Fractals

This work is devoted to studying the transmission dynamics of CoV-2 under the effect of vaccination. The aforesaid model is considered under fractional derivative with variable order of nonsingular kernel type known as Atangan–Baleanue–Caputo (ABC). Fundamental properties of the proposed model including equilibrium points and $R_0$ are obtained by using nonlinear analysis. The existence and uniqueness of solution to the considered model are investigated via fixed point theorems due to Banach and Krasnoselskii. Also, the Ulam–Hyers (UH) approach of stability is used for the said model. Further numerical analysis is investigated by using fundamental theorems of AB fractional calculus and the iterative numerical techniques due to Adams–Bashforth. Numerical simulations are performed by using different values of fractional-variable order $\varrho (𝜗)$ for the model. The respective results are demonstrated by using real data from Saudi Arabia for graphical presentation.

This paper will introduce novel techniques for a fractional-order model of the human liver involving the Atangana–Baleanu, Atangana–Toufik, and the Fractal fractional method with the nonsingular kernel. These techniques give more accurate and appropriate results. Existence and uniqueness have been developed with the help of fixed-point theory results. Numerical simulations are done from the developed algorithm of the model to elaborate the effect of fractional values and justify the theoretical results. Such kind of analysis will be useful for further investigation of epidemic diseases, and also provide a better understanding of disease dynamics to overcome the effect of disease in a community.

This paper deals with a class of fractional variational problems involving Atangana–Baleanu (AB) derivative. The problem under consideration is a graceful combination of AB derivative with indefinite integral. A proposed numerical technique based on the shifted Vieta–Lucas polynomials is utilized for obtaining the solutions to the given problem. Our numerical process is principally based on the operational matrix (OM) of fractional integral in the AB sense in order to approximate solutions of the problem under study. By utilizing the OM along with the Lagrange multiplier method, the given problems are converted into a problem easier to solve. The resultant problem can then be solved using standard methods. Convergence of the presented method is investigated. In order to ensure and display the efficiency and accuracy of the proposed technique, some examples are given.

In the cultivation of Capsicum annuum (C. annuum), the major obstacles are triggered by the Yellow virus (Gemini virus). The virus is spread through insects, namely Bemisia tabaci (B. tabaci). To reduce the spreading of the virus in C. annuum, the entomopathogenic fungi (Verticillium lecanii) are used. We have analyzed the fractional-order model of chili plants with Atangana–Baleanu derivative (AB-derivative). Also, we calculate the numerical values to illustrate the results.

The main objective of this study is to provide a new computational procedure for extracting approximate and exact solutions of the temporal-fractional Swift–Hohenberg (S–H) equations in the context of conformable derivatives using the conformable natural transform (CNT) and Daftardar–Jafari method (DJM). We refer to it as the “natural conformable Daftardar–Jafari method” (CNDJM). The three types of errors are assessed in order to gauge the efficiency and consistency of the proposed method. Furthermore, 2D and 3D graphics are used to compare the exact and approximate solutions. This method offers a considerable benefit over homotopy analysis and Adomian decomposition methods in terms of computational work because it does not require Adomian and He’s polynomials. The procedure is quick and easy to use.

Recently, Atangana proposed new operators by combining fractional and fractal calculus. These recently proposed operators, referred to as fractal–fractional operators, have been widely used to study complex dynamics. In this paper, the COVID-19 model is considered via Atangana–Baleanu fractal-fractional operator. The Lyapunov stability for the model is derived for first and second derivative. Numerical results have developed through Lagrangian-piecewise interpolation for the different fractal–fractional operators. We develop numerical outcomes through different differential and integral fractional operators like power-law, exponential law, and Mittag-Leffler kernel. To get a better outcome of the proposed scheme, numerical simulation is made with different kernels having the memory effects with fractional parameters.

