Applying the Adomian Method to Solve the Fokker–Planck Equation: A Case Study in Astrophysics

Abstract

The objective of this study is to model astrophysical systems using the nonlinear Fokker–Planck equation, with the Adomian method chosen for its iterative and precise solutions in this context, applying boundary conditions relevant to data from the Rossi X-ray Timing Explorer (RXTE). The results include analysis of 156 X-ray intensity distributions from X-ray binaries (XRBs), exhibiting long-tail profiles consistent with Tsallis q-Gaussian distributions. The corresponding q-values align with the principles of Tsallis thermostatistics. Various diffusion hypotheses—classical, linear, nonlinear, and anomalous—are examined, with q-values further supporting Tsallis thermostatistics. Adjustments in the parameter α (related to the order of fractional temporal derivation) reveal the extent of the memory effect, strongly correlating with fractal properties in the diffusive process. Extending this research to other XRBs is both possible and recommended to generalize the characteristics of X-ray scattering and electromagnetic waves at different frequencies originating from similar astronomical objects.

1. Introduction

The 20th century was a pivotal period for astronomy and stellar astrophysics [1]. Photometric and spectroscopic observations unveiled stellar properties that only became fully comprehensible through more recent theoretical studies. However, certain aspects, such as the precise number of binary stars in our galaxy, still require further investigation [1,2]. While many studies suggest that the majority of stars belong to multiple systems, others propose that solitary stars are more numerous. This research is crucial, given the significance of multiple stellar systems in advancing our understanding of stellar dynamics and the broader universe [2].

Binary star systems are of significant importance because they allow us to infer specific parameters, such as radius, surface temperature, mass, and rotation periods, through indirect techniques related to the system’s rotational, precessional, and other physical properties [1,3]. Among these systems are those that emit X-rays, known as X-ray binary systems (XRBSs). It is plausible to assume that the ‘light curves’ of X-ray emissions from these binary pairs can be analyzed and accurately modeled using classical hypotheses [2,3], as well as with considerations based on a Tsallian framework. This approach helps to explain their diffusion behaviors—both normal and anomalous—even under the influence of intense gravitational forces [4,5,6].

Preliminary analyses of data collected by the All Sky Monitor instrument aboard the Rossi X-ray Timing Explorer (RXTE-ASM), which are publicly available, suggest a diffusion process that can be modeled using standard diffusion considerations. This process occurs spontaneously, with each photon in the system following a trajectory influenced by numerous linear collisions, allowing it to be treated as a stochastic process. Consequently, it is reasonable to assume that the probability of an X-ray photon’s movement can be modeled as either a Markovian process (where dependencies exist between successive events) or a non-Markovian process (where such dependencies in the probability distribution are not considered) [3,4].

The scenario discussed here involves a sequence of hypotheses regarding the diffusive process, including the possibility of anomalous diffusion, represented by a generalized probability distribution within the framework of Fokker–Planck equations (FPEs). These equations are fundamental to the study of stochastic phenomena and are applied in various physical contexts to model position- and time-dependent behaviors. They achieve this by incorporating both restorative (drift) and diffusive (random) forces, which influence the systems and shape the probability distribution of their states and their spatiotemporal evolution [4,5,6,7].

Alongside this scenario is the recognition that statistical physics, through various hypotheses, models the diffusive and anomalous behavior of electromagnetic spectra, particularly X-rays, in binary star systems—scenarios that remain underexplored yet offer significant potential for investigation. Nonlinear diffusive equations, especially the Fokker–Planck equation and its various adaptations, provide legitimate physical representations that can be effectively modeled in this context.

To legitimize this process, 156 XRBS X-ray intensity distributions with long-tail distribution and high correlation to the diffusive process proposals predicted in the q-Gaussian Tsallian model are presented. However, these models, until now, have been hypothesized but not yet confronted, with the phenomenological legitimization of data with representation in the range of diffusive analyses in the X-ray spectrum.

To validate the underlying hypotheses in the Tsallian framework, classical diffusion hypotheses are analyzed, including those with constant diffusion coefficients as well as linear, nonlinear, and anomalous cases. The analysis focuses on the coefficient values (q), which are derived from the principles of Tsallis thermostatistics.

Each analyzed hypothesis supports the prediction of the anomalous diffusion phenomenon as described by Tsallis statistics. This includes fractional derivation analyses with adjustments reflecting the extent of the associated memory effect, which strongly correlate with the presence of fractal properties in the diffusion process, as predicted by the Tsallis model.

This is why the Fokker–Planck equation (FPE) was initially employed to explain Brownian motion, which generates unpredictable paths due to oscillations that prevent precise determination of particle locations [5]. Over time, it became possible to assess the probability of locating particles within a given region by analyzing various probabilistic distributions (both linear and nonlinear) based on hypotheses about the forces acting on X-ray binary systems (XRBSs) [5,6,7,8,9]. Consequently, entropic analysis emerges as a likely tool for modeling this complex system. It is combined with a range of physical and phenomenological interpretations to validate the collected numerical data [6,8,9].

The numerical data displayed indicate a generalized behavior that is verified as well correlated with the q-exponential [5,6,7], which is obtained by maximizing the Tsallian entropy, as long as it is subjected to appropriate links that reflect the hypotheses of the governing forces of the system. Hence, the nonlinear Fokker–Planck equation with Tsallis thermostatistics is used [8,9].

This article results from a comprehensive research effort aimed at analyzing X-ray spectral series from binary astrophysical systems to validate a diffusion model using a Fokker–Planck-type equation. The study employed both analytical and iterative methods proposed by Adomian [5,9,10,11,12,13,14,15,16]. The specific objectives included conducting a statistical analysis of the collected numerical series and modeling X-ray diffusion in regions surrounding these objects using the Fokker–Planck equation (FPE). Boundary conditions were also established based on numerical data from the Rossi X-ray Timing Explorer (RXTE-ASM) [8,9].

2. Materials and Methods

The methodology involves a thorough and systematic review of the literature across astrophysics, statistical mechanics, and contemporary theories of differential equation solutions. This review led to the modeling of the phenomenon based on physical and mathematical considerations. To ensure phenomenological validity, boundary conditions were proposed and then addressed through both analytical and iterative solutions using Adomian’s method [5,10,11,12,13,14,15,16]. This approach was applied to diffusive phenomena within the framework of Tsallis’s statistical mechanics [5,6,7,8].

To achieve the objectives, it is crucial to identify the governing equations of the analyzed diffusive phenomenon, along with the boundary conditions for the derived Fokker–Planck equation (FPE) and the exact method proposed [9,10] for solving it. Validating these hypotheses with numerical data enabled the identification of viable ensembles for the studied binary pairs.

