_1\left(t\right)\right)=I^{ϵ}\left({\lambda }_1{\left(N_1\right)}_0\left(t\right)-{\lambda }_2{\left(N_2\right)}_0\left(t\right)\right),& {\left(N_2\right)}_0\left(0\right)=0,\\ ⋮& ⋮\\ I^{ϵ}\left({\mathfrak{D}}_t^{ϵ}{\left(N_j\right)}_1\left(t\right)\right)=I^{ϵ}\left({\lambda }_{j-1}{\left(N_{j-1}\right)}_0\left(t\right)\right),& {\left(N_j\right)}_0\left(0\right)=0.\end{array}\right.$$

(53)

This leads us to obtain the subsequent iteration in the following form:

$$\left\{\begin{array}{c}{\left(N_1\right)}_1\left(t\right)=z-{\displaystyle \frac{z{\lambda }_1{t}^{ϵ}}{\Gamma (ϵ+1)}},\\ {\left(N_2\right)}_1\left(t\right)={\displaystyle \frac{z{\lambda }_1{t}^{ϵ}}{\Gamma (ϵ+1)}},\\ {\left(N_3\right)}_1\left(t\right)=0,\\ ⋮\\ {\left(N_j\right)}_1\left(t\right)=0.\end{array}\right.$$

(54)

The subsequent step in the iteration process can be determined and expressed as follows:

$$\left\{\begin{array}{cc}{\Xi }_1\left({\left(N_1\right)}_2\left(t\right)\right)={\Lambda }_1\left({\left(N_1\right)}_1\left(t\right)\right)+p_1\left(t\right),& {\left(N_1\right)}_1\left(0\right)=z,\\ {\Xi }_2\left({\left(N_2\right)}_2\left(t\right)\right)={\Lambda }_2\left({\left(N_2\right)}_1\left(t\right)\right)+p_2\left(t\right),& {\left(N_2\right)}_1\left(0\right)=0,\\ ⋮& ⋮\\ {\Xi }_j\left({\left(N_j\right)}_2\left(t\right)\right)={\Lambda }_j\left({\left(N_j\right)}_1\left(t\right)\right)+p_j\left(t\right),& {\left(N_j\right)}_1\left(0\right)=0.\end{array}\right.$$

(55)

Next, by integrating both sides of the above equation and using the fundamental characteristics of definition (1), we obtain

$$\left\{\begin{array}{cc}{I}^{ϵ}\left({\mathfrak{D}}_t^{ϵ}{\left(N_1\right)}_2\left(t\right)\right)=I^{ϵ}\left(-{\lambda }_1{\left(N_1\right)}_1\left(t\right)\right),& {\left(N_1\right)}_1\left(0\right)=z,\\ I^{ϵ}\left({\mathfrak{D}}_t^{ϵ}{\left(N_2\right)}_2\left(t\right)\right)=I^{ϵ}\left({\lambda }_1{\left(N_1\right)}_1\left(t\right)-{\lambda }_2{\left(N_2\right)}_1\left(t\right)\right),& {\left(N_2\right)}_1\left(0\right)=0,\\ ⋮& ⋮\\ I^{ϵ}\left({\mathfrak{D}}_t^{ϵ}{\left(N_j\right)}_2\left(t\right)\right)=I^{ϵ}\left({\lambda }_{j-1}{\left(N_{j-1}\right)}_1\left(t\right)\right),& {\left(N_j\right)}_1\left(0\right)=0.\end{array}\right.$$

(56)

This leads us to obtain the subsequent iteration in the following form:

$$\left\{\begin{array}{c}{\left(N_1\right)}_2\left(t\right)=z-{\displaystyle \frac{z{\lambda }_1{t}^{ϵ}}{\Gamma (ϵ+1)}}+{\displaystyle \frac{z{\lambda }_1^2{t}^{2ϵ}}{2!\Gamma (2ϵ+1)}},\\ {\left(N_2\right)}_2\left(t\right)={\displaystyle \frac{z{\lambda }_1{t}^{ϵ}}{\Gamma (ϵ+1)}}-{\displaystyle \frac{z{\lambda }_1{t}^{2ϵ}}{2!\Gamma (2ϵ+1)}}\left({\lambda }_1+{\lambda }_2\right),\\ {\left(N_3\right)}_2\left(t\right)={\displaystyle \frac{z{\lambda }_1{\lambda }_2{t}^{2ϵ}}{2!\Gamma (2ϵ+1)}},\\ {\left(N_4\right)}_2\left(t\right)=0,\\ ⋮\\ {\left(N_j\right)}_2\left(t\right)=0.\end{array}\right.$$

