Fractals

Currently, the world is facing a devastating pandemic of a novel coronavirus, which started as an outbreak of pneumonia of unknown cause in Wuhan city of China in December of 2019. According to the recent report of the World Health Organization (WHO), 210 countries convicted badly 1.8 million infections and almost 200,000 causalities. Due to the non-availability of the vaccination, delay strategies such as community distancing, travel restrictions, extension in breaks, use of face-mask, and self-quarantine are the effective treatments to control the pandemic of coronavirus. So, we proposed the delayed susceptible-exposed-infected-recovered model with a nonlinear incidence rate to study the effective role of control strategies. For this analysis, we discussed three types of equilibria of the model such as trivial, coronavirus free, and coronavirus existence with delay terms. The local and global stabilities are investigated by using well-posed notations like the Lasalle invariance principle, Routh–Hurwitz criterion, and Lyapunov function. In the end, some useful replications are presented.

Among other hospitalization causes and cases, the clinical emergency is a critical case and the data of the reporting patients are biased as well as poorly managed due to the chaotic situation. The world has faced chaos over the past year due to the frequent waves of COVID-19 and the resulting emergencies. The data banks, linked with the clinical emergencies require serious quantitative and qualitative analysis to drive interpretable conclusions for necessary future emergency measures and to develop explainable artificial intelligence tools. This important procedure involves the clear understanding of the data patterns and topologies, which is a great challenge for the multidimensional data sets. Mathematically, the topological mapping can resolve this problem by mapping higher-dimensional data to two-dimensional representation, based on the overall association. Proper data mining and pattern recognition can help in improving the rapid patients admission, in providing the medical resources timely and in proper patient administration. In this paper, the importance of self-organizing maps, to interpret the hospital data, particularly for the COVID-19 epidemic is discussed in detail. Important variables are identified with the aid of networks and mappings.

In this paper, a numerical algorithm is presented to obtain approximate solution of distributed order integro-differential equations. The approximate solution is expressed in the form of a polynomial with unknown coefficients and in place of differential and integral operators, we make use of matrices that we deduce from the shifted Legendre polynomials. To compute the numerical values of the polynomial coefficients, we set up a system of equations that tallies with the number of unknowns, we achieve this goal through the Legendre–Gauss quadrature formula and the collocation technique. The theoretical aspects of the error bound are discussed. Illustrative examples are included to demonstrate the validity and applicability of the method.

The aim of this study is to consider solving an important mathematical model of fractional order ($2+1$)-dimensional breaking soliton (Calogero) equation by Khater method. The derivatives are in the local fractional derivative sense. The fractional transformation equation is utilized to convert the proposed nonlinear fractional order differential equation into nonlinear ordinary differential equation. The Khater method is used to construct the closed-form traveling wave solutions of the said fractional differential equation. In addition, many new exact solutions are constructed. This shows that the Khater method is more convenient, powerful, and easy to solve the nonlinear fractional differential equation arising in mathematical physics.

Higher-order correlations measured in lead collisions examined the geometrical strength as well as the coherence–chaotic characteristics of the particles creation region. The three- and four-particle quantum correlations can delineate the system of pions at small relative momentum region and observe a significant innovative penchant about the granular sources with various parameters. Such conglomerate of correlation functions and their corresponding genuine correlations are rarely analyzed at higher-order quantum computation via invariant momenta to search the granular structure of pion emanation sources. We evince and examine the pions excretion from a dispersed collection of fluid droplets with coherence snippets that produce quantum interferences to explore the nature of the fluid and the characteristics of the chaos fraction in collisions of heavy nuclei. The cumulant correlations and the normalized correlators with their intercepts contained the existence of amalgamation facet of hadronic particles, and the partons plasma. Specifically, we analyze such hybrid fluid with various correlations functions via droplet numbers to probe the trifle of chaos–coherence fraction of pions condensate by using the source with spherical structure droplets.

This paper discussed a new operator known as a fractal-fractional (FF), considered in the Caputo sense. We have investigated the fractional mathematical model of Tuberculosis (TB) disease under FF Caputo derivative. We have provided the existence and uniqueness for the appropriate system by using Banach and Leray–Schauder theorem and for the stability of the model, we have used the Ulam–Hyers approach. Applying the methods of basic theorems of FF calculus (FFC) and the iterative numerical techniques of fractional Adams–Bashforth method for approximate solution. For the simulation of the model, we have considered different values for fractional order $\alpha$ and fractal dimension $\beta$ and compared the results with integer order for real data. The FFC technique is applied as a beneficial technique to know about the real-world problem and also to control the whole world situation of the aforesaid pandemic in the different continents and territories of the world. This new operator, FF in the form of Caputo derivative, gives better results than ordinary integer order. Several results have been discussed by taking the different fractal dimensions and arbitrary order.

Mathematical epidemiology holds prime importance for comprehending the dynamics of infectious diseases. Consequently, mathematical model of hepatitis B with fractional-order derivative under Caputo sense is primarily focused in this research. The analysis of the required solution is qualitatively derived by applying the fixed-point theory approach. By perturbing the proposed model, the Ulam–Hyer’s stability techniques are further derived. To achieve the iterative series solution of the proposed system of hepatitis, the modified Euler method like Taylor’s series method is utilized. For validation and importance of the fractional operators, sufficient significant numerical results at various fractional orders are presented and compared them with the integer order. It is inferred from this research that, by using the fractional-order method, the transmission mechanism of hepatitis B disease can be acutely revealed. This study may provide positive theoretical support for the prevention and treatment of hepatitis B disease.

Smoking can cause numerous diseases like lung cancer, mouth diseases, bronchitis, stroke, heart diseases, etc. To overcome such and other related diseases, we need to avoid the smoking habit from the population at large scale. The key purpose of this work is to discuss and analyze different kinds of smoking phases which are very common in our society. The model is formulated in the form of a system of ordinary differential equations keeping in view the existing literature. The model is investigated for possible equilibria besides positivity and boundedness of solution. The model exhibits six equilibrium points and the stability analysis of each fixed point is investigated. The local stability analysis of each equilibrium point is carried out with the help of linearization approach and it is shown that these equilibria are locally asymptotically stable under certain conditions. We also checked the fixed points for global analysis with the help of Lyapunov stability theory and stability criteria were obtain therein. We also perform the local sensitivity analysis. To verify the theoretical results, a scheme is developed using non-standard finite difference method. Simulations suggest that the obtained graphical results support our analytical findings with a high degree of accuracy.