2.1. Physical Context and Modeling of the Fokker–Planck Equation

Considering the description and physical characteristics determined by the governing equation in the context of the research, the phenomenon of diffusion was modeled, considering that it occurs analogously to a fluid with a nonuniform concentration of suspended particles (X-ray photons). As the average number of particles per unit of volume depends on the position at each instant, $n_{\left(x,t\right)}$, in a one-dimensional analysis, the total number of photons in the entire volume remains constant. This occurs due to the principle of energy conservation, even without the action of external elements, leading the system to a state of higher entropy [5] and providing the diffusive process of electromagnetic radiation, the objective of X-ray data analysis.

Still regarding the methodological aspects, it was considered that each photon constituting the system performs a random trajectory, resulting in a completely irregular individual movement, causing the set to diffuse in such a way that this collective photon emissions, contrasted with individual movement, have a regular behavior, following well-defined dynamic laws [6,7].

In this context, a mathematical relation can be inserted that considers $P(x,t)$, the probability distribution, and $W(x,t+\Delta t|x^{\prime },t)$, the transition probability, between the amplitude states, so that the sum of the distribution products for each of their respective ones results can be shown as

$$P(x,t+\Delta t)={\displaystyle \int W(x,t+\Delta t|x^{\prime },t).P(x^{\prime },t)d{x}^{\prime }}$$

(1)

By expanding the expression in Taylor’s series, the following can be obtained:

$$W(x,t+\Delta t|x^{\prime },t).P(x^{\prime },t)$$

(2)

Until the second order, the equation can be written as

$$W(x,t+\Delta t|x^{\prime },t).P(x^{\prime },t)=\mathrm{\Omega }(x,t)={\displaystyle \sum _{n=0}^{\infty }\left[{\mathrm{\Omega }}_{(x_0,t_0)}^{(n)}\frac{{(x_0-x)}^n}{n!}\right]}\cong {\mathrm{\Omega }}_{(x_0,t_0)}^{(0)}{\displaystyle \frac{{(x_0-x)}^0}{0!}}+{\mathrm{\Omega }}_{(x_0,t_0)}^{(1)}{\displaystyle \frac{{(x_0-x)}^1}{1!}}$$

(3)

If we consider Newton’s binomial expansion to replace the posited expansion, its integration also provides us with an expansion to its quadratic term, with a generalized approximation. Therefore, the integrative calculus, term by term until the second term, gives us, in its sums, the root mean square variation of x to obtain the finite format of the linear FPE. We consider the limit of the incremental ratio in time presented to us by the Fokker–Planck equation, which indicates the probability of occurrence of a certain amplitude of X-rays diffused in the form [5,9].

$${\displaystyle \frac{\partial P\left(x,t\right)}{\partial t}}=-{\displaystyle \frac{\partial }{\partial x}}\left[A(x,t)P\left(x,t\right)\right]+{\displaystyle \frac{{\partial }^2[B(x,t)P\left(x,t\right)]}{\partial x^2}}$$

(4)

This equation is used here to model the probability distributions of the X-ray diffusive process based on the time series arising from XRBSs in its linear form. The term A(x,t) corresponds to an external restoring force associated with a confining potential and contrary to the diffusive process due to the gravitational confident potential arising from the analyzed binary pair. B(x,t) is the diffusive term, which may have some characteristics that fit, when modeled, in stellar systems, according to the process of statistical distribution of X-rays [5,10].

In FPE, the probability distribution obtained in the solution maximizes the Boltzman–Gibbs entropy, enabling the extraction of all statistical and thermodynamic characteristics of the system, as they are anomalous systems, where fluctuations occur disproportionately to time with the presence of far-reaching interactions between the constituent parts, which is the case with XRBSs [7,8].

Due to the presupposed physical considerations, it is possible to arrive, in an analogous way, to nonlinear Fokker–Planck [16] equations following the same modeling scheme with considerations that predispose an anomalous diffusive process. In this, it is possible to consider and incorporate other spatiotemporal distributions, considering a nonlinear diffusion equation [5,9] and analytically describing a class of anomalous diffusive processes, as elucidated in the following equation [5,14,15]:

$${\displaystyle \frac{{\partial }^{\alpha }P\left(x,t\right)}{\partial t^{\alpha }}}=-{\displaystyle \frac{\partial }{\partial x}}\left[A(x,t)P\left(x,t\right)\right]+B(x,t){\displaystyle \frac{{\partial }^2\left[P\left(x,t\right)\right]}{\partial x^2}}$$

(5)

α is a real parameter that can assume values depending on the environment where the phenomenon occurs and can be adjusted to better correlate with numerical data that show the diffusive process (normal or anomalous).

As α is adjusted as a function of the system and the coverage of the memory effect of the initial conditions in the time horizon, in the case of XRBSs, each binary pair must have the coefficient of best adjustment to its diffusive process. For example, for α = 1, there is a distribution of probabilities associated with nonextensive systems that maximizes the entropy proposed by Tsallis and which is a generalization of the Boltzmann–Gibbs entropy, enabling a reduction in the classical type:

$${\displaystyle \frac{{\partial }^{\alpha =1}P\left(x,t\right)}{\partial t^{\alpha =1}}}=-{\displaystyle \frac{\partial }{\partial x}}\left[A(x,t)P\left(x,t\right)\right]+B(x,t){\displaystyle \frac{{\partial }^2\left[P\left(x,t\right)\right]}{\partial x^2}}$$

(6)

The restorative term A(x,t) may be associated with an external restorative force associated with a confining Tsallian potential and contrary to the diffusive process. This is due to the gravitational confining potential arising from the analyzed XRBSs, whose diffusive term is B(x,t) [5,10].

2.2. Phenomenology to Support Research Hypotheses

For this research, some possible configurations of the terms A(x,t) and B(x,t) were simulated in the analysis of the diffusive behavior of X-rays from XRBSs, and for this, 142 XRBSs were studied and categorized by groups thus distributed [2,8,17]:

• Group 1: BHC + HM (black hole candidate + high mass);
• Group 2: BHC + LM (black hole candidate + low mass);
• Group 3: HM + PSR (migh mass + pulsar);
• Group 4: LM + NS (low mass + star neutron);
• Group 5: SNR + PSR (supernova remnant + pulsar) [2,8,9].

The signal intensity behavior in the modeled results demonstrated a q-Gaussian distribution as the best-fit distribution, which characterizes its phenomenological Tsallian character.

This result allows us to model the data by determining specific values of the q parameter for each set of specific XRBSs, as seen in Figure 1.

The importance of modeling is due to the fact that in systems whose elements are strongly correlated, Tsallian entropy becomes nonadditive for an appropriate value of the parameter q, known as the entropic index, and its determination provides important characteristics for each one of the respective systems, allowing us to infer spatiotemporal behavior in its historicity.