(57)

Every time $N_i\left(t\right),i=1,2,\cdots ,j$ are repeated, an approximate solution to Equation (1) is found, according to Equation (10). The analytical solution gets closer to the exact solution as the iterations grow in number. Following this method, the structure of the approximate solution series can be constructed as follows:

$$\left\{\begin{array}{c}{N}_1\left(t\right)=\underset{m\to \infty }{lim}{\left(N_1\right)}_m\left(t\right)\hfill \\ N_2\left(t\right)=\underset{m\to \infty }{lim}{\left(N_2\right)}_m\left(t\right)\hfill \\ N_3\left(t\right)=\underset{m\to \infty }{lim}{\left(N_3\right)}_m\left(t\right)\hfill \\ ⋮\hfill \\ N_j\left(t\right)=\underset{m\to \infty }{lim}{\left(N_j\right)}_m\left(t\right),\hfill \end{array}\right.$$

(58)

then

$$\begin{array}{ccc}\hfill {\left(N_1\right)}_n\left(t\right)& =& z-{\displaystyle \frac{z{\lambda }_1{t}^{ϵ}}{\Gamma (ϵ+1)}}+{\displaystyle \frac{z{\lambda }_1^2{t}^{2ϵ}}{2!\Gamma (2ϵ+1)}}-{\displaystyle \frac{z{\lambda }_1^3{t}^{3ϵ}}{3!\Gamma (3ϵ+1)}}+{\displaystyle \frac{z{\lambda }_1^4{t}^{4ϵ}}{4!\Gamma (4ϵ+1)}}-{\displaystyle \frac{z{\lambda }_1^5{t}^{5ϵ}}{5!\Gamma (5ϵ+1)}}+\cdots ,\hfill \\ \hfill {\left(N_2\right)}_n\left(t\right)& =& {\displaystyle \frac{z{\lambda }_1{t}^{ϵ}}{\Gamma (ϵ+1)}}-{\displaystyle \frac{z{\lambda }_1{t}^{2ϵ}}{2!\Gamma (2ϵ+1)}}\left({\lambda }_1+{\lambda }_2\right)+{\displaystyle \frac{z{\lambda }_1{t}^{3ϵ}}{3!\Gamma (3ϵ+1)}}\left({\lambda }_1^2+{\lambda }_1{\lambda }_2+{\lambda }_2^2\right)\hfill \\ & & -{\displaystyle \frac{z{\lambda }_1{t}^{4ϵ}}{4!\Gamma (4ϵ+1)}}\left({\lambda }_1^3+{\lambda }_1^2{\lambda }_2+{\lambda }_1{\lambda }_2^2+{\lambda }_2^3\right)+{\displaystyle \frac{z{\lambda }_1{t}^{5ϵ}}{5!\Gamma (5ϵ+1)}}\left({\lambda }_1^4+{\lambda }_1^3{\lambda }_2+{\lambda }_1^2{\lambda }_2^2+{\lambda }_1{\lambda }_2^3+{\lambda }_2^4\right)+\cdots ,\hfill \\ \hfill {\left(N_3\right)}_n\left(t\right)& =& {\displaystyle \frac{z{\lambda }_1{\lambda }_2{t}^{2ϵ}}{2!\Gamma (2ϵ+1)}}-{\displaystyle \frac{z{\lambda }_1{t}^{3ϵ}}{3!\Gamma (3ϵ+1)}}\left({\lambda }_1{\lambda }_2+{\lambda }_2^2+{\lambda }_2{\lambda }_3\right)+{\displaystyle \frac{z{\lambda }_1{t}^{4ϵ}}{4!\Gamma (4ϵ+1)}}\left({\lambda }_1^2{\lambda }_2+{\lambda }_1{\lambda }_2^2+{\lambda }_1{\lambda }_2{\lambda }_3+{\lambda }_2^3+{\lambda }_2{\lambda }_3^2+{\lambda }_2^2{\lambda }_3\right)+\cdots ,\hfill \\ \hfill ⋮& ⋮& ⋮\hfill \\ \hfill {\left(N_j\right)}_n\left(t\right)& =& {\displaystyle \frac{z{t}^{(j-1)ϵ}}{(j-1)!\Gamma \left[(j-1)ϵ+1\right]}}\stackrel{[}{m}=1]j-1\prod {\lambda }_m.\hfill \end{array}$$

