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This study delves into the exploration of the dynamics of (3+1)-dimensional Painlevé integrable generalized model from different perspectives, which delineates the evolution of nonlinear phenomena in three spatial dimensions and one temporal dimension, displaying the remarkable Painlevé integrability property. The ϕ6-expansion technique is applied to extract traveling wave solutions in the form of Jacobi elliptic functions. To give physical insights of obtained solutions, we present these solutions through graphs such as 2D, 3D and contour plots. Further, to understand the planar dynamical system, we employ the concepts of bifurcation, chaos theory and sensitivity analysis. Bifurcation analysis reveals the dependence on the solution of a planar dynamical system at critical points. Additionally, the detection of chaotic movements in the perturbed dynamical system is achieved by detecting tools. Also the sensitivity analysis of the model is investigate by three distinct initial conditions. The findings are innovative, valuable and captivating for the readers in exploring this model.

The current paper studies the influence of the temperature ratio of ion-to-electron α, dust concentration μ, and κ- deformed parameter on dust ion acoustic solitary waves in an unmagnetized plasma with Kaniadakis distributed electrons. More precisely, the reductive perturbation technique is utilized to extract the Korteweg-de Vries and modified Korteweg-de Vries equations. Both compressive and rarefactive Korteweg-de Vries solitons are found to exist in the ranges 0 < μ ≤ 0.677 and 0.677 < μ < 1, respectively, and only compressive modified Korteweg-de Vries solitons in the range 0 < μ ≤ 0.11. In an unmagnetized plasma with Kaniadakis distributed electrons, the influence of the ion-to-electron temperature ratio on dust ion acoustic solitary waves can have several fascinating applications and consequences in plasma physics and astrophysics.

This article analyzes and computes numerical solutions for the fractional isothermal chemical (FIC) model. This work suggested a Jacobi collocation method (JCM) to examine the FIC model. In the beginning, we constructed the operational matrices for fractional order derivatives for Jacobi polynomials. Then, by using operational matrices and the collocation technique, we converted the provided model into a set of algebraic equations. The approach that is used in this article is quicker and more effective than several other schemes. We solve the system of equations for various fractional orders of differentiation. Comparison of JCM and Newton polynomial interpolation (NPI) technique is present in this article. Furthermore, an error analysis of the proposed procedure is also given.

In this study, the sequential operator of mixed order is analysed on the domain [Formula: see text] with [Formula: see text]. Then, the positivity of the nabla operator is obtained analytically on a finite time scale under some conditions. As a consequence, our analytical results are introduced on a set, named [Formula: see text], on which the monotonicity analysis is obtained. Due to the complicatedness of the set [Formula: see text] several numerical simulations so are applied to estimate the structure of this set and they are provided by means of heat maps.

AbstractIn this article, we investigate a mathematical malaria-filariasis co-infection model with the assistance of the non-integer order operator. Using the fractal-fractional operator in the Caputo-Fabrizio (CF) sense, it has been possible to understand the dynamical behaviour and complicatedness of the malaria-filariasis model. An investigation of the existence and uniqueness of the solution employs fixed-point theory. Ulam-Hyers stability helps examine the stability analysis of the proposed co-infection model. The malaria-filariasis model has been investigated using the Toufik-Atanagana (TA), a sophisticated numerical method for these biological co-infection models. With the help of numerical procedures, we provide the approximate solutions for the proposed model. A variety of fractal dimension and fractional order options are utilized for the presentation of the results. When we adjust sensitive parameters like τ and γ, the graphical representation illustrates the system’s behaviour and identifies suitable parameter ranges for solutions. In addition, we evaluate the model along with the regarded operators and various β1 values using an exceptional graphical representation.

This paper introduces a novel numerical scheme, the conformable finite difference method (CFDM), for solving time-fractional gas dynamics equations. The method was developed by integrating the finite difference method with conformable derivatives, offering a unique approach to tackle the challenges posed by time-fractional gas dynamics models. The study explores the significance of such equations in capturing physical phenomena like explosions, detonation, condensation in a moving flow, and combustion. The numerical stability of the proposed scheme is rigorously investigated, revealing its conditional stability under certain constraints. A comparative analysis is conducted by benchmarking the CFDM against existing methodologies, including the quadratic B-spline Galerkin and the trigonometric B-spline functions methods. The comparisons are performed using $ {L}_{2} $ and $ {L}_{\infty } $ norms to assess the accuracy and efficiency of the proposed method. To demonstrate the effectiveness of the CFDM, several illustrative examples are solved, and the results are presented graphically. Through these examples, the paper showcases the capability of the proposed methodology to accurately capture the behavior of time-fractional gas dynamics equations. The findings underscore the versatility and computational efficiency of the CFDM in addressing complex phenomena. In conclusion, the study affirms that the conformable finite difference method is well-suited for solving differential equations with time-fractional derivatives arising in the physical model.