This paper proposes the shifted Legendre polynomial approximations-based stochastic operational matrix of integration method to solve persistent processes-based fractional stochastic differential equations. The operational matrix of integration, stochastic operation matrix and fractional stochastic operational matrix of the shifted Legendre polynomials are derived. The stochastic differential equation is transformed into an algebraic system of $(N+1)$ equations by the operational matrices. For the proposed approach, a thorough discussion of the error analysis in $L^2$ norm is provided. The proposed method’s applicability, correctness, and accuracy are examined using a few numerical examples. Comparing the numerical examples to the other methods discussed in the literature demonstrates the solution’s effectiveness and attests to the solution’s high quality. The error analysis also reveals the method’s superiority. A more accurate solution is obtained, thus maintaining a minimum error.

In this paper, we study a nonlinear mathematical model which addresses the transmission dynamics of COVID-19. The considered model consists of susceptible ($S$), exposed ($E$), infected ($I$), and recovered ($R$) individuals. For simplicity, the model is abbreviated as $SEIR$. Immigration rates of two kinds are involved in susceptible and infected individuals. First of all, the model is formulated. Then via classical analysis, we investigate its local and global stability by using the Jacobian matrix and Lyapunov function method. Further, the fundamental reproduction number ${\mathcal{ℛ}}_0$ is computed for the said model. Then, we simulate the model through the Runge–Kutta method of order two abbreviated as RK2. Finally, we switch over to the fractional order model and investigate its numerical simulations corresponding to different fractional orders by using the fractional order version of the aforementioned numerical method. Finally, graphical presentations are given for the approximate solution of various compartments of the proposed model. Also, a comparison with real data has been shown.

When it comes to human care service challenges, it is critical to understand the dynamic behavior of the associated contagious diseases because non-Markovian effects play a significant role in their spread. This investigation takes a look into an etiological model directly linked to the complexities of Lassa hemorrhagic fever disease in pregnant women in Africa. This is a bacteriostatic fever and an outbreak illness. The aforesaid ailment in expectant mothers has adverse implications that first popped up in Africa. The etiological model was constructed for the first time utilizing a recently designed fractal-fractional (FF) operator based on the power law, exponential decay, and Mittag-Leffler kernels with fractional order and fractal dimension. We construct three schemes of successive approximations based on FF operators using Lagrange polynomials, and the fundamental reproducing number is determined to be ${{ℛ}}_0$. The existence and uniqueness of the suggested model’s mathematical technique are examined. The endemic and disease-free equilibria are also calculated. Eventually, simulated results of the framework are performed using the suggested numerical approach, and the outcomes in graphical representations quantify the effects of the projected and integrated characteristics and demonstrate that the ailment can be considerably governed or exterminated if the outbreak propagation rate is lessened and the rate of intervention is improved.

Waterborne diseases are illnesses caused by pathogenic bacteria that spread through water and have a negative influence on human health. Due to the involvement of most countries in this vital issue, accurate analysis of mathematical models of such diseases is one of the first priorities of researchers. In this regard, in this paper, we turn to a waterborne disease model for solution’s existence, HU-stability, and computational analysis. We transform the model to an analogous fractal-fractional integral form and study its qualitative analysis using an iterative convergent sequence and fixed-point technique to see whether there is a solution. We use Lagrange’s interpolation to construct numerical algorithms for the fractal-fractional waterborne disease model in terms of computations. The approach is then put to the test in a case study, yielding some interesting outcomes.

In this paper, we analyzed the chaotic complexity of a financial mathematical model in terms of a new generalized Caputo fractional derivative. There are three components in this nonlinear financial model: price indexes, interest rates, and investment demand. Our analysis is based on applying the fixed point hypothesis to determine the existence and uniqueness of the solutions. The bifurcation of the proposed financial system has been analyzed at various parameters of the system. Dynamical phase portraits of the proposed financial model are demonstrated at various fractional-order values. We investigated the possibility of finding new complex dynamical behavior with generalized Caputo fractional derivative. This economic model is solved numerically using a predictor–corrector (PC) algorithm with a generalized Caputo derivative. This algorithm can be viewed as a non-integer extension of the classical Adams–Bashforth–Moulton (ABM) algorithm. Additionally, this numerical algorithm has been studied for stability. A number of diverse dynamic behaviors have been observed in numerical simulations of the system, including chaos. Over a broad range of system parameters, bifurcation diagrams indicate that the system behaves chaotically.