(59)

6. Numerical Results and Discussion

The analytical solutions of the Actinium series system, performed using both methods, are tested numerically in this section. The Actinium series which involves alpha, beta, and gamma decays is studied in this section. The Actinium radioactive series, in which multiple decay steps are considered, begins from 238-U nuclei and reaches 206-Pb.

The theoretical formalism studied for the Actinium (4n + 3) nuclear radioactive series is presented computationally. The necessary numerical procedures were implemented to carry out the calculations, and the results are presented in this section.

The values of half-life and decay constants of four cases of the Actinium series are given in

Table 1. The nuclei half-life and decay constants.

	Nucleus	${\mathit{t}}_{\frac{1}{2}}$	$\mathit{\lambda }$
1	${}_{\mathrm{92 }}{}^{239}U$	$1.40700000\times {10}^3$	$4.9253731\times {10}^{-4}$
2	${}_{\mathrm{93 }}{}^{239}\mathit{Np}$	$2.03558000\times {10}^5$	$3.4044351\times {10}^{-6}$
3	${}_{\mathrm{94 }}{}^{239}\mathit{Pu}$	$7.6083748\times {10}^{11}$	$9.1083840\times {10}^{-13}$
4	${}_{\mathrm{92 }}{}^{235}U$	$2.2216075\times {10}^{16}$	$3.1193628\times {10}^{-17}$
5	${}_{\mathrm{90 }}{}^{231}\mathit{Th}$	$9.20520000\times {10}^4$	$7.5283535\times {10}^{-6}$
6	${}_{\mathrm{91 }}{}^{231}\mathit{Pa}$	$1.0278090\times {10}^{12}$	$6.7424973\times {10}^{-13}$
7	${}_{\mathrm{89 }}{}^{227}\mathit{Ac}$	$6.87057390\times {10}^8$	$1.0086493\times {10}^{-9}$
8	${}_{\mathrm{90 }}{}^{227}\mathit{Th}$	$1.61542100\times {10}^6$	$4.2899033\times {10}^{-7}$
9	${}_{\mathrm{88 }}{}^{223}\mathit{Ra}$	$9.87552000\times {10}^5$	$7.0173520\times {10}^{-7}$
10	${}_{\mathrm{86 }}{}^{219}\mathit{Rn}$	3.9600000000	$1.7500000\times {10}^{-1}$
11	${}_{\mathrm{84 }}{}^{215}\mathit{Po}$	$1.7810000\times {10}^{-3}$	$3.8910724\times {10}^2$
12	${}_{\mathrm{82 }}{}^{211}\mathit{Pb}$	$2.16600000\times {10}^3$	$3.1994460\times {10}^{-4}$
13	${}_{\mathrm{83 }}{}^{211}\mathit{Bi}$	$1.28400000\times {10}^2$	$5.3971963\times {10}^{-3}$
14	${}_{\mathrm{84 }}{}^{211}\mathit{Po}$	$5.1600000\times {10}^{-1}$	$1.3430233\times {10}^0$
15	${}_{\mathrm{82 }}{}^{207}\mathit{Pb}$	STABLE
16	${}_{\mathrm{87 }}{}^{227}\mathit{Fr}$	$1.48200000\times {10}^2$	$4.6761134\times {10}^{-3}$
17	${}_{\mathrm{81 }}{}^{207}\mathit{Ti}$	$2.86200000\times {10}^2$	$2.4213836\times {10}^{-3}$

[8].

Here, four cases of the Actinium series are considered, where certain nuclei decay into different new nuclei.