Dealing with nonsingular kernels is not an easy task due to their restrictions at origin. In this short paper, we suggest an extension of the fractional operator involving the Mittag-Leffler kernel which admits integrable singular kernel at the origin. New solutions of the related differential equations were reported together with some perspectives from the modelling viewpoint.

In this paper, the dynamical behavior of Middle East respiration syndrome coronavirus (MERS-CoV) via a sense of Caputo fractal-fractional order system of differential equation is established. A novel approach of fractional operator known as fractal-fractional Riemann–Liouville derivative is applied to the model considered. Moreover, fractal-fractional derivative is applied to the said problem. Furthermore, the existence and uniqueness of the solution of the considered model are verified by the fixed-point theory approach. The local and global stability of developed system are investigated with the help of the Ulam–Hyers stability technique from nonlinear functional analysis. The fractal-fractional type Adams–Bashforth iterative method is used to establish the numerical solution of said problem. The proposed model is simulated by considering different fractal dimensions $(𝜃)$ and fractional order $(\delta )$, converging to the integer order. Hence, it is evident that all the compartmental quantities possess convergence and stability in fractal-fractional form. Moreover, the Fractal-fractional techniques may also be used as a powerful technique/tool to investigate the global dynamics of the disease. However, it can be concluded from the results that quick recovery is possible for human population in the pandemic if there is no animal interaction with humans. In future, we are planning to extend the reported analysis to other fractional (fractal-fractional) operators. Furthermore, the behavior of different infectious models can also be analyzed with the help of newly developed scheme at different fractional orders lying between 0 and 1.

This paper studies the well-known mathematical model’s analytical wave solutions (modified Benjamin–Bona–Mahony (BBM) equation), which demonstrates the propagation of long waves in the nonlinear dispersive media in a visual illusion. Six recent analytical and semi-analytical schemes (extended simplest equation (ESE) method, modified Kudryashov (MKud) method, sech–tanh expansion method, Adomian decomposition (ADD) method, El Kalla (EK) expansion method, variational iteration (VI) method) are applied to the considered model for constructing abundant analytical and semi-analytical novel solutions. This variety of solutions aims to investigate the analytical techniques’ accuracy by calculating the absolute error between analytical and semi-analytical solutions that shows the matching between them. The analytical results are sketched through two-dimensional (2D), three-dimensional (3D), contour plot, spherical plot and polar plot. The stability characterization of the analytical solutions is investigated through the Hamiltonian system’s features. The originality and novelty of this paper are discussed, along with previously published papers.

An age-structured susceptible–exposed–infected–recovered–susceptible (SEIRS) endemic model is proposed in this analysis utilizing the tools of partial differential equations. Because of different outflows and inflows that are lopsided by migration and demographics factors, the population is supposed to be not constant. To demonstrate that the model is well-posed, an abstract Cauchy problem is developed from the proposed system. The simple reproduction number $R_0$ is used to analyze the local and global behavior of the disease-free equilibrium. The disease present equilibrium point is shown to exist and be stable locally under appropriate assumptions and conditions. We consider the age-free parameters and the problem is converted into an ordinary differential equations (ODEs) model. The ODEs model is investigated for disease-free and endemic equilibria and the global stability of each equilibrium is presented therein. A few simulations are carried out and discussed at the end of the paper to explain the central theorem of the study.

In this paper, some new fractal versions of Fejér–Hermite–Hadamard (FHH) type variants for generalized Raina $(\mathrm{\Psi },\hslash )$-convex mappings are established benefiting from Raina’s function and fractal set ${ℝ}^{\phi },0<\phi <1$. By means of three integral identities coupled with Raina’s function and local differentiation, we established some bounds for the difference between the left and central parts and also the difference between the center and right parts in FHH inequality. Besides that, some illustrative examples and noted special cases are apprehended. Additionally, we developed various generalizations for random variables, cumulative distribution functions, and special function theory as applications of local fractional integrals. The consequences established can provide contribution to inequality theory, fractional calculus and probability theory from the viewpoint of application to establish the other associated classes of functions. With the aid of these methodologies, it is promising to comprise further bounds of other type of variants which involve local fractional techniques.

The purpose of this paper is to explore novel variants for convex functions via discrete ${𝒜}{ℬ}$-fractional operator in the frame of time scale calculus $\hslash \mathbb{Z}$ with $0<\alpha <1$ and $0<\hslash \le 1.$ Based on a comparison with the integral inequalities, we have provided new discrete fractional inequalities having $\hslash$-discrete generalized Mittag-Leffler function in the kernel which generates several known results and can be utilized as handy tools in the investigation of qualitative and quantitative properties of solutions of certain classes of difference equations. Several new special cases are also apprehended in the setting of time scale $\mathbb{Z}$ and $𝕋.$ As the application aspect, we provide an illustrated example to show the effectiveness of our new criteria in fractional $\hslash$-difference type initial value problem.

In this paper, we have taken into consideration the most dangerous form of leishmaniasis known as visceral leishmaniasis, which has the highest fatality rate and is also famous with the name of kala-azar. The analysis is carried out using Atangana–Baleanu–Caputo (ABC) operator with a convex incidence rate. Using fixed point theorems the existence and uniqueness for the solutions of the fractional leishmania epidemic model have been established. All the basic properties of the given model are studied along with stability analysis. Newton polynomial and Adams–Bashforth methods are taken into account to find the numerical solution.

One of the most severe and troubling diseases these days is COVID-$19$ pandemic. The COVID-$19$ pandemic’s dangerous effects are extremely rapid, and infection normally results in death within a few weeks. As a consequence, it is important to delve deeper into the complexities of this elusive virus. In this study, we propose a Caputo-based model for increasing COVID-19 strains. The memory effect and hereditary properties of the fractional variant for the model enable us to fully comprehend the dynamics of the model’s features. The existence of unique solution using the fixed-point theorem and Arzelá–Ascoli principle as well as the stability analysis of the model by means of Ulam–Hyer stability (UHS) and generalized Ulam–Hyer stability (GUHS) have been discussed. Furthermore, the parameters of the model are estimated using $3$ months data points chosen from Nigeria using the nonlinear least-squares technique. The best-suited parameters and the optimized Caputo fractional-order parameter $\alpha$ are obtained by running simulations for both models. The proposed model is shown to comprehend the dynamical behavior of the virus better than the integer-order version. In addition, to shed more light on the model’s characteristics, various numerical simulations are performed using an efficient numerical scheme.