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This paper suggests using fractional Euler polynomials (FEPs) to solve the fractional diffusion-wave equation in Caputo’s sense. We present the fundamental characteristics of Euler polynomials. The method for building FEPs is discussed. By basically converting fractional partial differential equations into a system of polynomial equations, these qualities enable us to come near to solving the original problem. A conventional numerical method is then used to solve the resulting system of equations. Theoretical analysis for our proposed strategy is also established, including the convergence theorem and error analysis. The proposed technique’s error bound is confirmed for the test problems as well. The method’s applicability and validity are examined using a variety of instances. The acquired solution is contrasted with other approaches’ solutions described in the literature. This method is better in terms of implementation, adaptability and computing efficiency for solving other partial differential equations as a result of the comparison of the proposed method to existing methods used to solve the fractional diffusion-wave equation.

In this paper, we study time-fractional diffusion equations such as the time-fractional Kolmogorov equations (TF–KEs) and the time-fractional advection–diffusion equations (TF–ADEs) in the Caputo sense. Here, we have developed the operational matrices (OMs) using the Hosoya polynomial (HP) as basis function for OMs to obtain solution of the TF–KEs and the TF–ADEs. The great benefit of this technique is converting the TF–KEs and the TF–ADEs to algebraic equations, which can be simply solved the problem under study. We provide error bound for the approximation of a bivariate function using the HP. Furthermore, comparison of the numerical results obtained using the proposed technique with the exact solution is done. The results prove that the proposed numerical method is most relevant for solving the TF–KEs and the TF–ADEs and accurate.

In this paper, we establish some new Bullen-type inequalities for differentiable convex functions using the generalized fractional integrals. The main advantage of the inequalities and operators used to obtain them is that these inequalities can be turned into some existing inequalities for Riemann integrals and new inequalities for Riemann–Liouville fractional integral inequalities and $k$-fractional integrals. Finally, we add some applications of special means of real numbers using the newly established inequalities to make these results more interesting.

This paper formulates the L1 implicit difference scheme (L1IDS) for the time-fractional Kuramoto–Sivashinsky equation (TFKSE) with non-smooth solution. The TFKSE is one of useful descriptions for modeling flame-propagation, viscous flow problems, and reaction–diffusion systems. The proposed method approximates the unknown solution by using two main stages. At the first stage, the L1 method with nonuniform meshes and the general centered difference method is adopted to discretize the Caputo fractional derivative and the spatial derivative, respectively. In the second stage, the fully-discrete L1IDS is established with the help of the Galerkin scheme based on piecewise linear test functions. Meanwhile, an iterative algorithm is adopted to solve the nonlinear systems. Furthermore, the convergence and stability of the proposed method are both demonstrated and confirmed numerically. Finally, three numerical examples highlight the accuracy and efficiency of the proposed strategy.

In this paper, we investigate a terminal value problem for stochastic fractional diffusion equations with Caputo–Fabrizio derivative. The stochastic noise we consider here is the white noise taken value in the Hilbert space $W$. The main contribution is to investigate the well-posedness and ill-posedness of such problem in two distinct cases of the smoothness of the Hilbert scale space $W_{\nu }$ (see Assumption 3.1), which is a subspace of $W$. When $W_{\nu }$ is smooth enough, i.e. the parameter $\nu$ is sufficiently large, our problem is well-posed and it has a unique solution in the space of Hölder continuous functions. In contract, in the different case when $\nu$ is smaller, our problem is ill-posed; therefore, we construct a regularization result.