The first case of the Actinium series is represented by the following equation

$$\begin{array}{c}\hfill {}_{\mathrm{92 }}{}^{235}\mathrm{U}{\to }_{90}^{231}\mathrm{Th}{\to }_{91}^{231}\mathrm{Pa}{\to }_{98}^{227}\mathrm{Ac}{\to }_{90}^{227}\mathrm{Th}{\to }_{88}^{223}\mathrm{Ra}{\to }_{86}^{219}\mathrm{Rn}{\to }_{84}^{215}\mathrm{Po}{\to }_{82}^{211}\mathrm{Pb}{\to }_{84}^{211}\mathrm{Bi}{\to }_{84}^{211}\mathrm{Po}{\to }_{82}^{207}\mathrm{Pb}.\end{array}$$

(60)

where the ${}_{\mathrm{98 }}{}^{227}\mathrm{Ac}$ decay to ${}_{\mathrm{90 }}{}^{227}\mathrm{Th}$ with ratio 98% and the ${}_{\mathrm{84 }}{}^{211}\mathrm{Bi}$ decay to ${}_{\mathrm{84 }}{}^{211}\mathrm{Po}$ with ratio 3% [26]. The behavior of these nuclei is represented in Figure 1. And in fractional order in Figure 2.

The maximum values of all nuclei and the corresponding times at which these maxima occur are provided in

Table 2. The maximum values of all nuclei and the corresponding times at which these maxima occur for the first and second cases are tabulated in Table.

Case 1			Case 2
Nucleus	t	Parameter	Nucleus	t	Parameter
${}^{235}U$	1.000000000	$4.08\times {10}^8$	${}^{235}U$	1.000000000	$4.08\times {10}^8$
${}^{231}\mathit{Th}$	$4.13564\times {10}^{-11}$	$4.08\times {10}^8$	${}^{231}\mathit{Th}$	$4.13564\times {10}^{-11}$	$4.08\times {10}^8$
${}^{231}\mathit{Pa}$	$4.62893\times {10}^{-4}$	$1.23\times {10}^{13}$	${}^{231}\mathit{Pa}$	$4.62893\times {10}^{-4}$	$1.23\times {10}^{13}$
${}^{227}\mathit{Ac}$	$3.08595\times {10}^{-7}$	$1.23\times {10}^{13}$	${}^{227}\mathit{Ac}$	$3.08595\times {10}^{-4}$	$1.23\times {10}^{13}$
${}^{227}\mathit{Th}$	$7.25226\times {10}^{-10}$	$1.23\times {10}^{13}$	${}^{223}\mathit{Fr}$	$5.925033\times {10}^{13}$	1.23 × ${10}^{13}$
${}^{218}\mathit{Ra}$	$4.42116\times {10}^{-10}$	$1.23\times {10}^{13}$	${}^{218}\mathit{Ra}$	$4.42116\times {10}^{-10}$	$1.23\times {10}^{13}$
${}^{219}\mathit{Rn}$	$1.78669\times {10}^{-15}$	$1.23\times {10}^{13}$	${}^{219}\mathit{Rn}$	$1.78669\times {10}^{-15}$	$1.23\times {10}^{13}$
${}^{215}\mathit{Po}$	$7.99034\times {10}^{-19}$	$1.23\times {10}^{13}$	${}^{215}\mathit{Po}$	$7.99034\times {10}^{-19}$	$1.23\times {10}^{13}$
${}^{211}\mathit{Pb}$	$9.49814\times {10}^{-13}$	$1.23\times {10}^{13}$	${}^{211}\mathit{Pb}$	$9.49814\times {10}^{-13}$	1.23 × ${10}^{13}$
${}^{211}\mathit{Bi}$	$5.79047\times {10}^{-14}$	$1.23\times {10}^{13}$	${}^{211}\mathit{Bi}$	$5.79047\times {10}^{-14}$	$1.23\times {10}^{13}$
${}^{207}\mathit{Tl}$	$1.28485\times {10}^{-13}$	$1.23\times {10}^{13}$	${}^{207}\mathit{Tl}$	$1.28485\times {10}^{-13}$	$1.23\times {10}^{13}$
${}^{207}\mathit{Pb}$	0.998000000	$2.00\times {10}^{16}$	${}^{207}\mathit{Pb}$	0.998000000	$2.00\times {10}^{16}$

.