2.3. Adomian’s Decomposition

Adomian’s method of decomposition [9,10,11,12,13,14,15,16,17,18] was adopted, as it is an algorithm that provides the solution of a partial differential equation in a numerical series quickly and it is always converging, leading to a solution presented in a closed and exact form.

The choice of the Adomian method [19,20,21,22] was due to the advantage of combining two other powerful methods in obtaining exact solutions of linear and nonlinear partial differential equations [13,14]. Initially, and for the sake of generality, we applied the following method to a nonlinear differential equation:

$$FP\left[P\left(x,t\right)\right]=L_{FP}\left[P\left(x,t\right)\right]=0$$

(7)

FP, or L_FP, is a nonlinear differential operator that is decomposed into two parts: a linear one, composed of an easily invertible term (L) and a proper operator of the equation (R), and a nonlinear one (N) [14,15,16]:

$$FP\left[P\left(x,t\right)\right]=\left[L+R+N\right]P\left(x,t\right)$$

(8)

This can be rewritten as

$$FP\left[P\left(x,t\right)\right]=LP\left(x,t\right)+RP\left(x,t\right)+NP\left(x,t\right)=0$$

(9)

If we apply L⁻¹ to the members of this equation, we can obtain

$$L^{-1}LP\left(x,t\right)=P\left(x,t\right)-P\left(x_o,t_o\right)=-L^{-1}RP\left(x,t\right)-L^{-1}NP\left(x,t\right)$$

(10)

where

$$P(x,t)={\displaystyle \sum _{i=0}^{\infty }{P}_i}=P_0+P_1+\dots +P_i=P(x_o,t_o)-L^{-1}R\left[P_0+P_1+\dots +P_i\right]-L^{-1}\left[A_0+A_1+\dots +A_i\right]$$

(11)

This gives rise to what are conventionally called Adomian polynomials [14,15,16,20,21,22]. Considering that the general solution is expressed according to the generalized Taylor series expansion [14] of each term of the operators of the originating equation, terms of the same order can be equalized, generating equality, and the generating formula provides us with the following equation:

$$P_{n+1}=-L^{-1}R{P}_n-L^{-1}\left[A_n\right]$$

(12)

The nonlinear part of NP is written as the series

$$NP(x,t)=N\left({\displaystyle \sum _{i=0}^{\infty }{P}_i}\right)={\displaystyle \sum _{n=0}^{\infty }{A}_n}$$

(13)

The A_n in P₀, P₁, P₂,..., P_n is known as an Adomian polynomial [14,15,16], the calculation of which is provided by the formula

$$N\left(P_0+\sum _{i=1}^{\infty }{\lambda }^i{P}_i\right)=A_0+\sum _{i=1}^{\infty }{\lambda }^n{A}_n.$$

(14)

Considering the general expression

$$A_n(P_0,P_1,P_2,\dots ,P_n)={\displaystyle \frac{1}{n!}}{\left\{{\displaystyle \frac{d^n}{d{\lambda }^n}}\left[N\left({\displaystyle \sum _{i=0}^{\infty }{\lambda }^i{P}_i}\right)\right]\right\}}_{\lambda =0},$$

(15)

it should be considered that

$$N\left(P\right)=\sum _{n=0}^{\infty }\frac{{\left(P-P_0\right)}^n}{n!}{N}^{\left(n\right)}\left(P_0\right)=\sum _{n=0}^{\infty }\left[\frac{{\left(P-P_0\right)}^n}{n!}{N}^{\left(n\right)}\left(P_0\right)\right]=\sum _{n=0}^{\infty }{A}_n=A_0+A_1+\dots +A_n$$

(16)

This can be written as

$$N\left(P\right)=N\left(P_0\right)+{\left(P_1+\dots +P_n\right)}^1{N}^{\left(1\right)}\left(P_0\right)+{\displaystyle \frac{{\left(P_2+\dots +P_n\right)}^2{N}^{\left(2\right)}\left(P_0\right)}{2!}}+\dots +{\displaystyle \frac{{\left(P_n\right)}^n{N}^{\left(n\right)}\left(P_0\right)}{n!}}$$

(17)

The generating formulas of the Adomian polynomials can be written in four different ways [19,20,21,22]. We will look at each of them in the subtopics below.

3. Results

Preliminary applications of FPE modeling have enabled the consideration of variations based on astrophysical hypotheses regarding the terms of the restorative external force and the diffusive term. These hypotheses allow for the exploration of multiple modeling approaches within the possible scenarios in XRBSs, yielding results consistent with the behavior observed in experimentally collected data.

Before presenting the models and their proposed solutions, it is also important to consider that, due to the distance from the source of formation of the analyzed diffusive processes as well as the long formation time, the stationary proposal is adopted at the time in which the terms A(x,t), the restoring external force, and B(x,t), the diffusive term, are considered constant and normalized. This is the case except for in the proposal of its linear growth hypothesis (in Section 3.5), with the initial conditions also in several hypotheses outlined in Section 3.1 and Section 3.5, which substantiates arguments for a physically linear interaction between the barycenter orbits with very long distances, as recommended in the phenomenology between some of the analyzed XRBSs [3,6].

On the other hand, the classical modeling proposals for the emission, expansion, and decay of electromagnetic waves include those with hyperbolic behavior of the diffusion and restoration terms (Section 3.2), those with an initial condition of normalized exponential decay (Section 3.3) and adjustable exponential decay (Section 3.4), those with hypotheses of fractional FPE in time, Caputo’s memory effect (α), and constant linear and diffusive restorative terms (Section 3.6), and those with hypotheses for the initial conditions of behavior of a q-Gaussian distribution (Section 3.7), which may be characteristic of simple binary systems, that is, those with only two stars. There is a lack of knowledge about the characteristics of binary systems, as they have long orbital periods (of several centuries or millennia), so the characteristics of their orbits and other variables are little known or are not accurately known [3,6,8].

It is important to note that, in addition to orbital singularities, there are phenomena such as coronal mass ejections, the emission of gases from outer atmospheric regions, the compaction of nearby small stars, the random capture of dark matter, and the accumulation of degenerate matter. These events occur alongside the immense energy released by fusion, which causes sudden increases in brightness across various spectra. Additionally, gravitational disruptions in the protoplanetary disk may lead to the expansion of zones with stable orbits, potentially increasing the accretion rate of any protostars in the vicinity. Although this scenario is plausible, it is difficult to detect, as such protostars are rarely observed. For instance, the singularities in the white dwarf–pulsar binary PSR B1620-26b, the red dwarf subgiant Gamma Cephei, and the white dwarf–red dwarf system NN Serpentis, among others already cataloged [6], remain unexplained.