Detection of relationship between two time series is so important in different scientific fields. Most common techniques are usually sensitive to stationarity or normality assumptions. In this research, a new copula-based method (cyclocopula) is introduced to detect the relationship between two cylostationary time series with fractional Brownian motion (fBm) errors. The performance of the proposed method is studied by employing numerous simulated datasets. The applicability of the introduced approach is also investigated in real-world problems. The numerical and applied studies verify the performance of the introduced technique.

The new extended direct algebraic technique is used to attain the new exact solitary wave solutions of the nonlinear fractional couple Drinfeld–Sokolov–Wilson system (DSWS). This DSWS has a lot of applications in water waves, fluid mechanics, quantum mechanics, chemistry, and biology. This system is developed in the perspective of fractional theory of calculus, for this purpose Riemann–Liouville fractional derivative (RLFD), $\beta$-fractional differential operator and Atangana–Baleanu fractional derivative (ABFD) are utilized. The attained solutions contain some new families such as trigonometric functions, hyperbolic functions, rational functions, logarithmic functions, and exponential functions. The graphically analysis and comparison of solutions are plotted by taking appropriate values of the constant parameters to enhance the understanding of physical real-life phenomena. The approach is said to be compatible and effective as a scientific approach for the examination of the various spacetime fractional models in engineering and physics concerning real-life issues.

In 2017, Atangana proposed more generalized operators depending on two parameters: one is fractional order (FO) and other is fractal dimension. The novel operators are defined with three different kernels. These operators produced excellent dynamics of the chaotic systems. In this paper, the Caputo fractal-fractional operator is used to explore a chaotic system which contains only one signum function. The existence theory is developed by using the fixed-point result of Leray–Schauder to prove that the considered chaotic system possesses at least one solution. The proposed chaotic system has a unique solution, according to Banach’s fixed-point theorem. We demonstrate that the suggested chaotic system is Ulam–Hyres (UH) stable under the novel operator of power law kernel by employing nonlinear functional analysis. The Adams–Bashforth technique is used to evaluate the numerical outcomes of the considered model. We show the complex structure of numerical solutions for different FO and fractal dimension values.

This paper investigates the newly constructed fractional order (FO) smoking model with relapse and harmonic mean type incidence rate under the fractional operator ${𝒜}{ℬ}{𝒞}$. Furthermore, for the model considered, the existence results and uniqueness of the solution are established with the help of the fixed point theory approach. The stability of the model is proven with the help of Ulam–Hyres approach by considering small perturbation in the problem. The iterative FO Adams–Bashforth technique is considered for obtaining the numerical solution. Moreover, the established scheme is simulated against the available data for some smoking communities where various types of potential, moderate and quit smokers are living. However, different fractional orders are used for simulation and it is evident through the results that FO converges to integer order as order is increased. Moreover, smoking may also be controlled through education, employment and the awareness of its harmfulness by doctors and their dosage of some drugs.

This paper investigates the solitons and breathers wave solutions of the generalized nonlinear Schrödinger equation (NLGSE) of third order through new generalized computational (generalized Khater (GK)) technique. These solutions are employed to construct the initial and boundary conditions to apply a new numerical scheme known by the trigonometric Quintic spline. This well-known mathematical model in ultra-short pulses in the engineering and applied sciences, especially optical fibers, where the obtained solutions can be used in distinct applications. The NLGSE is the basis of the mechanical wave model, which helps in studying the structure of the atom, and as the name suggests, it shows all the wave-like properties of matter. Also, it is considered a state function of a quantum-mechanical system that describes the atoms’ dynamical and physical behavior, or transistors where it determines the atomic particle at a specific position and time. In fact, this approach is sometimes regarded as a platform for quantum aspects. The accuracy of our solutions is checked by evaluating the absolute error value between functional and numerical solutions. In addition, the physical representations of the model under consideration are represented in some of the solutions obtained graphically. The consequences of empirical and computational methods are confirmed by the accuracy and uniqueness of their tests.

This research paper investigates the numerical solutions of the nonlinear fractional Ostrovsky equation through five recent numerical schemes (Adomian decomposition (AD), El Kalla (EK), Cubic B-Spline (CBS), extended Cubic B-Spline (ECBS), exponential Cubic B-Spline (ExCBS) schemes). We investigate the obtained computational solutions via the generalized Jacobi elliptical functional (JEF) and modified Khater (MK) methods. This model is considered as a mathematical modification model of the Korteweg–de Vries (KdV) equation with respect to the effects of background rotation. The solitary solutions of the well-known mathematical model (KdV equation) usually decay and are replaced by radiating inertia gravity waves. The obtained solitary solutions show the localized wave packet as a persistent and dominant feature. The accuracy of the obtained numerical solutions is investigated by calculating the absolute error between the exact and numerical solutions. Many sketches are given to illustrate the matching between the exact and numerical solutions.

In this paper, a conformable fractional time derivative of order $\alpha \in (0,1]$ is considered in view of the Lax-pair of nonlinear operators to derive a fractional nonlinear evolution system of partial differential equations, called the Fractional-Six-Wave-Interaction-Equations, which is derived in terms of one temporal plus one and two spatial dimensions. Further, an ansatz consisting of linear combinations of hyperbolic functions with complex coefficients is utilized to obtain an infinite set of exact soliton solutions for this system. Certain numerical examples are introduced to show the effectiveness of the ansatz method in obtaining exact solutions for similar systems of nonlinear evolution equations.

This paper deals with the fractional mathematical model for the spread of polio in a community with variable size structure including the role of vaccination. The considered model has been extended with help of Atangana–Baleanu in the sense of the Caputo (ABC) fractional operator. The positivity and boundedness of solution (positively invariant region) are presented for the ABC-fractional model of polio. The fixed-point theory has been adopted to study the existing results and uniqueness of the solution for the concerned problem. We also investigate the stability result for the considered model using the Ulam–Hyers stability scheme by taking a small perturbation in the beginning. Numerical simulation is obtained with the help of the fractional Adams–Bashforth technique. Two different initial approximations for all the compartments have been tested for achieving stability to their same equilibrium points. The control simulation is also drawn at fixed infection and exposure rates at various fractional orders. The comparison at different available rates of infection and exposition is also plotted to show the decrease in the infection by decreasing these rates. Various graphical presentations are given to understand the dynamics of the model at various fractional orders.

