**1. Introduction**

In the present era, fractional-order derivatives have become widespread due to their wide interdisciplinary applications and implementation in various fields of science and technology, such as solid mechanics, fluid dynamics, financial mathematics, social sciences, and other areas of science and engineering (see References [1–5]). As the solutions of non-integer order differential equations are more complicated than integer-order differential equations, computationally efficient and reliable numerical methods need to be developed to handle these. Authors have written different books (see References [6–10]) in which various studies and analyses on fractional calculus may be found that will support the authors for better understanding of the concepts of fractional calculus.

The hypothesis of entropy has been connected formerly with thermodynamics only; however, in present-day, it has additionally been utilized in different areas like data hypothesis, psychodynamics, biophysical financial aspects, human relations, etc. The second law of thermodynamics expresses that entropy increases with time. It demonstrates the unpredictability of a structure over some time if there is nothing to balance out it. Likewise, in human interactions, every day, various associations lead to some turmoil. Recently, the discussion of the titled model has been attaining recognition throughout the past few years. Relational relations emerge from numerous points of view, for instance, marriage, blood relations, close attachments, work, clubs [11,12], and so forth. Many authors have studied various research related to FDMM. The nonlinear coupled fractional FDMM was first investigated by Ozalp and Koca [13]. In that paper, they performed a balance situation for equilibrium points. Khader and Alqahtani [14] applied the Bernstein collocation method for obtaining the solution of a nonlinear FDMM, and they also compared their results with the Runge–Kutta fourth-order method. They defined the fractional derivative in the Riemann–Liouville sense, and via the utilization of Bernstein polynomials, they converted the FDMM to a system of nonlinear algebraic equations, which were solved using Newton's iterative method. Khader et al. [15] also solved the same model by implementing the Legendre spectral collocation method and a ffirmed the natural behavior of the present system. Singh et al. [16] implemented a q-homotopy analysis method coupled with Sumudu transform and Adomian decomposition method to solve FDMM and comparison results with the existing literature are also included. Goyal et al. [17] studied the FDMM utilizing a variation iteration method and a homotopy perturbation transform method.

Few authors have scientifically investigated the causes of extramarital interactions in marriage. It is essential and challenging to find out why some wedded couples separate, while a few couples do not. Moreover, among wedded couples, a few are fulfilled, while some are not fulfilled with each other. As such, the number of divorce cases are increasing every day all over the globe. A survey inside the U.S. uncovered that inside a forty-year interval, the probability of a first marriage finishing in separation are roughly 50 to 67 percent. The record is 10 percent higher for a second marriage. Around the world, the U.S. has the highest divorce rate. In this regard, experiments may be tough to conduct and may also be restricted for personal concerns, and so a mathematical model happens to be advantageous. As such, recently, researchers are investigating di fferent dynamical models for interpersonal relations.

The most recent model of marriage is the Romeo and Juliet model [18]. Assume that at any moment *t*, we need to determine Romeo's adoration or loathing for Juliet, *R*(*t*) and Juliet's a ffection or hate for Romeo, *J*(*t*). Positive estimations of these propose love, and negative values specify hate.

The presumption about this model is that the change in Romeo's adoration for Juliet is a small amount of his present love in addition to a small amount of her present love. Also, Juliet's a ffection for Romeo will change by a small amount of her present love for Romeo and a small amount of Romeo's adoration for her. This presumption prompts the model as given below [18,19]:

$$\begin{array}{l} \frac{dR}{dt} = aR(t) + bJ(t). \\ \frac{dI}{dt} = cR(t) + dJ(t). \end{array} \tag{1}$$

where *a*, *b*, *c*, and *d* are constants.

Gottman et al. [20] studied the discrete dynamical model to characterize the connection between them. Since the layouts of research in those fields are cumbersome and restrained through the moral reflections, mathematical models may furthermore have a fundamental influence in considering the elements of relations and conduct highlights. A few models are present for describing the romantic relationship; however, they may be limited to integer-order di fferential equations.

An integer order mathematical model of love is given as follows:

$$\begin{array}{l} \frac{d\psi}{dt} = -a\_1 \psi + b\_1 \xi \big( 1 - \delta \xi^2 \big) + c\_1. \\\frac{d\xi}{dt} = -a\_2 \xi + b\_2 \psi \big( 1 - \delta \psi^2 \big) + c\_2. \end{array} \tag{2}$$

Here, variables ψ and ξ measure the adoration of a man or woman for his/her partner. The parameters *ai*, *bi*, *ci*(<sup>1</sup> ≤ *i* ≤ 2) denote the oblivion, reaction, and attraction constants. We have measured that the decay of the feelings for one's partner occurs exponentially quickly within the absence of a partner. The parameter *ai* indicates the degree to which one is stimulated by way of one's personal feeling. It is used as a level of dependency along with fretfulness regarding other's affirmation in relationships. The parameter *bi* represents the level to which one is supported by one's partner and additionally anticipates him/her to be useful. It measures the tendency to keep away from or seek closeness in a relationship. The term −*ai*ψ and −*ai*ξ state that one's adoration measure decays exponentially without one's partner, 1/*ai* suggests the time needed for love to diminish and δ is a compensatory constant.