Finally, these possible scenarios, uniquely or concomitantly, start to justify the hypotheses put forward for the behavior of the restorative diffusive terms and the initial conditions that reflect current beliefs about X-ray emission processes of XRBSs, as the components of a system with XRBSs can be close enough together that they can mutually gravitationally distort their partner’s atmosphere or even transfer gravitational action from one star to another, generating unusual or even impossible behavior in a lone star, thus supporting the demand for knowledge of these terms in the FPE that proposes to model the respective XRBS [3,6].

It is important to note that the solutions obtained using the Adomian method are exact, either as direct functional expressions or as convergent series, and can be easily composed into their original functions. This potential is particularly valuable for exploring new models for differential equations, especially nonlinear partial ones, and for singular modeling, which is the specific focus of our research. The protost of Tsalian behavior, however, has not yet been explored within the scenarios hypothesized in this study. In this way, the proposed modeling records, in their unique results, a nonclassical behavior of the distribution. Once resolved, Tsallis’s mechanics are legitimized.

The proposed modeling is inherently unique; however, future work will explore additional models to develop new solutions and distribution proposals. These efforts will consider both the stationary state, where current data are focused, and the dynamic state, to enable new predictions.

Table 1. Ensembles considered in the analysis.

Term		Initial Condition ${\mathit{P}}_{(\mathit{x},0)}={\mathit{P}}_0$	Considerations and Hypotheses for the Emission of X-rays of XRBSs (Ensemble Physics)	Subchapter
Restorer A(x,t)	Diffusive B(x,t)
−1	1	$P_{\left(x,0\right)=x}$	VET with first-order approximation in the description of the phenomenon, and with assumptions in constant and normalized values with restorative and diffusive terms.	Section 3.1
$e^t\cosh (x)$	$e^t\cosh (x)$	$P_{\left(x,0\right)=senh\left(x\right)}$	FPE with classical proposals for the emission, expansion, and decay of electromagnetic waves with hyperbolic behavior of the diffusion and restoration terms (Markovian event).	Section 3.2
1	1	$P_{\left(x,0\right)=e^{-x}}$	FPE with unitary and constant restorative and diffusive rates (normalized) and with initial condition with normalized exponential decay (Markovian event).	Section 3.3
1	1	$P_{\left(x,0\right)=e^{-ax}}$	FPE with unitary and constant restorative and diffusive rates (normalized) and with initial condition with adjustable exponential decay.	Section 3.4
$-x$	1	$P_{\left(x,0\right)=x}$	Fractional time FPE to rescue the memory effect of the initial scenarios, recommended under the assumed initial conditions, with the fractional derivation of Caputo (α), with a normalized and negative linear restorative term, with a unitary and normalized constant diffusive term, and with a hypothesis for the conditions of initial linear expansion.	Section 3.5
$-x$	1	$P_{(x,0)}={\left[1-\left(1-q\right)\left(x^2\right)\right]}^{{\displaystyle \frac{1}{1-q}}}$	Fractional time FPE, with memory effect with Caputo fractional derivation (α), with normalized and negative linear restorative terms and unitary and normalized constant diffusive terms, and with hypotheses for the initial conditions of behavior of a q-Gaussian distribution.	Section 3.6
−1/x	$x$	$P_{(x,0)}={\left[1-\left(1-q\right)\left(x^2\right)\right]}^{{\displaystyle \frac{1}{1-q}}}$	Fractional time FPE, with Caputo memory effect (α), with attractive restorative terms inversely proportional to distance and linear diffusive, and with hypotheses for the initial conditions of behavior of a q-Gaussian distribution.	Section 3.7

presents some possible considerations regarding the function of restorative and diffusive terms, boundary conditions, and descriptions of the hypothetical set for the linear and nonlinear Fokker–Planck equations for modeling the behavior of the XRBSs analyzed. These suggestions have not yet been properly modeled because, in all cases, the proposed EFP did not record an immediate analytical solution until the proposal to use the Adomian method for its solutions, including analytical and exact solutions, as the method recommends, according to the description of the physical characteristics of the proposed modeling in the fourth column of Table 1.

The final column provides a summary of each specified item, with deductions from the solutions that will be discussed next. This allows for the modeling of different perspectives based on the related physical considerations, which will be elaborated upon in the subsequent discussions of this research.

It is also worth noting that a stationary solution is being explored, as satellite data suggest stationary behavior, which supports the proposed models and the solutions analyzed. This includes correlations with current hypotheses regarding the diffusive behavior of X-rays in binary systems.

With this foundation, we will present the modeling proposals and their solutions, followed by comments and considerations in the discussion section.

3.1. Constant and Normalized Restorative and Diffusive Term and Linear Stationary Term

Considering a restoring force to be constant and negative and one diffusive term to be a positive constant with linear initial behavior in the initial conditions, we can obtain a very simple phenomenological scenario that can simulate some cases of XRBSs with little observation time because this type of behavior emulates conditions that can be represented by constant values that greatly simplify the modeling, as can be seen below [3]:

$${\displaystyle \frac{\partial P\left(x,t\right)}{\partial t}}=-{\displaystyle \frac{\partial }{\partial x}}\left[A(x,t)P\left(x,t\right)\right]+{\displaystyle \frac{{\partial }^2\left[B(x,t)P\left(x,t\right)\right]}{\partial x^2}}=-L_{FP}P(x,t)$$

(18)

where

$$A(x,t)=-1$$

(19)

and

$$B(x,t)=1,$$

(20)

while

$$P(x,0)=P_0=x$$

(21)

And its graphic composition is evidenced in Figure 2 below.

Thus,

$$L_{FP}={\displaystyle \frac{\partial }{\partial x}}A(x,t)-{\displaystyle \frac{{\partial }^2}{\partial x^2}}B(x,t)=-{\displaystyle \frac{\partial }{\partial x}}-{\displaystyle \frac{{\partial }^2}{\partial x^2}}$$

(22)

as

$$L_t\left\{P(x,t)\right\}=-L_{FP}\left\{P(x,t)\right\}$$

(23)

Applying the inverse operation to the temporal operator, we obtain

$$L_t^{-1}\left\{L_t\left\{P(x,t)\right\}\right\}=-L_t^{-1}\left\{L_{FP}\left\{P(x,t)\right\}\right\}$$

(24)

where

$$P(x,t)=P(x,0)-L_t^{-1}\left\{L_{FP}\left\{P(x,t)\right\}\right\},$$

(25)

which is a recursive and immediate way of calculating the terms of the general solution given by

$$P(x,t)={\displaystyle \sum _{n=0}^{\infty }{P}_n}=P_0+P_1+\dots +P_n=x-t+0+0+\dots +0=x+t,$$

(26)

whose solution is

$$P(x,t)=x+t$$

(27)

The graphic composition is shown in Figure 3 below.