In this study, a synchronization problem for spatio-temporal partial differential systems is addressed and researched within a subjectivist framework. In light of Lyapunov direct method and some proposed nonlinear controllers, a new scheme is established to accomplish a full synchronization between two reaction–diffusion systems of integer- and fractional-order. In particular, a novel vector-valued control law is analytically derived to attain the desired synchronization between two chemical models, namely, the Lengyel–Epstein and Gray–Scott models. To validate the obtained theoretical results, further numerical simulations are carried out in 2D and 3D configurations.

Mathematical modeling can be utilized to find out how the coronavirus spreads within a population. Hence, considering models that can precisely describe natural phenomena is of crucial necessity. Besides, although one of the most significant benefits of mathematical modeling is designing optimal policies for battling the disease, there are a few studies that employ this beneficial aspect. To this end, this study aims to design optimal management policies for the novel coronavirus disease 2019 (COVID-19). This is a pioneering research that designs optimal policies based on multi-objective evolutionary algorithms for control of the fractional-order model of the COVID-19 outbreak. First, a fractional-order model of the disease dynamic is presented. The impacts of the fractional derivative’s value on the modeling and forecasting of the disease spread are considered. After that, a multi-objective optimization problem is proposed by considering the rate of communication, the transition of symptomatic infected class to the quarantined one, and the release of quarantined uninfected individuals. Numerical results clearly corroborate that by solving the proposed multi-objective problem, governments can control the massive disease outbreak while economic factors have reasonable values that prevent economic collapse.

In this study, a novel heuristic computing technique is presented to solve bioinformatics problem for the corneal shape model of eye surgery using Morlet wavelet artificial neural network optimized by the global search schemes, i.e. genetic algorithm (GA), local search technique, i.e. sequential quadratic programming (SQP) and the hybrid of GA-SQP. To measure the performance of the design network configuration, different cases based on nonlinear second-order differential equations governing the corneal model have been solved effectively. The numerical procedure of Adams method is implemented for the comparison purpose of the presented outcomes of the stochastic solver, which shows the worth of the present scheme based on accuracy and convergence with negligible values of absolute error in the range 10${}^{-7}$ to 10${}^{-8}$. Furthermore, statistical measures are presented based on “mean absolute error”, “root mean square error” and “coefficient of Theil’s inequality” which additionally endorsed consistently accurate performance of integrated intelligent computing framework for solving the corneal shape model.

This study focuses on the prevalence of COVID-19 disease along with vaccination in the United States. We have considered the daily total infected cases of COVID-19 with total vaccinated cases as exogenous input and modeled them using light/heavy tailed auto-regressive with exogenous input model based on the innovations that belong to the flexible class of the two-piece scale mixtures of normal (TP–SMN) family. We have shown that the prediction of COVID-19 spread is affected by the rate of vaccine injection. In fact, the presence of exogenous input variables in time series models not only increases the accuracy of modeling, but also causes better and closer approximations in some issues including predictions. An Expectation-Maximization (EM) type algorithm has been considered for finding the maximum likelihood (ML) estimations of the model parameters, and modeling as well as predicting the infected numbers of COVID-19 in the presence of the vaccinated cases in the US.

In this paper, we solve a nonlinear fractional-order model for analyzing the dynamical behavior of vector-borne diseases within the frame of Caputo-fractional derivative. The proposed mathematical model advances the existing integer-order model on transmission and cure of vector-borne diseases. The existence and uniqueness of the solutions of the fractional-order model are proved using the Banach contraction principle. We investigate the local asymptomatic stability for the obtained disease-free equilibrium point and global stability for the proposed model in the sense of Ulam–Hyers stability criteria, respectively. Besides that, we obtain a numerical solution for the projected model using the Corrector-Predictor algorithm. Finally, to illustrate the obtained theoretical results, we perform numerical simulations for different values of fractional-order derivative and make a comparison with the results of the integer-order derivative.

In this work, we study the norovirus (NoV) epidemic model with random perturbations and a time delay. First of all, the existence and uniqueness of the global positive solution are obtained. Then, we derive sufficient conditions for the extinction of the disease. Moreover, by establishing appropriate Lyapunov function, the existence of a stationary distribution is discussed. Some numerical simulations are given to illustrate our analytical results.

In the current research community, certain new fractional derivative ideas have been successfully applied to examine several sorts of mathematical models. The fractal fractional derivative is a novel concept that has been proposed in recent years. In the presence of heat generation, however, it is not employed for the free convection Couttee flow of the Casson fluid model. The core interest of the present analysis is to examine the Casson fluid under the influence of heat generation and magnetic field. The flow of the Casson fluid has been considered in between two vertical parallel plates. The distance between the plates is taken as $l$. The linear coupled governing equation has been developed in terms of classical PDEs and then generalized by employing the operator of the fractal-fractional derivative with an exponential kernel. The numerical solution of the proposed problem has been found employing the finite-difference technique presented by Crank–Nicolson. The Crank–Nicolson finite difference scheme has the advantage of being unconditionally stable and can be applied directly to the PDEs without any transformation to ODEs. This technique in sense of exponential memory has been revealed to be unreported in the literature for such a proposed problem. For graphical analysis, the graphs of velocity profile and thermal field have been plotted in response to several rooted parameters. For comparative analysis, the graphs for the parameter of fractal-fractional, fractional, and classical order have also been plotted. From the analysis, it has been found that the fractal-fractional order model has a large memory effect than the fractional-order and classical model due to the fractal order parameter.

The infection of coronavirus (COVID-19) is a dangerous and life-threatening disease which spread to almost all parts of the globe. We present a mathematical model for the transmission of COVID-19 with vaccination effects. The basic properties of fractional calculus are presented for the inspection of the model. We calculate the equilibria of the model and determined the reproduction number ${{ℛ}}_0$. Local asymptotic stability conditions for the disease-free are obtained which determines the conditions to stabilize the exponential spread of the disease. The nonlinear least-square procedure is utilized to parameterize the model from actual cases reported in Pakistan. By fixed point theory, we prove the existence of a unique solution. We also present numerical results to simulate virus transmission and compare the results with those of the Caputo derivative. We study the solution pathways of the COVID-19 system to provide effective control policies for the infection. Significant changes have been noticed by lowering the order of fractional derivative.