In the fuzzy calculus, the study of fuzzy differential equations (FDEs) created a proper setting to model real problems which contain vagueness or uncertainties factors. In this paper, we consider a class fuzzy differential equations (FFDEs) with non-integer or variable order (VO). The variable order derivative is defined in the Atangana–Baleanu–Caputo sense on fuzzy set-valued functions. The main problem under the fuzzy initial condition is converted to a new problem by the $r$-cut representation of fuzzy-valued function. For solving the new problem, we use the operational matrices (OMs) based on the shifted Legendre polynomials (SLPs). By approximating the unknown function and its derivative in terms of the SLPs and substituting these approximations into the equation, the main problem is converted to a system of nonlinear algebraic equations. An error estimate of the numerical solution is proved. Finally, an example is considered to confirm the accuracy of the proposed technique.

In this study, we examine a modified heat equation with memory and nonlinear source. The source function is considered under two different conditions, the global Lipschitz and the exponential growth functions. For the first condition, a special weighted Banach space is applied to deduce a desired result without any assumption on sufficiently small time and initial data. For the second condition of exponential growth, we apply the Moser–Trudinger inequality to cope with the source function, and a special time-space norm to deduce the unique existence of a global solution in regard to sufficiently small data. The main objective of this work is to prove the global existence and uniqueness of mild solutions. Besides the solution techniques, our main arguments are also based on the Banach fixed point theorem and linear estimates for the mild solution. The highlight of this study is that it is the first work on the global well-posedness for the mild solution of the fractional heat conduction with memory and nonlinear sources.

This paper is to investigate the extent and speed of the spread of the coronavirus disease 2019 (COVID-19) pandemic in the United States (US). For this purpose, the fractional form of the susceptible-exposed-infected-recovered-vaccinated-quarantined-hospitalized-social distancing (SEIR-VQHP) model is initially developed, considering the effects of social distancing, quarantine, hospitalization, and vaccination. Then, a Monte Carlo-based back analysis method is proposed by defining the model parameters, viz. the effects of social distancing rate ($\alpha$), infection rate ($\beta$), vaccination rate ($\rho$), average latency period ($\gamma$), infection-to-quarantine rate ($\delta$), time-dependent recovery rate ($\lambda$), time-dependent mortality rate ($\kappa$), hospitalization rate ($\xi$), hospitalization-to-recovery rate ($\psi$), hospitalization-to-mortality rate ($\varphi$), and the fractional degree of differential equations as random variables, to obtain the optimal parameters and provide the best combination of fractional order so as to give the best possible fit to the data selected between January 20, 2020 and February 10, 2021. The results demonstrate that the number of infected, recovered, and dead cases by the end of 2021 will reach 1.0, 49.8, and 0.7 million, respectively. Moreover, the histograms of the fractional order acquired from back analysis are provided that can be utilized in similar fractional analyses as an informed initial suggestion. Furthermore, a sensitivity analysis is provided to investigate the effect of vaccination and social distancing on the number of infected cases. The results show that if the social distancing increases by 25% and the vaccination rate doubles, the number of infected cases will drop to 0.13 million by early 2022, indicating relative pandemic control in the US.

In this paper, we study the continuity problem by an order of derivative for conformable parabolic equations. The problem is examined in both the linear and nonlinear cases. For the input data in suitable Hilbert scale spaces, we consider the continuity problem for the linear problem. In the nonlinear case, we prove the existence of mild solutions for a class of conformable parabolic equations once the source function is a global Lipschitz type in the $L^s$ space sense. The main results are based on semigroup theory combined with the Banach fixed point theorem and Sobolev embeddings. We also inspect the continuity problem for the nonlinear model, and prove the convergence of the mild solution to the nonlinear problem as $\alpha$ tends to $1^{-}$.

In this research, we examine the nonsingular memory effect when implementing the Atangana–Baleanu (AB) fractional derivative in the Caputo sense to the Hepatitis C virus (HCV) model. For this purpose, first, the sufficient conditions for the existence and uniqueness of the solutions under the AB fractional derivative of the model are expressed and proved. Therewithal, in order to show the response of the AB derivative on the system dynamics practically, a new version of the predictor–corrector method is used with the newly estimated model parameters from the literature. Finally, we provide some simulations of the results.

