Numerical investigations of the fractional order derivative-based accelerating universe in the modified gravity

1. Introduction

In recent years, an enhanced universe development is considered one of the challenging factors in the cosmology. The network phenomenon is explained in a better way by using the universal relativity, extensions, and corrections, which are required to extend the gravity theory. The analysis based on the data is obtained at the end of last century using the distance quantities of Ia Supernovae type, which leads to discovering the universe expansion rate.^1,2,3 The Nobel prize discovered in 2011, and the prima facie indication is presented based on the universe, which is used for the accelerated growth.⁴ This bizarre and unexpected performance was supposed due to a density: a mysterious energy named as dark energy.⁵ This energy works as a driving force using the cosmic form of the acceleration, which is known as one of the most stimulating systems in cosmology. It keeps an enormous negative value of the pressure $p=-\rho$ as it represents the invalidating of the gravitational communication, which produces the cosmic growth rate.⁶ The current observational statics is accrued from diverse sources that support the convincing evidence in the favor of late-time acceleration.^7,8,9

The simplest form to justify the dark energy effects is supposed as a cosmological positive constant value that approves the negative value of the pressure $p=-\rho$. According to the general relativity, the typical cosmological system is recognized as a Big-Bang based on the start of the universe.^10,11,12 Price rises, an early segment of the accelerated growth just after the Big-Bang, were suggested in 1980 for the cosmological system model governed with typical Big-Bang networks.^13,14,15,16 A hypothetical dark energy is caused through the late accelerated growth occurred as a portion of the standard system in the last century. An extensively recognized parameterization of the typical Big-Bang cosmology is a model of six-parameter, which shows the best data fitness using the current annotations, nevertheless, with some tests recently faced.¹⁷

The general theory of classical relativity cannot be generalized in time. It gives a report of classical cosmology and labels the matter-dominated eras and radiation evolution. Alternatively, the inflationary point is planned to happen in the universe preceding to the domination of the radiation epoch, which is used in the quantum account of cosmology.^18,19,20 The cosmological accelerated growth is estimated to establish the experimental indication.

The mathematical systems have been proposed in various investigations for solving the nonlinear and stiff problems. Few well-known mathematical applications of the SIR epidemic systems with same death as well as birth rates,²¹ stochastic computing schemes have been suggested for the HIV system,²² atherosclerosis model,²³ parabolic form of the nonlinear dynamical behavior of wave system,²⁴ nonlinear model based on dengue fever,²⁵ atherogenesis: Atheroprotective mathematical system,²⁶ fractional SITRS discrete dynamical system,²⁷ modeling of a phase-field of fracture spread in poroelastic structure,²⁸ dynamic analysis of porous functionally categorized beam through the sinusoidal theory of shear deformation,²⁹ the immune behavior in immunogenetic tumor cells using the fractional order (FO) non-singular derivative³⁰ and few more investigations have been cited in Refs. 31–38.

This work aims to present a design of mathematical model with Liouville–Caputo FO for accelerating the universe in the modified gravity (AUMG), i.e. FO-AUMG is presented to get more accurate solutions. The approximate numerical outcomes of the nonlinear dynamics of FO-AUMG are portrayed with the help of stochastic Levenberg–Marquardt backpropagated (LMB) scheme-based neural networks.

The other/remaining sections of this paper are given as follows: Section 2 for the design of FO-AUMG, the detailed procedures are shown in Sec. 3. The LMB scheme-based neural network is shown in Sec. 4. The simulations of the FO-AUMG are narrated in Sec. 5, while concluding inferences are shown in the final section.

2. Mathematical Design of Nonlinear FO-AUMG

The design of FO-AUMG along with the dynamics of five classes is portrayed here. The mathematical structure of the nonlinear mathematical AUMG with corresponding initial conditions (ICs) is shown as³⁹

where $l_1$, $l_2$, $l_3$, $l_4$, and $l_5$ indicate the ICs of the nonlinear mathematical AUMG model given in system (1), whereas $y_1$, $y_2$, $y_3$, $y_4$, and $y_5$ represent the model-dependent values. The mathematical nonlinear AUMG system has been numerically solved by the artificial intelligence (AI) methodologies-based LMB neural networks. The structure of the FO-AUMG model is signified as where $\alpha$ represents the FO of the FO-AUMG model and the fractional derivative is a type of Liouville–Caputo of order $\alpha =(0,1]$ defined in Ref. 31where ${}_0^C{D}_t^{\alpha }$ represents the Liouville–Caputo fractional derivative.