This paper studies the optical soliton wave solutions of the fractional complex paraxial wave dynamical (FPWD) model with Kerr media. This model investigates the frequency of the beam obeying that reveals non-dispersive and non-diffractive Spatio-temporal localized wave envelopes promulgating in optical Kerr media. This model is equivalently the well-known nonlinear Schrödinger equation. These two models concentrate on the nonlinear visual medium to increase its strength and increase the nonlinear optical operation’s efficiency. Three recent computational schemes are employed to investigate the considered model’s abundant wave solutions; these solutions are then used to evaluate the initial and boundary conditions. The solutions’ accuracy is investigated by calculating the absolute error between solutions exact and approximate solutions. Additionally, the solution’s stability is checked by employing Hamiltonian system properties. The results’ physical interpretation is explained through some different sketches in 2D, 3D, and density plots.

Measles is a highly transmissible disease in children around the world. According to the World Health Organization (WHO), 73% of deaths of children were due to measles in 2018. This study describes the physical solution of the SIQR model for measles spread under the effect of natural delay amongst different compartments. By three different numerical techniques, the efficacies of solutions of the underlying system have been compared and a clear preference of nonstandard finite-difference (NSFD) scheme over the rest has been established. It has also been observed, on principle, that the NSFD formulation recovers all the essential traits of a continuous model namely the boundedness, positivity and stability of equilibriums of populations. The numerical results have also been supported by a very strong classical analysis of the model where the existence of a solution vector in explicit subsets of the function spaces has been guaranteed which leads to optimization of fixed-point methods.

This study mainly investigates new techniques for obtaining numerical solutions of time-fractional diffusion equations. The fractional derivative term is represented in the Lagrange operational sense. First, we describe the temporal direction of the considered model using the Legendre orthogonal polynomials. Moreover, to archive a full discretization approach a type of nonlocal method has been applied that is known as the nonlocal peridynamic differential operator (PDDO). The PDDO is based on the concept of peridynamic (PD) interactions by proposing the PD functions orthogonal to each term in the Taylor Series Expansion (TSE) of a field variable. The PDDO for numerical integration uses the vicinity of each point (referred to as the horizon, which does not need background mesh). The PDDO is exclusively described in terms of integration (summation) throughout the interaction domain. As a result, it is unsusceptible to singularities caused by discontinuities. It does, however, need the creation of PD functions at each node. We numerically investigate the stability, the convergence of the scheme, which verifies the validity of the proposed method. Numerical results show the simplicity and accuracy of the presented method.

One of the fascinating aspects of sports rivalry is that anything can happen. The significant difficulty is that computer-aided systems must address how to record and analyze many game events, and fractal AI plays an essential role in dealing with complex structures, allowing effective solutions. In table tennis, we primarily concentrate on two issues: ball tracking and trajectory prediction. Based on these two components, we can get ball parameters such as velocity and spin, perform data analysis, and even create a ping-pong robot application based on fractals. However, most existing systems rely on a traditional method based on physical analysis and a non-machine learning tracking algorithm, which can be complex and inflexible. As mentioned earlier, to overcome the problem, we proposed an automatic table tennis-aided system based on fractal AI that allows solving complex issues and high structural complexity of object tracking and trajectory prediction. For object tracking, our proposed algorithm is based on structured output Convolutional Neural Network (CNN) based on deep learning approaches and a trajectory prediction model based on Long Short-Term Memory (LSTM) and Mixture Density Networks (MDN). These models are intuitive and straightforward and can be optimized by training iteratively on a large amount of data. Moreover, we construct a table tennis auxiliary system based on these models currently in practice.

The purpose of this research is to analyze the general equations of double diffusive magneto-free convection in a rate type fluid presented in non-dimensional form, and apply to a moving heated vertical plate as in the boundary layer flow up, with existence of externally magnetic fields which are either moving or fixed consistent with the plate. Thermal transport phenomenon is discussed in the presence of constant concentration coupled with first-order chemical reaction with exponential heating. An innovative definition in power law (CF) and Mittag-Leffler (ABC) kernels form time fractional operators are implemented to hypothecate the constitutive mass, heat and momentum equations. The results based on special functions are obtained by using the technique of Laplace transformation to tackle the non-dimensional equations for velocity, mass and energy. The contribution of mass, thermal and mechanical components on the dynamics of fluid is presented and discussed independently. An interesting property regarding the behavior of the fluid velocity is found when the movement is observed in the magnetic intensity along with the plate. In that situation, the fluid velocity is not to be zero when it is far and away from the plate. Moreover, the heat transfer aspects, flow dynamics and their credence on the parameters are drawn out by graphical illustrations along with some special cases for the movement of the plate which are also studied.

In this paper, we are interested in studying the initial value problem for parabolic problem associated with the Caputo–Fabrizio derivative. We deal the problem in two cases: linear inhomogeneous case and nonlinearity source term. For the linear case, we derive the convergence result of the mild solution when the fractional order $\alpha \to 1^{-}$ under some various assumptions on the initial datum. For the nonlinear problem, we show the existence and uniqueness of the mild solution using Banach fixed point theory. We also prove the convergence result of the mild solution when the fractional order $\alpha \to 1^{-}$.

The solitary wave solutions of the $(2+1)$-dimensional nonlinear Chiral Schrödinger ($(2+1)$-D CNLS) equation by making use of the modern extended direct algebraic approach are investigated in this paper. Such a nonlinear model’s different categories of solitary waves solutions are found. The division of semi-dark solitons, singular dark-pitch solitons, single solitons of type 1 along with 2, intermixed hyperbolically, trigonometrically and rational solitons are developed and evaluated using software tools for numerical calculations. Detailed graphical interpretation of results with deliberation is also discussed. This paper effectively depicts 3D- and 2D-contour plots to provide further detail about the physical characteristics of the obtained solutions. We compare our solutions to previously reported findings along with study motivation and future directions. The fractional quantum hall effect’s edge states are defined by the discussed model.

The security of mobile wireless sensor networks has captivated extensive attention of researchers because of their wide range of applications and vulnerability to attacks caused by malware. In this paper, we investigate a delayed malware propagation model on mobile wireless sensor network incorporating nonlinear incidence rate, logistic growth rate and recovery rate. Local asymptotic stability of the endemic equilibrium and existence of Hopf bifurcation at crucial value of the time delay are analyzed. Then, properties of Hopf bifurcation are explored. Specifically, global exponential stability is investigated via linear matrix inequality. An example is presented finally to underline the effectiveness of findings in our paper numerically and graphically.