In this paper, we derive a new version of L1-Predictor–Corrector (L1-PC) method by using some previously given methods (L1-PC for single delay, PC for non-delay, and decomposition algorithm) to solve multiple delay-type fractional differential equations. The Caputo fractional derivative with singular type kernel is used to establish the results. Some important remarks related to the delay term estimation and error analysis are mentioned. In order to check the accuracy and correctness of our method, we solve a neural network system with two delay parameters. A number of graphs are given to justify the role of delays as well as the accuracy of the algorithm. The given method is fully novel and reliable to solve multiple delay type fractional order systems in Caputo sense.

In this paper, we deal with the blow-up problems for Fujita-type parabolic system involving time-dependent coefficients on graphs. Under appropriate conditions, we prove that the nonnegative solution of the parabolic system blows up in a finite time on finite graphs and locally finite graphs, respectively. The results obtained extend some previous results of [Y. Lin and Y. Wu, Blow-up problems for nonlinear parabolic equations on locally finite graphs, Acta Math. Sci. Ser. B 38(3) (2018) 843–856; Y. Wu, Local existence and blow-up of solutions to Fujita-type equations involving general absorption term on finite graphs, Fractals 30(2) (2022) 2240053].

In this paper, we present new definitions of generalized fractional integrals and derivatives with respect to another function and derive some of their properties, such as their inter-relationship and semigroup law. Caputo-type generalized fractional derivative with respect to another function is also defined and its properties are derived. A Cauchy problem involving the new Caputo-type generalized fractional derivative is also studied. We also provide an expansion formula for Caputo-type derivative and apply it to solve a fractional-order problem.

In this paper, two algebraic methods are applied for solving a class of conformable fractional partial differential equations (FPDEs). We use these methods for the time-fractional Radhakrishnan–Kundu–Lakshmanan equation. With these methods, further solutions can be obtained compared with other approaches and techniques. The exact particular solutions include the exponential solution, trigonometric function solution, rational solution and hyperbolic function solution. These methods are very effective to obtain exact solutions of many fractional differential equations.

An efficient high-order computational procedure is going to be created in this paper to determine the solution to the mobile–immobile advection–dispersion model (MIAD) of temporal fractional order $0<\alpha \le 1$, which can be employed to model the solute forwarding in watershed catchments and floods. To do it, the temporal-first derivative of MIAD is discretized by using the finite-difference technique’s first-order precision and the linear interpolation’s temporal-fractional derivative. On either side, the space derivative is simulated using a collocation approach based on the Legendre basis to generate the full-discrete method. The order of MIAD-convergence for the implicit numerical structure is explained. Additionally, a basic conceptual discussion of the temporal-discretized stability mechanism is included in this paper. Finally, two models are provided to show the reliability and excellence of the organized approach.

Uses of time-fractional diffusion wave model (TFDWM) in its singular case with damping-reaction terms are widely seen in classical physics applications, e.g. for the quantitative measurement of activity diagnoses light-mechanical waves resulting from many physical experiments. The goal and importance of this paper are to predict and build accurate and convincing numerical solutions for TFDWM in its singular version by employing the collective cubic uniform B-spline approach (CUBSA) and standard finite difference approach (SFDA). The fractional Caputo time derivative (FCTD) has been estimated and broken down using SFDT, whilst the standard splines will be utilized upon realizing spatial discretization. To study the prediction error of our approach, some convergence and bound results are given under certain constraints. We demonstrate applications of our collective algorithm to a couple of fractional singular-type models appearing in fluid dynamics and electromagnetics. Detailed analysis, delegate tables, and representative graphs are displayed and offered in different dimensions to handle the crossover meaning for several order values of FCTDs. Some conclusions, observations, recommendations, and future issues were briefly raised in the final section of this paper.