This work presents a numerical approach for handling a fractional Lienard equation (FLE) arising in an oscillating circuit. The scheme is based on the Vieta Lucas operational matrix of the fractional Liouville-Caputo derivative and the collocation method. This methodology involves a systematic approach wherein the operational matrix aids in expressing the fractional problem in terms of non-linear algebraic equations. The proposed numerical approach utilizing the operational matrix method offers a vital solution framework for efficiently tackling the fractional Lienard equation, addressing a key challenge in mathematical modeling. To analyze the fractional order system, we derive an approximate solution for the FLE. The solutions are explained graphically and in tabular form.

This study explores the foundational iterative processes of fractional calculus, focusing on Θ-conformable fractional derivatives (Θ-CFD). We introduce novel fractional operators and define their associated function spaces. Additionally, we establish a series of theorems that enhance our understanding of these operators within the context of fractional calculus.

This document elaborates on a newly introduced analytical method known as the “Variable Coefficient Generalized Abel Equation Method,” as proposed by Hashemi in Hashemi (2024), designed specifically for addressing the two-mode Cahn–Allen equation. Diverging from conventional techniques that heavily rely on constant coefficient ordinary differential equations and auxiliary ordinary differential equations, our method innovatively incorporates variable coefficient ordinary differential equations within a sub-equation framework. Demonstrating its versatility, we apply this innovative technique to the two-mode Cahn–Allen equation, showcasing its effectiveness and efficiency through the derivation of analytical solutions. Notably, this method emerges as a promising tool for tackling complex nonlinear partial differential equations prevalent in fluid dynamics and wave propagation scenarios. Beyond merely expanding the repertoire of available analytical tools, our approach contributes to advancing solutions for various models within the realm of mathematical physics. Various forms of exact solutions, including exponential-type solutions, Kink solitons, dark solitons, and bright soliton solutions, are obtained for the model under consideration. Moreover, we delve into the analysis of bifurcation, chaotic behavior, and sensitivity within the context of the two-mode Cahn–Allen model, further enhancing the depth and breadth of our study. Three equilibria are analyzed across various classifications, including center point, focus point, saddle point, and node point. Chaotic behavior of the corresponding dynamical system is considered by adding the function ω1sin(ω2ζ). Lastly, sensitivity analysis of the system is conducted by examining different parameters of the model and imposing noise to the initial conditions.

In this paper, the Burgers’ equations using the fractional derivative of Atangana-Baleanu sense are investigated and discussed. A Laplace variational iteration approach is used to demonstrate the fractional model's mathematical solution. The solution's existence and uniqueness are examined using fixed point theory. Several numerical simulations that enhance the efficacy of the employed derivative are presented and discussed.

Hepatitis B is a worldwide viral infection that causes cirrhosis, hepatocellular cancer, the need for liver transplantation, and death. This work proposed a mathematical representation of Hepatitis B Virus (HBV) transmission traits emphasizing the significance of applied mathematics in comprehending how the disease spreads. The work used an updated Atangana-Baleanu fractional difference operator to create a fractional-order model of HBV. The qualitative assessment and well-posedness of the mathematical framework were looked at, and the global stability of equilibrium states as measured by the Volterra-type Lyapunov function was summarized. The exact answer was guaranteed to be unique using the Lipschitz condition. Additionally, there were various analyses of this new type of operator to support the operator's efficacy. We observe that the explored discrete fractional operators will be $ \chi^2 $-increasing or decreasing in certain domains of the time scale $ \mathbb{N}_j: = {j, j + 1, ... } $ by looking at the fundamental characteristics of the proposed discrete fractional operators along with $ \chi $-monotonicity descriptions. For numerical simulations, solutions were constructed in the discrete generalized form of the Mittag-Leffler kernel, highlighting the impacts of the illness caused by numerous causes. The order of the fractional derivative had a significant influence on the dynamical process utilized to construct the HBV model. Researchers and policymakers can benefit from the suggested model's ability to forecast infectious diseases such as HBV and take preventive action.

This paper introduced an efficient method to obtain the solution of linear and nonlinear weakly singular kernel fractional integro-differential equations (WSKFIDEs). It used Riemann-Liouville fractional integration (R-LFI) to remove singularities and approximated the regularized problem with a combined approach using the generalized fractional step-Mittag-Leffler function (GFSMLF) and operational integral fractional Mittag matrix (OIFMM) method. The resulting algebraic equations were turned into an optimization problem. We also proved the method's accuracy in approximating any function, as well as its fractional differentiation and integration within WSKFIDEs. The proposed method was performed on some attractive examples in order to show how their solutions behave at various values of the fractional order $ \digamma $. The paper provided a valuable contribution to the field of fractional calculus (FC) by presenting a novel method for solving WSKFIDEs. Additionally, the accuracy of this method was verified by comparing its results with those obtained using other methods.

In this article, the Montgomery identity and Ostrowski inequality are established for univariate first-order diamond-alpha differentiable functions. We also investigate the generalization of Ostrowski-type inequalities for bivariate functions with bounded second-order diamond-alpha derivatives by applying integration by parts for $ \diamondsuit_\alpha $-integrals. Moreover, some extensions of dynamic trapezoid- and Grüss-type inequalities are also obtained by using the Montgomery identity.