Wireless communication has become ubiquitous in almost every sector, both in military and commercial systems. Communicating objects are getting smaller and smaller. Therefore, the antennae which are the fundamental elements of this circuit must meet the requirements of their integrating environment in terms of bandwidth, multiple frequencies, smaller size and affordability. Microstrip patch antenna (MPA), a priori, should have been the preferred candidate, but the narrowness of its bandwidth is a limitation. Therefore, many techniques have been explored including the use of fractal shapes for improving the performance of this type of antenna. Multiform fractal shapes have been used in various works to achieve the results expected in specific cases to effectively meet the requirements of wireless communication. In this paper, we will do an overview by highlighting the various kinds of fractal geometries which have been the most used in recent years by clearly specifying that the options of miniaturization of microstrip antennae by the theories of Sierpinski and Minkowski are ideal methods and therefore, will drive our future work on miniaturized antenna designs.

A fast and precise prediction of stock market crashes is an important aspect of economic growth, fiscal and monetary systems because it facilitates the government in the application of suitable policies. Many works have examined the behavior of the fall of stock markets and have built models to predict them. Nevertheless, there are limitations to the available research, and the literature calls for more investigation on the topic, as currently the accuracy of the models remains low and they have only been extended for the largest economies. This study provides a comparison of quantum forecast methods and stock market declines and, therefore, a new prediction model of stock market crashes via real-time recession probabilities with the power to accurately estimate future global stock market downturn scenarios is achieved. A 104-country sample has been used, allowing the sample compositions to take into account the regional diversity of the alert warning indicators. To obtain a robust model, several alternative techniques have been employed on the sample under study, being Quantum Boltzmann Machines, which have obtained very good prediction results due to their ability to remember features and develop long-term dependencies from time series and sequential data. Our model has large policy implications for the appropriate macroeconomic policy response to downside risks, offering tools to help achieve financial stability at the international level.

This paper deals with the global dynamics of deterministic-stochastic COVID-19 mathematical model with quarantine class and incorporating a preventive vaccination. Lyapunov functions are utilized for the global stability of disease free equilibrium point and the graph theoretic method is used for the construction of Lyapunov function for positive equilibrium point. The stability of model is discussed regarding the reproductive number. Utilizing the non-standard finite difference scheme for the numerical solution of the deterministic model, the obtained results are shown graphically. Further, environmental noises are added to the model for description of stochastic model. Then we take out the existence and uniqueness of positive solution with extinction for infection. Finally, we solve numerically the stochastic model using Newton Polynomial scheme and present the results graphically.

We aim in this research to show the global behavior of each equilibrium of a heroin epidemic model with distributed delays. We obtained that the global behavior of the considered system is completely governed by the value of the basic reproduction number ${\Re }_0$. For the case when ${\Re }_0$ is less than one, we established the global stability of the drug-free equilibrium using the Lyapunov direct method. However, for ${\Re }_0$ greater than one, the addiction persists and the drug equilibrium is globally asymptotically stable, which is shown also by the Lyapunov direct method. Moreover, we obtained that by controlling the density of occasional heroin users we can control the heroin epidemic spread, and this measure is more effective than the treatment measure.

Smoking poses an enormous threat to public health in the globe and continues to be one of the main cause of health problems. In this paper, a delayed smoking model with reversion class is proposed. Local stability of the smoking equilibrium and Hopf bifurcation is analyzed by taking the time delay as the bifurcation parameter. In particular, global exponential stability is proved by using the linear matrix inequality method. The theoretical results obtained are validated numerically and graphically.

The path of the Lévy process can be considered for prices of options such as a Rainbow or Basket option on two assets which leads to a 2D Black–Scholes model. The generalized model of this type of equation can be referred to as a 2D spatial-fractional Black–Scholes equation. The analytical solution of this kind is very complex and difficult and can even be said to be unattainable. For this reason, a numerical method has been proposed to solve it via the collocation method based on the Chebyshev orthogonal basis. Moreover, based on the derivatives in the called model, we approximated the derivative operator by using this type of base. Then we first obtained the temporal discrete form and finally the full-discrete form and turned it into a system of linear equations with the help of Chebyshev base roots.

Love is said to be pure, terrible, sweet, and horrible all at the same time. Love is, in fact, a basic requirement in everyone’s life. To live a normal and healthy life, everyone requires love. Love encompasses a wide range of emotions, sentiments, and attitudes. For some, love entails more than a physical attraction; it also includes an emotional bond. However, it is commonly believed that “Mathematics is the language in which God has written the universe”, as evidenced by the transformation of every phenomenon into mathematical equations. On this basis, this study aims to express the feelings among Romeo and Juliet via mathematical tools. The love among Romeo and Juliet is shown as a coupled system of ODEs. The fractal fractional differential operator with the Mittag-Leffler function further generalizes the classical differential equations. Some theoretical analysis has been done for the considered problem. The graphical solution is obtained through a numerical scheme with the help of MATLAB software. The impact of the fractional-order parameter and fractal dimension parameter is shown on the feelings of both individuals. Furthermore, the impact of various physical parameters on the love or hate of Romeo and Juliet is displayed and discussed in detail. As a concern to the most sensitive parameter, it is observed that spending or saving money among both individuals has the ability to tend love into hate and vice versa.

In today’s sophisticated global marketplace, supply chains are complex nonlinear systems in the presence of different types of uncertainties, including supply-demand and delivery uncertainties. Though up to now, some features of these systems are studied, there are still many aspects of these systems which need more attention. This necessitates more research studies on these systems. Hence, in this study, we propose a variable-order fractional supply chain network. The dynamic of the system is investigated using the Lyapunov exponent and bifurcation diagram. It is demonstrated that a minor change in the system’s fractional-derivative may dramatically affect its behavior. Then, distributed consensus tracking of the multi-agent network is studied. To this end, a control technique based on the sliding concept and Chebyshev neural network estimator is offered. The system’s stability is demonstrated using the Lyapunov stability theorem and Barbalat’s lemma. Finally, through numerical results, the proposed controller’s excellent performance for distributed consensus tracking of multi-agent supply chain network is demonstrated.

These days the whole world is facing a serious problem of infectious maladies and how to control the endemic of these diseases. Testing correctly is one of the most important procedures in preventing the spread of infectious diseases, as incorrect testing can turn a susceptible person into an infected person. In this paper, we study the dynamics of imperfect testing and diagnostics of infectious diseases model by replacing the integer order derivative in the sense of fractional order Atangana–Baleanu operator coupled with Caputo operator. This AB-fractional operator is the generalization of classical derivative and gives more data of the factors of the nonlocal dynamical frameworks. Fixed point theorems have been used for the verification of existence results and Picard’s stability technique utilized for the stability study of a fractional order imperfect testing infectious disease (ITID) model. Finally, numerical computations are implemented for the fractional order ITID model to illustrate the results graphically.