This is the solution proposed by the method of Adomian [9,10,11,18,19,20], as proof of the method for proposing exact solutions.

3.2. FPE with Classical Hyperbolic Proposals for the Emission, Expansion, and Decay of Electromagnetic Waves

In the proposal of modeling with simplified hypotheses, one can consider a restorative and diffusive force with increasing exponential behavior shaped by hyperbolic terms; that is,

$${\displaystyle \frac{\partial P\left(x,t\right)}{\partial t}}=-{\displaystyle \frac{\partial }{\partial x}}\left[A(x,t)P\left(x,t\right)\right]+{\displaystyle \frac{{\partial }^2\left[B(x,t)P\left(x,t\right)\right]}{\partial x^2}}=-L_{FP}P(x,t),$$

(28)

with a restorative term

$$A(x,t)=e^t\coth (x)\cosh (x)+e^{t}senh(x)-\coth (x)$$

(29)

and

$$B(x,t)=e^t\cosh (x)$$

(30)

whose boundary condition could be

$$P(x,0)=P_0=senh(x)$$

(31)

These conditions could shape some types of behavior in line with hyperbolic functions, which end up modeling the listed attractions, decays, and diffusion scenarios well; however, these are classic solutions of linear differential equations, despite their various techniques [3,5,10], which justify their use.

Thus, it is possible to write the following equation:

$$L_t\left\{P(x,t)\right\}=-L_{FP}\left\{P(x,t)\right\}$$

(32)

where

$$P_0=P(x,0)=senh(x)$$

(33)

Thus,

$$P_{n+1}=-L^{-1}{L}_{FP}{P}_n={\displaystyle \frac{t^{n+1}}{(n+1)!}}senh(x).$$

(34)

This is a recursive and immediate way of calculating the terms of the general solution given by

$$\begin{array}{l}P(x,t)={\displaystyle \sum _{n=0}^{\infty }{P}_n}=\cosh (x)+tsenh(x)+{\displaystyle \frac{t^2}{2!}}senh(x)+{\displaystyle \frac{t^3}{3!}}senh(x)+\dots {\displaystyle \frac{t^n}{n!}}senh(x)\\ \dots +P_n=x-t+0+0+\dots +0=\left(1+t+{\displaystyle \frac{t^2}{2!}}+{\displaystyle \frac{t^3}{3!}}+\dots {\displaystyle \frac{t^n}{n!}}\right)senh(x)=e^{t}senh(x)\end{array}$$

(35)

whose closed solution is

$$P(x,t)=e^{t}senh(x)$$

(36)

This is the solution proposed by the method of Adomian [18,19,20,21,22,23]. It models a feasible scenario and its solution function is part of the hypothesized term for the initial conditions, which, in itself, legitimizes the fact that the proposal, reflected in the phenomenology of the data and its adjustment to the initial functions, unfolds as a constitutive part of the final solution of the associated FPE itself [3].

3.3. FPE with Unitary and Constant Restorative and Diffusive Rates with Initial Condition with Normalized Exponential Decay

In an attempt to model in accordance with physically feasible hypotheses and mathematically analytical solutions, a constant restoring force and diffusive term are considered viable, with boundary conditions in exponential decay [3]; thus, the following equation is obtained:

$${\displaystyle \frac{\partial P\left(x,t\right)}{\partial t}}=-{\displaystyle \frac{\partial }{\partial x}}\left[A(x,t)P\left(x,t\right)\right]+{\displaystyle \frac{{\partial }^2\left[B(x,t)P\left(x,t\right)\right]}{\partial x^2}}=-L_{FP}P(x,t)$$

(37)

where

$$A(x,t)=1$$

(38)

and

$$B(x,t)=1,$$

(39)

as

$$P(x,0)=P_0=e^{-x}$$

(40)

The graphic composition is shown in Figure 4 below.

From this, we can obtain the following:

$$L_{FP}={\displaystyle \frac{\partial }{\partial x}}A(x,t)-{\displaystyle \frac{{\partial }^2}{\partial x^2}}B(x,t)={\displaystyle \frac{\partial }{\partial x}}-{\displaystyle \frac{{\partial }^2}{\partial x^2}}.$$

(41)

It is possible to write

$$L_t\left\{P(x,t)\right\}=-L_{FP}\left\{P(x,t)\right\}$$

(42)

And applying the inverse operation to the temporal operator, we can obtain the following:

$$L_t^{-1}\left\{L_t\left\{P(x,t)\right\}\right\}=-L_t^{-1}\left\{L_{FP}\left\{P(x,t)\right\}\right\},$$

(43)

where

$$P(x,t)=P(x,0)-L_t^{-1}\left\{L_{FP}\left\{P(x,t)\right\}\right\}.$$

(44)

Thus,

$$P_0=P(x,0)=e^{-x}$$

(45)

which allows us to write that

$$P_n=-L^{-1}{L}_{FP}{P}_{n-1}=0$$

(46)

As such, the following closed solution can be obtained:

$$P(x,t)=(1+2t)e^{-x}$$

(47)

The graphic composition is shown in Figure 5 below.

This is the solution proposed by the method of Adomian [10,11,12,13,14,15,16,18,19,20], which comes closer to the objective reality of exponential adjustments of the numerical data collected on the satellite; however, the flexibility put into the different XRBSs is not verified [3].

3.4. FPE with Unitary and Constant Restorative and Diffusive Rates (Normalized) and with Initial Condition with Adjustable Exponential Decay

By considering a constant restoring force and diffusive term, with an adjustable exponential decay boundary condition, the method used for the previous item is repeated with the belief that the adjustment of the exponential coefficient would allow us to adjust the curves of the different XRBSs; however, these adjustments do not maximize the results, as can be seen in the solution. So, in this attempt, we use the following equation:

$${\displaystyle \frac{\partial P\left(x,t\right)}{\partial t}}=-{\displaystyle \frac{\partial }{\partial x}}\left[A(x,t)P\left(x,t\right)\right]+{\displaystyle \frac{{\partial }^2\left[B(x,t)P\left(x,t\right)\right]}{\partial x^2}}=-L_{FP}P(x,t)$$

(48)

where

$$A(x,t)=1$$

(49)

and

$$B(x,t)=1,$$

(50)

with an adjustable boundary condition in exponential parameter, represented as

$$P(x,0)=P_0=e^{-ax}$$

(51)

The graphic composition in the register is analogous to that shown in Figure 4 as a function of the values of a. So, the recursive formula is used:

$$P_{n+1}=-L^{-1}R{P}_n-L^{-1}{A}_n,$$

(52)

where

$$P_0=P(x,0)=e^{-ax}$$

(53)

which allows us to obtain the following equation:

$$P(x,t)=e^{-ax}\left[{\displaystyle \sum _{n=1}^{\infty }\left[a^{n-1}{(1+a)}^{n-1}\frac{t^{n-1}}{(n-1)!}\right]}\right].$$

(54)

This is an open solution, even though it is notoriously convergent. It is an exact solution, which is the solution proposed by the method of Adomian [9,10,11,12,13,14,15,16,17,18,19,20], and as previously announced, it is not completely modeled on the behavior of the phenomenology of the different XRBSs analyzed.