3. Novel Stochastic Solver Features

This section shows the numerical treatment of the nonlinear FO-AUMG system by using the process of LMB neural networks. The computing stochastic performances-based global and local operators are recently applied/executed for the solution of broad nonlinear, complex, and stiff differential models.^40,41,42 In recent decades, these submissions are effectively applied to solve the nonlinear Lane–Emden model,⁴³ singular fractional-order differential equations,^44,45,46 nonlinear functional differential models,⁴⁷ differential form of the periodic models⁴⁸ and differential delayed models.⁴⁹

There are various submissions associated with the FO derivatives, which represent the system conditions. Some FO derivatives indicate the real-world applications that are reported in Refs. 50 and 51. The novel structures of the stochastic designed LMB neural networks for the nonlinear FO-AUMG system are selected for the following salient features:

•	A construction of the AUMG mathematical model involving fractional-order terms is presented.
•	The computing numerical-based stochastic measures have not been proposed before to solve the nonlinear FO-AUMG mathematical model.
•	The numerical representation through the stochastic computing solvers is successfully investigated using the nonlinear FO-AUMG mathematical model.
•	The contruction of the LMB neual network using the AI procedures is available for the nonlinear fractional mathematical model.
•	Variants of the nonlinear FO-AUMG mathematical system are presented numerically to authenticate the constancy based on the LMB neural networks.
•	The result comparisons based on the obtained results through the LMB neural networks and the reference (Adams numerical scheme) results signify the accuracy viably.
•	The convergence and accuracy of the LMB neural networks are performed through the absolute error (AE), which is proficient in good order for the nonlinear FO-AUMG system.
•	The correlation, regression index, STs parameters and MSE measures support the dependability and reliability of LMB neural networks for the FO-AUMG mathematical model.

4. Proposed Method: LMB Neural Networks

This section of the study shows the LMB neural networks for the nonlinear FO-AUMG mathematical model. The design of the proposed LMB neural networks is categorized into two segments. First, the designed performances of the LMB neural networks for the nonlinear FO-AUMG mathematical model are provided. While the executions of LMB neural networks are provided in the second section. Figure 1 illustrates the procedures of multi-layer through the optimization performances based on the stochastic LMB neural networks for the nonlinear FO-AUMG mathematical model. In the first part of Fig. 1, the mathematical model based on the FO-AUMG is presented, a layer construction based on the proposed scheme is provided in the second part of the Fig. 1, while the obtained results are depicted in the last half of Fig. 1. The statistical LMB neural network performances are derived by using the “nftool” command (a built-in procedure in Matlab. The statics for the FO-AUMS is used as 72%, 16% and 12% for training, authorization, and testing.

5. Numerical Results Through the LMB Neural Networks Scheme

In this section, the nonlinear FO-AUMG mathematical model along with its five dynamics is presented. The proposed LMB neural network scheme is exploited to evaluate the numerical outcomes of the nonlinear FO-AUMG mathematical system. The mathematical expression for each variant is indicated as

Case 1: Suppose a nonlinear FO-AUMG model by using $\alpha =0.5$, $m=10$, $l_1=0.1$, $l_2=0.15$, $l_3=0.2$, $l_4=0.25$ and $l_5=0.3$ is given as

Case 2: Consider a nonlinear FO-AUMG model by using $\alpha =0.7$, $m=10$, $l_1=0.1$, $l_2=0.15$, $l_3=0.2$, $l_4=0.25$ and $l_5=0.3$ given as

Case 3: Consider a nonlinear FO-AUMG model by using $\alpha =0.9$, $m=10$, $l_1=0.1$, $l_2=0.15$, $l_3=0.2$, $l_4=0.25$ and $l_5=0.3$ given as

The numerical simulations/experimentations for the nonlinear FO-AUMG system are accomplished through the stochastic LMB neural networks with 20 numbers of neurons. The statics for the FO-AUMS is used for the nonlinear FO-AUMG, which is proposed as 72%, 16% and 12% for training, authorization, and testing. The neuron-based hidden layers, output, and input layers are shown in Fig. 2.