The cable equation plays a prominent role in biological neuron models, for instance, in spiking neuron models and electrophysiology. Thus, the current investigation scrutinizes the 2D time nonlinear multi-term fractional cable equation. By adopting a valid meshfree technique, the nonlinear multi-term time-fractional cable equations (NM-TTFCEs) that consist of government equations and their boundary conditions are transformed into the boundary value problems. For this purpose, the finite difference method is derived for temporal discretization such that the considered NM-TTFCEs can be transformed into a sequence of boundary value problems in inhomogeneous Helmholtz-type equations. The dual reciprocity method (DRM) is implemented to obtain a particular solution and the improved singular boundary method (ISBM) is employed to evaluate the homogeneous solution. Moreover, we apply the meshless method for solving two-dimensional NM-TTFCEs on regular and irregular distribution points with several computational domains. The numerical results vouch for the accuracy and high efficiency of the proposed method. Finally, we will conclude that the ISBM/DRM method can be considered a potential alternative to existing meshless strong form approaches in solving multi-term fractional equation problems with complex geometries.

In this paper, a new computing technique is investigated for the cause of Mycobacterium ulcerans (MU) with the help of differential equations through the SIR model. This study is carried out to find the cause between the ulcer and the environment for the spread of Buruli ulcer (BU). Fractional-order SIR model is applied to discuss in detail the BU disease. The construction and development of this given numerical scheme are performed for the SIR epidemiological model concerning the transmission dynamics of BUs. Mathematical modeling of the presented scheme with the help of stability techniques is accomplished. Further, the behavior of the scheme is accessed by the evaluation of the different parameters of interest. The attained numerical results are observed for the illustration of the benefit of introducing a fractional model. Finally, we profound that the presented numerical scheme maintains all required control measuring features of the corresponding dynamical system and decreases the ancillary cost when studied over a long time. With the help of a given novel and unique approach, a significant observation is noted and recommended for other epidemiological fractional-order models. In future, this novel fractal technique will be appertained in chemical and biomedical engineering applications.

The theoretical study focuses on the examination of the convective flow of Oldroyd-B fluid with ramped wall velocity and temperature. The fluid is confined on an extended, unbounded vertical plate saturated within the permeable medium. To depict the fluid flow, the coupled partial differential equations are settled by using the Caputo (C) and Caputo Fabrizio (CF) differential time derivatives. The mathematical analysis of the fractionalized models of fluid flow is performed by Laplace transform (LT). The complexity of temperature and velocity profile is explored by numerical inversion algorithms of Stehfest and Tzou. The fractionalized solutions of the temperature and velocity profile have been traced out under fractional and other different parameters considered. The physical impacts of associated parameters are elucidated with the assistance of the graph using the software MATHCAD 15. We noticed the significant influence of the fractional parameter (memory effects) and other parameters on the dynamics of the fluid flow. Shear stress at the wall and Nusselt number also are considered. It’s brought into notice the fractional-order model (CF) is the best fit in describing the memory effects in comparison to the C model. An analysis of the comparison between the solution of velocity and temperature profile for ramped wall temperature and velocity and constant wall temperature and velocity is also performed.

In this paper, we construct a scheme of fractional-order mathematical model for the population infected by diarrhea disease by using the four compartments S, I, T and R. The fractal-fractional derivative operator (FFO) with generalized Mittag-Leffler kernel is employed to obtain the solution of the proposed system. The system is analyzed qualitatively as well as verify non-negative unique solution. The existence and uniqueness results of fractional-order model under Atangana–Baleanu fractal-fractional operator have been proved by fixed point theory. Also error analysis has been made for the proposed fractional-order model. Simulation has been carried out for derived fractional-order scheme to check the effectiveness of the results which will help, how to prevent and control such type of epidemic in society.

Since ecological systems are history-dependent, incorporating fractional calculus and especially variable order ones could significantly improve the emulation of these systems. Nonetheless, in the literature, no study considers ecological processes by variable-order fractional (VOF) model. This study is motivated by this issue. At first, we propose to extend a predator–prey mathematical model with VOF derivatives. The underlying assumption in the proposed model lies in considering values of fractional derivatives as time-varying functions instead of constant parameters. Some system’s dynamic features are investigated, and then the control of the proposed system is studied. To this end, a nonlinear model predictive control is offered for the VOF system. The necessary optimality and sufficient conditions for solving the nonlinear optimal control problem in the form of fractional calculus with variable-order derivative are formulated, and the controller’s design procedure is delineated. Finally, numerical simulations are performed to demonstrate the developed control technique’s effectiveness and performance for the VOF predator–prey model.

In this paper, we introduce the $q$-Shehu transform. Further, we define the generalized hyper-Bessel function. Also, we state the $q$-Shehu transform for some elementary functions. The present aim in this paper is to obtain the solutions of the $q$-fractional kinetic equations in terms of the established generalized hyper-Bessel function by applying the established $q$-Shehu transform. Also, we give some special cases of our main results. At the end of this paper, we give the numerical values and the graphical representations of these solutions by using the software MATLAB.

In this paper, a numerical and analytical investigation of a Hepatitis C virus (HCV) transmission concept is described in the context of fractional order. The model is an extension of the classical model to a fractional order. The existence, uniqueness, Hyers–Ulam-type stability, and numerical results are all discussed in the work. Lagrange’s interpolation polynomial technique is used for the numerical outcomes. The proposed method has a high level of precision and a low computing cost. We observe that the numerical results for the fractional-order model also contain the dynamics of the previous integer-order model as a special case. Finally, numerical solutions are implemented for the fractional-order HCV model to demonstrate the results graphically.

In this work, we focus on an initial value problem for a class of 2D time-fractional diffusion evolution equations with Riemann–Liouville fractional derivative. There are three new results in this paper. First of all, the existence and ill-posedness result (in the sense of Hadamard) in three cases which are consisting of homogeneous, inhomogeneous, and nonlinear problems are considered. Next, by using the Fourier truncation method, we show that regularized problems are well-posed. Finally, some demonstrated examples are presented to test the proposed method.