The interrelationship between predator populations and prey populations is a central problem in biology and mathematics. Setting up appropriate predator–prey models to portray the development law of predator populations and prey populations has aroused widespread interest in many scholars. In this work, we propose a new fractional order predator–prey system involving Michaelis–Menten-type functional response and both unequal delays. Utilizing the contraction mapping theorem, we prove the existence and uniqueness of the solution to the considered fractional order predator–prey system. By virtue of some mathematical analysis techniques, nonnegativeness of the solution to the involved fractional order predator–prey system is analyzed. By constructing a suitable function, the boundedness of the solution to the considered fractional order predator–prey system is explored. Making use of Laplace transform, we derive the characteristic equation of the involved fractional order predator–prey system, then by means of the stability principle and the bifurcation theory of fractional order dynamical system, a series of novel delay-independent stability criteria and bifurcation conditions ensuring the stability of the equilibrium point and the creation of Hopf bifurcation of the considered fractional order predator–prey system, are built. The global stability of the involved fractional order predator–prey system is analyzed in detail. The role of time delay in controlling the stability and the creation of Hopf bifurcation is revealed. To check the legitimacy of the derived key results, software simulation results are effectively presented. The obtained results in this work are completely novel and play a significant role in maintaining ecological balance.

This paper focuses on the theoretical and computational investigation of the innovative nonlinear time-fractional PDEs incorporating the Caputo and Atangana–Baleanu fractional derivatives in the Caputo context using the $\widehat{q}$-homotopy analysis transform method (HATM). The expected strategy employs a combination of $\widehat{q}$-HATM and the Jafari transform with the assistance of Caputo and Atangana–Baleanu fractional derivative operators to obtain the methodology permits of PDEs with proportional delay. The fractional operators are employed in this research to demonstrate how crucial they are in generalizing frames involving singular and nonsingular kernels. The proposed series of solutions are closely in agreement with an exact solution. Several important challenges can be addressed to illustrate the validity of the proposed method. The outcomes of the proposed framework are displayed and assessed using numerical and graphical outputs. Furthermore, the results of our suggested strategy were compared to earlier outcomes. The proposed method requires less computation and has significantly better performance. Finally, the analysis shows that the enhanced technique is both reliable and meticulous when evaluating the impact of nonlinearities in science and technology.

Due to the increasing growth of economy in the last few years, environmental problems have become a prominent difficulty in Chinese economic growth and social development. Therefore, in order to respond to the environmental protection policy of Anhui Province, all prefecture-level cities need to be evaluated for their environmental efficiency. A study of environmental efficiency in Anhui Province is presented in this paper. The time period is from 2015 to 2020. By exploiting MaxDEA software, the input–output system in different years is taken as the decision-making unit, and slacks-based model (SBM-DEA) which is based on an unexpected output is used to construct the index system by selecting two output indexes and four input indexes, so as to calculate the environmental efficiency of Anhui Province, including 16 prefecture-level cities, under the condition of constant non-radial and non-angle scale benefits. In addition, based on the persistence and anti-persistence theory of fractal theory, in this paper, we have analyzed the future trend of environmental efficiency of prefecture-level cities in Anhui Province by rescaled range analysis (R/S). The results show that Anhui province has 16 prefectures that have not achieved an optimal level of environmental performance and fluctuates downward from 2015 to 2020. The average value of comprehensive efficiency is 0.846. As for technical efficiency, the mean value is 0.954, and scale efficiency has an average value of 0.887. The technical progress of environmental efficiency development in Anhui Province is small and the room for improvement is large. Finally, according to the results of the study, some suggestions on environmental technology are put forward.

This research paper introduces a novel fractional Caputo-type simultaneous method for finding all simple and multiple roots of polynomial equations. Without any additional polynomial and derivative evaluations using suitable correction, the order of convergence of the basic Aberth–Ehrlich simultaneous method has been increased from three to $\alpha +3$. In terms of accuracy, residual graph, computational efficiency and computation CPU time, the newly proposed families of simultaneous methods outperforms existing methods in numerical applications.