The graphic illustrations are derived in Figs. 3–5 for the nonlinear FO-AUMG system by engaging the LMB neural networks. The illustrations of Figs. 3 and 4 are provided to validate the exhibitions and STs parameters, respectively. The information for MSE and EHs for training, best curves and authentication have been demonstrated in Fig. 4 to solve the nonlinear FO-AUMG system by applying the LMB neural network measures. The best performances on validation data for the nonlinear FO-AUMG mathematical model are illustrated/achieved at epochs 241, 279 and 317, with respective levels of $1.000\times 10^{-08}$, $5.6966\times 10^{-10}$ and $2.2493\times 10^{-10}$. Presentations of the gradient index are graphically portrayed in Fig. 3 for the nonlinear FO-AUMG mathematical system by applying the LMB neural networks. The gradient observations have been indicated as $7.431\times 10^{-06}$, $5.8626\times 10^{-07}$ and $9.9481\times 10^{-08}$ for respective cases 1–3. These graphs are presented to check the convergence of the LMB neural network for the nonlinear FO-AUMG mathematical model. The first part of Fig. 4 indicates the fitting curves for the nonlinear FO-AUMG mathematical system. These graphs also designate the similarity between reference and proposed results. The second portion of Fig. 4 signifies the EH illustrations. The performances indicate the calculated values as $1.71\times 10^{-04}$, $3.23\times 10^{-05}$ and $6.29\times 10^{-06}$ for 1, 2 and 3 cases of the nonlinear FO-AUMG mathematical system. The correlation performances are authenticated with regression curves in Fig. 5 for the nonlinear FO-AUMG system by applying the LMB neural networks. It is easy in understanding that the correlation is found as 1 that shows the perfect system. The training, authentication and testing, procedures indicate the precision of the nonlinear FO-AUMG mathematical model by applying the LMB neural networks. The MSE level for convergence achieved, complexity, substantiation and generations is demonstrated in Table 1 for the nonlinear FO-AUMG model by applying the LMB neural networks.

Figures 6 and 7 represent the assessment of the precision in the outcomes and comparison performance-based AE is provided in these plots. The numerical results substantiated the efficacy for the nonlinear FO-AUMG system by using the stochastic LMB neural network measures. The simulations based on the approximated and reference outcomes are given in Fig. 6, which indicate the close matching of the obtained and reference solutions. These precisely overlying solutions validate the precision of the stochastic LMB neural networks for the nonlinear FO-AUMG. The AE of each category of the nonlinear FO-AUMG is provided in Fig. 7. The AE of the first category $y_1$ is found as 10${}^{-04}$ to 10${}^{-05}$, 10${}^{-05}$ to 10${}^{-06}$ and 10${}^{-05}$ to 10${}^{-07}$ for 1, 2 and 3 cases. The AE of the first category $y_2$ is found as 10${}^{-03}$ to 10${}^{-05}$, 10${}^{-04}$ to 10${}^{-05}$ and 10${}^{-05}$ to 10${}^{-06}$ for 1, 2 and 3 cases. The AE of the first category $y_3$ is found as 10${}^{-04}$ to 10${}^{-05}$, 10${}^{-04}$ to 10${}^{-06}$ and 10${}^{-05}$ to 10${}^{-07}$ for 1, 2 and 3 cases. The AE of the first category $y_4$ is found as 10${}^{-05}$ to 10${}^{-06}$, 10${}^{-05}$ to 10${}^{-07}$ and 10${}^{-06}$ to 10${}^{-08}$ for 1, 2 and 3. The AE of the first category $y_5$ is found as 10${}^{-05}$ to 10${}^{-06}$, 10${}^{-05}$ to 10${}^{-07}$ and 10${}^{-06}$ to 10${}^{-08}$ for 1, 2 and 3 cases. The magnitude of AE representations signifies the perfectionism of LMB neural networks for solving the nonlinear FO-AUMG mathematical model.

6. Conclusion

This work represents the exhaustive simulations of the Liouville–Caputo fractional mathematical system governing the accelerating universe in the modified gravity. The fractional dynamics have been executed to find more reliable solutions of the nonlinear system. The nonlinear dynamics of the fractional-order accelerating universe in the modified gravity are classified into five dynamics. The stochastic simulations studies of fractional-order accelerating universe in the modified gravity system have never been implemented before by using the AI-based solvers. The statics for the FO-AUMS is applied for the nonlinear fractional-order accelerating universe with the modified gravity mathematical model as 72%, 16% and 12% for training, authorization, and testing along with the 20 numbers of neurons. The comparison of three different cases based on the fractional-order values of the nonlinear FO-AUMS is performed with dataset Adams method. To validate the exactness, and competence of LMB scheme-based neural networks, the obtained results of the state transitions, regression, correlation, error histograms and MSE have been exploited. The function fitness has been observed by taking different input and output values for each model’s case. The regression is obtained as one for each case of the model that shows the perfect model, while the negligible AE and MSE values have been reported for each case of the model. The LMBA neural network performance is stated in terms of constancy and dependability for solving the fractional-order nonlinear system.

In future, the LMB scheme-based neural network models should be implemented to describe the numerical trials for different still nonlinear models.^{52,53,54,55,56,57,58,59}

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through small group research project under grant number (RGP1/216/44).

ORCID

A. A. Alderremy   https://orcid.org/0000-0002-7401-0522

J. F. Gómez-Aguilar   https://orcid.org/0000-0001-9403-3767

Zulqurnain Sabir   https://orcid.org/0000-0001-7466-6233

Muhammad Asif Zahoor Raja   https://orcid.org/0000-0001-9953-822X

Shaban Aly   https://orcid.org/0000-0002-8286-8123