3.5. Fractional Time FPE with Caputo Fractional Derivation (α) and with Negative Linear Restorative Term and Unitary Constant Diffusive Term with Linear Initial Conditions

Nonlinear FPE with fractional time derivation can be deduced in the same way as nonlinear nonfractional FPE. Thus, due to nonlinearity in time, that is, more generally according to the Fokker–Planck equation, with considerations that give rise to an anomalous diffusive process, it becomes possible to incorporate other spatiotemporal distributions [3]. Thus, it is possible to consider a nonlinear diffusion equation, analytically describing a class of anomalous diffusive processes, represented by the following equation [5,14,15,21,22,23,24,25,26,27]:

$${\displaystyle \frac{{}^C\partial ^{\alpha }P\left(x,t\right)}{\partial t^{\alpha }}}=-{\displaystyle \frac{\partial }{\partial x}}\left[A(x,t)P\left(x,t\right)\right]+{\displaystyle \frac{{\partial }^2\left[B(x,t)P\left(x,t\right)\right]}{\partial x^2}};0<\alpha <1.$$

(55)

Here, the term fractional derivative in time rescues the memory effect of phenomenological evolution, better describing the system as a complex system with nonlocal effects as well as its fractality and better representing the exposed scenario and its contribution rates for each part of the phenomena that occurred [15,16].

Before applying the Adomian method to propose a solution to this equation, it is worth considering that many definitions of fractional calculus are used to solve fractional differential problems and the most frequently found ones include those of Riemann–Liouville, Caputo, Wely, and Rize [16,17,18]. In the application dealt with here, the proposal of Riemann–Liouville derivation and integration operators was adopted, and the fractional derivative of order α is defined as

$${}^{C}D^{\alpha }P\left(x,t\right)={\displaystyle \frac{d^{\alpha }P\left(x,t\right)}{d{x}^{\alpha }}}={\displaystyle \frac{1}{\mathrm{\Gamma }(n-\alpha )}}{\displaystyle \frac{d^n}{d{x}^n}}\left[{{\displaystyle {\int }_0^x(x-t)}}^{n-\alpha -1}P\left(x,t\right)dt\right];n-1\le \alpha <n$$

(56)

The integral operator $I^{\alpha }$, also called the Riemann–Liouville fractional integral, is defined as follows [20,21]:

$$I^{\alpha }P\left(x,t\right)={\displaystyle \frac{d^{-\alpha }P\left(x,t\right)}{d{x}^{-\alpha }}}={\displaystyle \frac{1}{\mathrm{\Gamma }(\alpha )}}{{\displaystyle {\int }_0^x(x-t)}}^{\alpha -1}P\left(x,t\right)dt.$$

(57)

We also use the relationships involving the modified fractional derivative, the Caputo derivative [20,21], which is defined as

$$D_C^{\alpha }P\left(x,t\right)={\displaystyle \frac{d^{\alpha }P\left(x,t\right)}{d{x}^{\alpha }}}={\displaystyle \frac{1}{\mathrm{\Gamma }(n-\alpha )}}\left[{{\displaystyle {\int }_0^x(x-t)}}^{n-\alpha -1}P\left(x,t\right)dt\right];n-1\le \alpha <n.$$

(58)

We also have a generalized exponential function, known as the Mittag–Leffler function, which is defined as [23]

$$E_B(x)={\displaystyle \sum _{r=0}^{\infty }\left(\frac{x^r}{\mathrm{\Gamma }(\beta r+1)}\right)};\beta >0$$

(59)

That said, it is possible to write, using the Adomian method, the following relations [10,11,12,13,14,15,16]:

$$D_{C_t}^{\alpha }P\left(x,t\right)=-{\displaystyle \frac{\partial }{\partial x}}\left[A(x,t)P\left(x,t\right)\right]+{\displaystyle \frac{{\partial }^2\left[B(x,t)P\left(x,t\right)\right]}{\partial x^2}}=-L_x\left\{A(x)P(x,t)\right\}+L_{xx}\left\{B(x)P(x,t)\right\}$$

(60)

Thus,

$$P\left(x,t\right)=P(x,0)+I^{\alpha }\left[-L_x\left\{A(x)P(x,t)\right\}+L_{xx}\left\{B(x)P(x,t)\right\}\right].$$

(61)

The α index is adjusted according to the analyzed system. In the case of XRBSs, with the best-fit coefficients in the diffusive process, we were able to relate them phenomenologically. For example, α = 1 corresponds to a distribution of probabilities associated with nonextensive systems, which maximize the entropy proposed by Tsallis, and the entire order (α) does not record effects of memory of the originating phenomenon; however, each XRBS must be better explained as a function of its age, originating phenomena, and distance from the source, this being best modeled for specific values of α different from the unit.

This, as stated above, is a generalization of the Boltzmann–Gibbs entropy, enabling a reduction in the classical type of FPE.

Thus, α is a real parameter that can assume values depending on the environment and the historicity of the phenomena since its formation, with the respective values of this parameter reflecting the transient effects of its formation so that it can be adjusted to enable a better correlation with the numerical data to reveal the “type” of diffusive process (normal or anomalous).

Parallel to this, the restorative term A(x,t) may be associated with an external restorative force, which in turn is associated with a Tsallian confining potential. This is contrary to the diffusive process, due to the confident gravitational potential arising from the XRBSs analyzed with the diffusive term B(x,t), with characteristics that fit when modeled in stellar systems [5,10].

The conditions to consider are as follows:

$$A(x,t)=-x$$

(62)

and

$$B(x,t)=1$$

(63)

and

$$P(x,0)=P_0=x$$

(64)

The graphic composition is shown in Figure 6 below.