This paper introduces some important dissipative problems that are recent and still of intermittent interest. The classical dynamics of Helmholtz and Kelvin–Helmholtz instability equations are modeled with the Riesz operator which incorporates the left- and right-sided of the Riemann–Liouville non-integer order operators to mimic naturally the physical patterns of these models arising in hydrodynamics and geophysical fluids. The Laplace and Fourier transform techniques are used to approximate the Riesz fractional operator in a spatial direction. The behaviors of the Helmholtz and Kelvin–Helmholtz equations are observed for some values of fractional power in the regimes, $0<\alpha \le 1$ and $1<\alpha \le 2$, using different boundary conditions on a square domain in 1D, 2D and 3D (spatial-dimensions). Numerical results reveal some astonishing and very impressive phenomena which arise due to the variations in the initial and source function, as well as fractional parameter $\alpha$, for subdiffusive and superdiffusive scenarios.

The primary goal of this research is to look into the dynamical behavior of a fractional and fractal-fractional (FF) order laser chaotic model. The model is first considered with Caputo’s fractional derivative. An iterative technique based on the Laplace transform and its inverse aspect is used to achieve a particular solution for the fractional model. The existence of a unique solution is examined for the fractional laser chaotic model. Next, the laser chaotic model is addressed with a FF approach in the Caputo sense. The approximate results are achieved for the proposed fractional and FF models by using the Atangana–Seda numerical methods. The dynamical structures are plotted at various values of fractional and FF parameters.

In this study, we model the fractal-fractional system of the Computer virus problem using the Atangana–Baleanu operator. Moreover, we have presented the existence and the uniqueness of the results under applying the Schauder fixed point and Banach fixed theorems. We have used the Atangana–Toufik technique to obtain the approximate solutions by choosing various values of orders. Different values of fractal-fractional orders along with different amounts of initial conditions are selected to examine the performance of the suggested numerical method on the new fractal-fractional system. Also, graphs in different dimensions are presented to exhibit the solutions, clearly.

In this paper, we investigate the dynamics of quantum memory-assisted entropic uncertainty, uncertainty-induced nonlocality and log-negativity entanglement of two nitrogen-vacancy centers (NVC). In particular, we are interested in analyzing two nitrogen-vacancy centered qubits coupled with two coherent nanocavities. We consider two different configurations of NVC qubits: when the qubits are prepared initially in separable and maximally entangled states. We demonstrate that coherent nanocavities can be successfully used to generate nonlocality and entanglement in the two NVC qubits while suppressing the associated entropic uncertainty. When the system is considered initially in separable two-NVC qubits, we show that the coupled coherent nanocavities successfully generate nonlocality and entanglement while avoiding the associated entropic uncertainty. The parameter’s adjustment determines the degree of entanglement, non-locality generation, and preservation. In comparison, we find entanglement more vulnerable to the decoherence and entropic uncertainty action between the two NVC qubits and nanocavities than the uncertainty-induced nonlocality. Furthermore, the NVC-qubits coupled with two coherent nanocavities promote entanglement’s sudden birth and death phenomena. This study contributes to the quantum mechanical protocols requiring the robust practical generation of entanglement and nonlocality as well as the related information preservation for the increased efficiency of quantum devices.

Bose–Einstein correlations of identical particles reveal the shape and size of the particle emitting source of the given boson. These correlations are pivotal for a better understanding of the source dynamics and developing techniques to examine the propagation of quantum chaos in the presence of coherence at a certain temperature and momentum. Femtoscopy is of the utmost importance during this transformation for data processing and coherence-chaos analysis discrepancies. We introduced an evolving source to describe chaotic information inside quantum entanglement and the impact of the coherence order is discussed in this research. We also investigate the role of source size parameter and particle number in the dynamics of the temperature profiles with the function of momentum variation. The influences of the modeling factors on the correlations with their normalized correlator in the perspective of source coherence have been evaluated and the orientation of the source has an effect on the diffusivity of the fluid under consideration. Such consequences can cause an increase or decrease the genuine correlations due to the distributions of velocity and temperature. The main findings of the paper have been illustrated using the graphical representations of the considered correlations according to the geometry of the expanding source. Such results reveal the lucrativeness in the field of engineering applications.

The aim of the study is to discuss the controllability results of Hilfer neutral non-instantaneous impulsive fractional integro-differential equations (HNNIIFIE). Total controllability results were discussed by employing set-contraction theory and Kuratowski measure of non-compactness. Also we derived the results on optimal control using appropriate conditions. An illustration is given to validate the outcomes.

Nonlinear functional-integral equations contain the unknown function nonlinearly, occurring extensively in theory developed to a certain extent toward different applied problems and solutions thereof. Nonlinear science serves to reveal the nonlinear descriptions of widely different systems, has had a fundamental impact on complex dynamics. Accordingly, this study provides the basic definitions and results regarding Banach spaces with the proof as well as applications. Besides, proving the existence and uniqueness of the solutions by the Banach theorem is carried out. The existence results of the equations are generally obtained based on fundamental methods by which the fixed-point theorems are frequently applied. Subsequently, the definition of the measure pertaining to noncompactness on Banach space along with properties is presented. Hence, the aim is to study nonlinear functional-integral equations in the Banach space $BC(R_{+})$ using the measure of noncompactness. In addition, the proof of the existence of global attractors and asymptotic stabilities, having at least one solution which pertains to $BC(R_{+})$ is looked into. As the last stage, the definitions of global attractivity and asymptotical stability for equations on Banach space $BC(R_{+})$ are given while proving the existence of solutions and global attractivity. Besides these, the existence and asymptotical stability are proven. Furthermore, the proof of the existence of global attractors and asymptotic stabilities, having at least one solution which pertains to $BC(R_{+})$ for an example nonlinear integral equation has been demonstrated by one example premediated and the existence of blow-up point has been examined in one and more than one points regarding the nonlinear integral equation. The example nonlinear integral equation as addressed in this study has been examined in terms of its blow-up point and asymptotic stability with computational complexity for the first time. By doing so, it has been demonstrated that the existence of blow-up point has been the case in one and more than one points and besides this, the examination of its asymptotic stability has shown that it is asymptotically stable at least in one point, yet has no blow-up point pertaining to $BC(R_{+})$. This has enabled thorough analysis of computational complexity of the related nonlinear functional-integral equations within the example both addressing dimension of input in Banach space and that of dataset with $N$ dimension. This applicable approach assures the accurate transformation of the complex problems towards optimized solutions through the generation of advanced mathematical models in nonlinear dynamic settings characterized by complexity.