The second case of the Actinium series is represented by the following equation

$$\begin{array}{c}\hfill {}_{\mathrm{92 }}{}^{235}\mathrm{U}{\to }_{90}^{231}\mathrm{Th}{\to }_{91}^{231}\mathrm{Pa}{\to }_{98}^{227}\mathrm{Ac}{\to }_{87}^{227}\mathrm{Fr}{\to }_{88}^{223}\mathrm{Ra}{\to }_{86}^{219}\mathrm{Rn}{\to }_{84}^{215}\mathrm{Po}{\to }_{82}^{211}\mathrm{Pb}{\to }_{84}^{211}\mathrm{Bi}{\to }_{84}^{211}\mathrm{Po}{\to }_{82}^{207}\mathrm{Pb}\end{array}$$

(61)

where the ${}_{\mathrm{98 }}{}^{227}\mathrm{Ac}$ decay to ${}_{\mathrm{87 }}{}^{227}\mathrm{Fr}$ with ratio 1.4% and the ${}_{\mathrm{84 }}{}^{211}\mathrm{Bi}$ decay to ${}_{\mathrm{84 }}{}^{211}\mathrm{Po}$ with ratio 3% [26]. The behavior of these nuclei is represented in Figure 3.

The third case of the Actinium series is represented by the following equation

$$\begin{array}{c}\hfill {}_{\mathrm{92 }}{}^{235}\mathrm{U}{\to }_{90}^{231}\mathrm{Th}{\to }_{91}^{231}\mathrm{Pa}{\to }_{98}^{227}\mathrm{Ac}{\to }_{90}^{227}\mathrm{Th}{\to }_{88}^{223}\mathrm{Ra}{\to }_{86}^{219}\mathrm{Rn}{\to }_{84}^{215}\mathrm{Po}{\to }_{82}^{211}\mathrm{Pb}{\to }_{84}^{211}\mathrm{Bi}{\to }_{81}^{207}\mathrm{Ti}{\to }_{82}^{207}\mathrm{Pb}\end{array}$$

(62)

where the ${}_{\mathrm{98 }}{}^{227}\mathrm{Ac}$ decay to ${}_{\mathrm{91 }}{}^{231}\mathrm{Pa}$ with ratio 98.6% and the ${}_{\mathrm{84 }}{}^{211}\mathrm{Bi}$ decay to ${}_{\mathrm{84 }}{}^{211}\mathrm{Po}$ with ratio 3% [26]. The behavior of these nuclei is represented in Figure 4.

The final case of the Actinium series is given by the following equation

$$\begin{array}{c}\hfill {}_{\mathrm{92 }}{}^{235}\mathrm{U}{\to }_{90}^{231}\mathrm{Th}{\to }_{91}^{231}\mathrm{Pa}{\to }_{98}^{227}\mathrm{Ac}{\to }_{87}^{227}\mathrm{Fr}{\to }_{88}^{223}\mathrm{Ra}{\to }_{86}^{219}\mathrm{Rn}{\to }_{84}^{215}\mathrm{Po}{\to }_{82}^{211}\mathrm{Pb}{\to }_{84}^{211}\mathrm{Bi}{\to }_{81}^{207}\mathrm{Ti}{\to }_{82}^{207}\mathrm{Pb}\end{array}$$

(63)

where the ${}_{\mathrm{98 }}{}^{227}\mathrm{Ac}$ decay to ${}_{\mathrm{91 }}{}^{231}\mathrm{Pa}$ with ratio 98.6% and the ${}_{\mathrm{84 }}{}^{211}\mathrm{Bi}$ decay to ${}_{\mathrm{81 }}{}^{207}\mathrm{Ti}$ with ratio 99.7% [26]. The behavior of these nuclei is represented in Figure 5.

The maximum values for the third and the fourth cases of the nuclei and the corresponding times at which these maxima occur are provided in

Table 3. The maximum values of all nuclei and the corresponding times at which these maxima occur.