In the present study, a time-fractional order dynamical system has been considered instead of its integer order system because fractional order equations are generalizations of integer order di fferential equations and fractional order models hold memory. Interpersonal relationships are influenced by memory, which makes the modeling more appropriate than the integer one for this kind of dynamical system. This fact confirms that fractional modeling is best suited for this kind of system. Hence, the investigation of the time-fractional systems is significant. The FDMM is given as:

$$\begin{array}{ll}\frac{d^a\psi}{dt^a} = -a\_1\psi + b\_1\xi(1 - \delta\xi^2) + c\_1. \\\frac{d^a\xi}{dt^a} = -a\_2\xi + b\_2\psi(1 - \delta\psi^2) + c\_2. \end{array} \tag{3}$$

where 0 < α ≤ 1 *ai* ≥ 0

with initial conditions (ICs):

$$
\psi(0) = 0 = \xi(0) \tag{4}
$$

It is observed that all the authors mentioned above have considered the parameters and variables involved in FDMM as crisp or precise. However, in real life, it may not always be possible to take crisp values due to errors in experiments, observations, and many other errors. Therefore, the parameters and variables may be considered as uncertain. Here, the uncertainties are considered as intervals/fuzzy. The parameters *ai*, *bi*, *ci*(<sup>1</sup> ≤ *i* ≤ 2) and δ denote the oblivion, reaction, attraction, and compensatory constants, respectively. As these parameters are related to attractions and reactions of the model, its values may not always be fixed. As such, the main targets of the authors are to consider these parameters as fuzzy and then solve this fuzzy fractional model using an e fficient method.

Let us consider the coupled fuzzy FDMM as given below:

$$\begin{aligned} \frac{d^2 \psi}{dt^4} &= -(a\_1 - 0.02, a\_1, a\_1 + 0.02)\widetilde{\psi} + (b\_1 - 0.02, b\_1, b\_1 + 0.02)\widetilde{\xi} \\ &\quad \left\{ 1 - (\delta - 0.01, \delta, \delta + 0.01)\widetilde{\xi}^2 \right\} + (c\_1 - 0.2, c\_1, c\_1 + 0.2). \end{aligned} \tag{5}$$
 
$$\begin{aligned} \frac{d^2 \widetilde{\xi}}{dt^4} &= -(a\_2 - 0.02, a\_2, a\_2 + 0.02)\widetilde{\xi} + (b\_2 - 0.02, b\_2, b\_2 + 0.02)\widetilde{\psi} \\ &\quad \left\{ 1 - (\delta - 0.01, \delta, \delta + 0.01)\widetilde{\psi}^2 \right\} + (c\_2 - 0.2, c\_2, c\_2 + 0.2). \end{aligned} \tag{6}$$

with fuzzy ICs

$$
\overline{\psi}(0) = \overline{\xi}(0) = (-0.1, 0, 0.1) \tag{6}
$$

where variables ψ and ξ describe the uncertain adoration of a man or woman for his/her partner.

The basic concepts of fuzzy variables were first presented by Chang and Zadeh [21], where they suggested the theory of a fuzzy derivative. The extensive analysis in Chang and Zadeh [21] was well-defined and studied by Dubois and Prade [22]. Kaleva [23] and Seikkala [24] studied the fuzzy di fferential equations (FDEs) and initial value problems. Various problems related to the di fferential FDEs are broadly studied by Chakraverty et al. (see References [25–27]). As fuzzy fractional di fferential equations (FFDEs) are quite challenging to solve as compared to fractional di fferential equations, computationally e fficient numerical methods should be developed. In this research, we have applied a fractional reduced di fferential transform method (FRDTM) along with imprecisely defined parameters involved in the FDMM in order to study this dynamical system. Also, the convergence analysis of the present solution has been discussed with an increasing number of terms of the solution. The double-parametric form of a fuzzy number is applied to find the solution of the fractional fuzzy dynamical model of marriage. This model has not ye<sup>t</sup> been studied using FRDTM. The main benefit of using this technique are: First, this procedure achieves the expansions of the solutions. Second, this technique does not require any discretization, perturbations, or modification of the ICs. Also, this technique needs fewer computations with high precision, as well as less time compared to other techniques. In view of the above literature, FFDEs are first changed to a di fferential equation using a double-parametric form (DPF). Then, the equivalent equation is solved using FRDTM to have an interval/fuzzy solution in terms of the DPF.

The remaining parts of the manuscript are arranged as follows. In the "Preliminaries" section, we give essential information related to fuzzy arithmetic, triangular fuzzy number, and double-parametric form of a fuzzy number. In section "Fractional Reduced Di fferential Transform Method," we discuss methodology and important theorems related to this technique. The double-parametric form-based solution of FDMM is given in section "Double-Parametric-Based Solution of Uncertainty FDMM Using FRDTM." Next, numerical outcomes and deliberations are given in the "Results and Discussions" section. Finally, conclusions are drawn.