Generally speaking, the following equation applies:

$$P_n=I^{\alpha }\left[-L_x\left\{A(x)P_{n-1}\right\}+L_{xx}\left\{B(x)P_{n-1}\right\}\right]={\displaystyle \frac{2^{n}x{t}^{n\alpha }}{\mathrm{\Gamma }(n\alpha +1))}}.$$

(65)

So, finally, the best option for the solution and the closed solution (in the limit of infinite terms) will produce the following equation:

$$P(x,t)={\displaystyle \sum _{n=0}^{\infty }\left[P_n\right]}=P_1+P_2+P_3+\dots +P_n={\displaystyle \sum _{n=0}^{\infty }\left[\frac{2^{n}x{t}^{n\alpha }}{\mathrm{\Gamma }(n\alpha +1))}\right]}$$

(66)

That is,

$$P(x,t)=x{\displaystyle \sum _{n=0}^{\infty }\left[\frac{{\left(2t^{\alpha }\right)}^n}{\mathrm{\Gamma }(n\alpha +1))}\right]}=x{E}_{\alpha }\left(2t^{\alpha }\right)$$

(67)

Thus,

$$P(x,t)=x{E}_{\alpha }\left(2t^{\alpha }\right)$$

(68)

The graphic composition is shown in Figure 7 below.

It should also be considered that the stationary solution falls to classical diffusion when the index α is an integer and can be adjusted as a function of experimental data so that each XRBS will be well represented with its respective value for the order of the fractional derivation which best applies to modeling and which fits the power law that governs the diffusion of the respective XRBS. This defines intrinsic characteristics of each XRBS, allowing us to generalize the modeling proposal for this category of astronomical objects.

3.6. Caputo Fractional FPE (α) with Normalized Linear Restorative Terms and Negative and Unitary Constant Diffusive Terms with Initial Conditions of Behavior of a q-Gaussian Distribution

The nonlinearity in time results in disruptive action and the accumulation of different weights at each moment of occurrence, keeping the memory effect in accordance with the more general Fokker–Planck equation, which gives rise to an anomalous diffusive process [3]. In this way, it becomes possible to incorporate other spatiotemporal distributions so that it is feasible to consider a nonlinear diffusion equation, analytically describing a class of anomalous diffusive processes with a hypothesis of Tsallian behavior for the initial conditions, as detected in the analyzed phenomenology [5,14,15,21,22,23,24,25,26,27] and properly represented by the following equation:

$${\displaystyle \frac{{}^C\partial ^{\alpha }P\left(x,t\right)}{\partial t^{\alpha }}}=-{\displaystyle \frac{\partial }{\partial x}}\left[A(x,t)P\left(x,t\right)\right]+{\displaystyle \frac{{\partial }^2\left[B(x,t)P\left(x,t\right)\right]}{\partial x^2}};0<\alpha <1$$

(69)

Here, the term fractional derivative in time rescues the memory effect of phenomenological evolution, better representing the exposed scenario and its contribution rates for each part of the phenomena that occurred [15,16].

That said, it is possible to write, using the Adomian method, the following general solution [11,12,13,14,15,16]:

$$P\left(x,t\right)=P(x,0)+I^{\alpha }\left[-L_x\left\{A(x)P(x,t)\right\}+L_{xx}\left\{B(x)P(x,t)\right\}\right]$$

(70)

The α index is adjusted according to the analyzed system, as already elucidated, and in this case, there is a better correlation of the boundary conditions, since the hypothesis of the Tsallian behavior has already been properly identified in the real data. Considering that the restorative term A(x,t) may be associated with an external restorative force, which in turn is associated with a Tsallian confining potential, the following can be determined:

$$A(x,t)=-x$$

(71)

and

$$B(x,t)=1$$

(72)

and

$$P_{(x,0)}={\left[1-(1-q)(x^2)\right]}^{{\displaystyle \frac{1}{1-q}}}$$

(73)

The graphic composition is shown in Figure 8 below.

A more complete modeling that is, therefore, closer to the phenomenological reality analyzed in the data, can be modelled for a closed solution as follows:

$$P(x,t)={\left(1+(-1+q)x^2\right)}^{{\displaystyle \frac{1}{1-q}}}{E}_{\alpha }\left({\displaystyle \frac{t^{\alpha }(1+q+3x-2qx+2x^2-3q{x}^2+q^2{x}^2)}{{\left(-1+(-1+q)x^2\right)}^2}}\right)$$

(74)

The graphic composition is shown in Figure 9 below.

The unit index α is related to the Tsallian case of the q-Gaussian distribution. For the other values of the α index, the stationary solution already used in the other models is obtained, plus a temporal term derived from the memory effect of the origin of the phenomenon’s formation. Depending on how it happened in each XRBS, there is a custom index for it.

3.7. Fractional Time FPE with Caputo Memory Effect (α) and with Attractive Restorative Terms Inversely Proportional to Distance and Linear Diffusive with q-Gaussian Initial Condition

Still considering the nonlinearity in time, that is, the proposal of an anomalous diffusive process with other hypotheses associated with the restorative and diffusive terms and with regard to initial conditions, it becomes possible to incorporate other spatiotemporal distributions. For example, a nonlinear diffusion equation, analytically describing a class of anomalous diffusive processes [3] and with a hypothesis of compound Tsallian behavior, is q-Gaussian in its composition and decays with the inverse square of the distance for the initial conditions. This hypothesis is consistent with some current theoretical proposals and with the phenomenology that is hypothesized in the regions of influence of XRBSs [5,14,15,21,22,23,24,25,26,27]. This way, the following can be obtained:

$${\displaystyle \frac{{}^C\partial ^{\alpha }P\left(x,t\right)}{\partial t^{\alpha }}}=-{\displaystyle \frac{\partial }{\partial x}}\left[A(x,t)P\left(x,t\right)\right]+{\displaystyle \frac{{\partial }^2\left[B(x,t)P\left(x,t\right)\right]}{\partial x^2}};0<\alpha <1$$

(75)

Analogously, the term fractional derivative in time rescues the memory effect of the evolution of the diffusive process, better representing the exposed scenario and its contribution rates for each part of the phenomena that occurred [15,16]. In this way, using the Adomian method, we have the following general solution [10,11,12,13,14,15,16]:

$$P\left(x,t\right)=P(x,0)+I^{\alpha }\left[-L_x\left\{A(x)P(x,t)\right\}+L_{xx}\left\{B(x)P(x,t)\right\}\right]$$

(76)

In this proposal, the restorative term A(x,t) may be associated with an external restorative force that decays with the inverse of the distance, consolidating a proposal that goes beyond the gravitational influence, as it adds possibilities associated with electrical and magnetic oscillations provided by coronal jets and other oscillations and anomalies that may originate from the stellar interaction scenario and which are hypothesized to be associated with a confining potential. As such,

$$A(x,t)=-1/x$$

(77)

and has a diffuse term of linear behavior where

$$B(x,t)=x$$

(78)

and an initial condition, as recommended in the Tsallis statistics and in line with the measured phenomenology, of

$$P_{(x,0)}={\left[1+(-1+q)(x^2)\right]}^{{\displaystyle \frac{1}{1-q}}}$$

(79)

The solution has been previously presented in Figure 8.