Case 3			Case 4
Nucleus	t	Parameter	Nucleus	t	Parameter
${}^{235}U$	1.000000000	$4.08\times {10}^8$	${}^{235}\mathit{U}$	1.000000000	$4.08\times {10}^8$
${}^{231}\mathit{Th}$	$4.13564\times {10}^{-11}$	$4.08\times {10}^8$	${}^{231}\mathit{Th}$	$4.13564\times {10}^{-11}$	$4.08\times {10}^8$
${}^{231}\mathit{Pa}$	$4.62893\times {10}^{-4}$	$1.23\times {10}^{13}$	${}^{231}\mathit{Pa}$	$4.62893\times {10}^{-4}$	1.23 $\times {10}^{13}$
${}^{227}\mathit{Ac}$	$3.08595\times {10}^{-7}$	$1.23\times {10}^{13}$	${}^{227}\mathit{Ac}$	$3.08595\times {10}^{-7}$	1.23 $\times {10}^{13}$
${}^{227}\mathit{Th}$	$7.25226\times {10}^{-10}$	$1.23\times {10}^{13}$	${}^{223}\mathit{Fr}$	$5.92503\times {10}^{-13}$	$1.23\times {10}^{13}$
${}^{218}\mathit{Ra}$	$4.42116\times {10}^{-10}$	$1.23\times {10}^{13}$	${}^{218}\mathit{Ra}$	$4.42116\times {10}^{-10}$	$1.23\times {10}^{13}$
${}^{219}\mathit{Rn}$	$1.78669\times {10}^{-15}$	$1.23\times {10}^{13}$	${}^{219}\mathit{Rn}$	$1.78669\times {10}^{-15}$	$1.23\times {10}^{13}$
${}^{215}\mathit{Po}$	$7.99034\times {10}^{-19}$	$1.23\times {10}^{13}$	${}^{215}\mathit{Po}$	$7.99034\times {10}^{-19}$	$1.23\times {10}^{13}$
${}^{211}\mathit{Pb}$	$9.49814\times {10}^{-13}$	$1.23\times {10}^{13}$	${}^{211}\mathit{Pb}$	$9.49814\times {10}^{-13}$	$1.23\times {10}^{13}$
${}^{211}\mathit{Bi}$	$5.79047\times {10}^{-14}$	$1.23\times {10}^{13}$	${}^{211}\mathit{Bi}$	5.79047 $\times {10}^{-14}$	$1.23\times {10}^{13}$
${}^{211}\mathit{Po}$	$2.58573\times {10}^{-16}$	$1.23\times {10}^{13}$	${}^{211}\mathit{Po}$	$2.58573\times {10}^{-16}$	$1.23\times {10}^{13}$
${}^{207}\mathit{Pb}$	0.998000000	$2.00\times {10}^{16}$	${}^{207}\mathit{Pb}$	0.998000000	$2.00\times {10}^{16}$

.

The behavior of each element in the four different cases of the Actinium series is described graphically over time; the maximum value of each element, along with the time at which it occurs, is tabulated.

It must be noted that the use of fractional calculus in this work helps us describe the radioactive decay model more accurately, especially in cases where classical equations fall short. Traditional models assume that decay occurs in a simple manner. But in reality, some nuclear processes are influenced by past behavior, which fractional derivatives can account for.

Also, not all decay processes follow the ideal exponential pattern. Some isotopes show slower or more complex decay rates, and using fractional-order equations gives us the flexibility to model those behaviors more realistically.

In addition, when comparing numerical results with experimental data, models based on fractional calculus often perform better. This makes them valuable tools in nuclear physics, especially when working with complicated decay series like the Actinium series.

7. Conclusions

The study of the Actinium series system of equations is performed in this work; the key findings of this research highlight using first analytical solutions using the Sumudu residual power series method and the Temimi–Ansari method. The uniqueness and existence of the solution to the proposed model using the Caputo operator are explored, and the fractional calculus techniques are introduced. Introducing fractional calculus techniques, which describe slower decay and capture long-term memory and unusual kinetics, produces results that often match experimental observations in complex systems better than the traditional exponential model. The numerical simulation of the Actinium series is tabulated and graphically figured out for four different scenarios for the Actinium series. Solving nuclear radioactive series analytically provides results that avoid the complexity of extensive experimental procedures. This work represents a step forward in the study of theoretical nuclear physics, where considering other nuclear radioactive series, and research focusing on incorporating spatial diffusion, are proposed as future works.