Thus, a more complete modeling that is, therefore, closer to the phenomenological reality of the data can be achieved:

$$P(x,t)={\displaystyle \sum _{n=0}^{\infty }\left[P_n\right]}=P_1+P_2+P_3+\dots +P_n$$

(80)

with closed solution like

$$P(x,t)={\left(1-(-1+q)x^2\right)}^{{\displaystyle \frac{1}{1-q}}}{E}_{\alpha }\left({\displaystyle \frac{t^{\alpha }(3+6x^2+\left(3-4q+q^2\right)x^4)}{{\left(-1-(-1+q)x^2\right)}^2}}\right)$$

(81)

The graphic composition is shown in Figure 10 below.

Furthermore, the α index can be adjusted as a function of the experimental data so that each XRBS will be well represented with its respective value for the order of the fractional derivation that best applies to the modeling. This defines intrinsic characteristics of each XRBS, allowing us to generalize the modeling proposal for this category of astronomical object [3,27,28,29,30,31,32,33].

4. Discussion

In general, based on the values of the entropic index, q, and its proposed physical interpretation within the dynamics of binary X-ray systems, as well as its kinetic foundation in Tsallis’s generalized thermostatistics, it is possible to support, through mathematical modeling and the Fokker–Planck equation (FPE), a nonextensive and turbulent behavior in X-ray scattering. The anomalies observed can be attributed to potential long-range interactions that suggest fractality in the governing system of the analyzed XRBS. To this end, fractional derivative modeling and associated FPE solutions were employed [8,34].

The solutions of FPE show a probabilistic behavior of the X-ray diffusion processes in these XRBSs, with boundary conditions that not only fit in the analyzed data but are also highly correlated with nonadditive conformal entropy, verified in isomorphic modeling in recent research in different areas [4,5,6,7,35,36].

In this research, there is a greater correlation with the probability distribution modeled with the real data in the proposals with initial condition hypotheses and a q-Gaussian process; in addition, these same assumptions end up encompassing the canonical switching relations in the diffusive and restorative relations, leaving only adjustments of the entropy index (q) for each analyzed XRBS.

A significant observation from the analysis of the X-ray spectrum is that X-rays are emitted only when the two stars are in close proximity, potentially leading to the filling of the Roche lobe, a region of critical gravitational potential. The Roche lobe opens up various hypotheses, as this area of gravitational influence remains largely unexplored. If a star expands beyond its Roche lobe, the material outside it will be drawn toward the other star, affecting the diffusion process and connecting it to this accretion scenario.

The fact is that these scenarios involve singular or anomalous turbulence and diffusion, particularly in the context of magnetohydrodynamics and its unique characteristics—many of which remain unexplored, unpredicted, or unexplained. This makes the Adomian decomposition method and its derivatives especially powerful in deriving exact solutions, as the tension forces within a magnetic field deviate from typical hydrodynamic behavior, potentially predicting various unusual phenomena. This justifies the use of a nonlinear Fokker–Planck equation (FPE) to describe this phenomenology, with adjustments to the relevant entropic indices and fractional derivative orders to capture long-range effects, memory, and other fractal characteristics of the complex system under analysis. This approach effectively generalizes the linear Ornstein–Uhlenbeck process and its associated drift coefficients [34,35,36,37,38].

This entire scenario justifies the numerous modeling attempts centered on the analyzed XRBS FPE solutions and the underlying hypotheses modeled in Section 3.1, Section 3.2, Section 3.3 and Section 3.4. These range from the simplest to the most complex systems, including those that consider nonadditive entropy scenarios (Section 3.5, Section 3.6 and Section 3.7). In each case, solutions are directly tied to the hypothesized boundary conditions and yield exact, often convergent, and sometimes closed solutions. These solutions consistently reflect X-ray diffusion as a function of the initial conditions, which are also hypothesized, while always adhering to the consensus of physical laws applicable within the regions of influence of the respective XRBS.

The use of the Adomian decomposition method, along with other related methods such as Laplace, He–Laplace, and others utilizing different kernels, significantly enhances the modeling’s alignment with the proposed phenomenological hypotheses. The primary advantage of the Adomian method is that it not only facilitates an analytical process but also produces a convergent series with rapid and continuous approximation, applicable to both closed and open solutions. Moreover, the Adomian method does not require any transformation, linearization, or discretization of variables, consistently converging to an analytical and exact solution [28,29,30].

Given the solution capabilities of the Adomian decomposition method, it is clear that simpler proposals (Section 3.1, Section 3.2, Section 3.3 and Section 3.4), despite some phenomenological support, can be set aside in favor of more elaborate proposals (Section 3.5, Section 3.6 and Section 3.7). Specifically, the proposed solution outlined in Section 3.6 and Section 3.7, as illustrated in Figure 8, Figure 9 and Figure 10, demonstrates that the hypothesis of an initial condition with a q-Gaussian distribution (Figure 8) aligns with results from previous studies [8,34]. Additionally, the advanced modeling using the Adomian decomposition method (and its derivatives), as shown in Figure 9 and Figure 10, indicates that the probabilistic behavior encompasses more than just anomaly recording and long-range interactions. It also reveals visible fractalities in the graphs, thereby reinforcing the relevance of Tsallis’s generalized thermostatistics.

Furthermore, the α index can be adjusted based on experimental data, allowing each XRBS to be accurately represented with the appropriate fractional derivative order for its specific modeling. This adjustment defines the intrinsic characteristics of each XRBS, enabling the generalization of the modeling approach for this category of astronomical objects.

Additionally, we emphasize that Tsallis’s physical proposals for thermostatistics, as well as the Adomian decomposition method and its derivatives for single or coupled differential equations, significantly advance the study and modeling of stellar physical systems. These methods offer valuable insights that can be further explored and explained in future research.

5. Conclusions

Regarding the results obtained from the studied phenomenon, several proposals for restorative and diffusive terms were employed, including first-order approximations with Markovian events, exponential rates aligned with restorative and diffusive rates, and both unitary and parameterized decay rates. The Tsallis hypothesis was applied to constant restorative and diffusive terms, alongside a nonlinear and fractional time Fokker–Planck equation (FPE), to facilitate comparisons between the models and the actual data for each binary star pair analyzed. Ultimately, a convergent solution for the FPE was proposed for XRBSs.

Preliminary results from analyzing 156 X-ray intensity distributions from XRBSs revealed that these distributions are long-tailed, indicating a q-Gaussian nature. The entropic index q exhibited characteristics consistent with Tsallis thermostatistics, suggesting that these gravitational systems can be effectively interpreted within the framework of generalized thermostatistics (GTS), providing strong evidence for the phenomenological support of the theory.

This reality motivates the enhancement of modeling through new proposals for analyzing the restorative and diffusive terms of the Fokker–Planck equation (FPE), which describes the phenomena of X-ray diffusion in stellar binary systems.