Abstract The Ruscheweyh derivative operator is used in this paper to introduce and investigate interesting general subclasses of the function class Σ m $\Sigma_{m}$ of m-fold symmetric bi-univalent analytic functions. Estimates of the initial Taylor-Maclaurin coefficients | a m + 1 | $\vert a_{m+1} \vert $ and | a 2 m + 1 | $\vert a_{2 m+1} \vert $ are obtained for functions of the subclasses introduced in this study, and the consequences of the results are discussed. Additionally, the Fekete-Szegö inequalities for these classes are investigated. The results presented could generalize and improve some recent and earlier works. In some cases, our estimates are better than the existing coefficient bounds. Furthermore, within the engineering domain, the utilization of the Ruscheweyh derivative operator can encompass a broad spectrum of engineering applications, including the robotic manipulation control, optimizing optical systems, antenna array signal processing, image compression, and control system filter design. It emphasizes the potential for innovative solutions that can significantly enhance the reliability and effectiveness of engineering applications.

The major goal of this research study is to solve the fractional order Wolbachia invasive model (FWIM) by developing a computational framework based on the Bayesian regularization backpropagation neural network (BRB-NN) approach. The population of mosquitoes is categorized into two classes, Wolbachia-infected mosquitoes and Wolbachia-uninfected mosquitoes. We also incorporate incomplete cytoplasmic incompatibility and imperfect maternal transmission. We investigate the effects of the fractional order derivative (α) and reproduction rate of Wolbachia-infected mosquitoes ϕw on the dynamics of mosquitoes. The proposed Bayesian regularization backpropagation scheme is applied to three distinct cases using 80% and 20% of the created dataset for training and testing, respectively, with 15 hidden neurons. Comparisons of the results are presented to verify the validity of the proposed technique for solving the model. The Bayesian regularization approach is used to lower the mean square error (MSE) for the fractional order Wolbachia invasive model. The achieved results are based on MSE, correlation, state transitions, error histograms, and regression analysis to confirm the effectiveness of the suggested approach. Additionally, the absolute error value modifies the designed approach’s accuracy.

ABSTRACTIn this study, we investigate novel fractional tuberculosis model with Caputo fractional derivative. The computational solution of the fractional tuberculosis disease is obtained with the help of the generalized Euler's method (GEM). This article considers two treatment strategies: protective treatment for latent populations and main treatment for infected ones. The compartmental structure of the six-dimensional model includes the susceptible, latent, infected, recovered, and treatment classes. To examine the nature of differential equations of fractional order that arise in biological illness, the findings produced utilizing the method under consideration are more accurate and straightforward. Additionally, the stability of the equilibrium point is also investigated. The fractional model has been graphically simulated using MATLAB22.

The novel coronavirus SARS-Cov-2 is a pandemic condition and poses a massive menace to health. The governments of different countries and their various prohibitory steps to restrict the virus's expanse have changed individuals' communication processes. Due to physical and financial factors, the population's density is more likely to interact and spread the virus. We establish a mathematical model to present the spread of the COVID-19 in worldwide. In this article, we propose a novel mathematical model (‘[Formula: see text]’) to assess the impact of using hospitalization, quarantine measures, and pathogen quantity in controlling the COVID-19 pandemic. We analyse the boundedness of the model's solution by employing the Laplace transform approach to solve the fractional Gronwall's inequality. To ensure the uniqueness and existence of the solution, we rely on the Picard-Lindelof theorem. The model's basic reproduction number, a crucial indicator of epidemic potential, is determined based on the greatest eigenvalue of the next-generation matrix. We then employ stability theory of fractional differential equations to qualitatively examine the model. Our findings reveal that both locally and globally, the endemic equilibrium and disease-free solutions demonstrate symptomatic stability. These results shed light on the effectiveness of the proposed interventions in managing and containing the COVID-19 outbreak.

In this work, we present a numerical scheme based on the operational matrix of fractional Caputo-Fabrizio (CF) integration for handling fractional Bloch equation (FBE) in nuclear magnetic resonance (NMR). The understanding of Bloch equation provides us a fundamental framework for describing magnetic resonance phenomena, facilitating breakthrough in diverse fields such as medical diagnostics, quantum computing and materials characterization. The non-integer order derivative and integration are presented in the Caputo-Fabrizio sense. To construct the operational matrix, Jacobi polynomial is used as a basis. The fractional Bloch equation is transformed into a set of algebraic equations by using the operational matrix. In order to examine the fractional order problem, we obtain an approximate solution for FBE and present the numerical results in graphical and tabular forms.

Time-delay fractional optimal control problems (OCPs) are an important research area for developing effective control and optimization strategies to address complex phenomena occurring in various natural sciences, such as physics, chemistry, biology, and engineering. By considering fractional OCPs with time delays, we can design control strategies that take into account the system's history and optimize its behavior over a given time horizon. However, applying the Pontryagin principle of maximization to solve these problems leads to a boundary value problem (BVP) that includes delay and advance terms, making analytical solutions difficult and demanding. To address this issue, this paper presents a precise finite difference formula to solve the aforementioned advance-delay BVP numerically. The suggested approximate method's error analysis and convergence properties are provided, and several illustrative examples demonstrate the applicability, validity, and accuracy of the proposed approach. Simulation results confirm the proposed technique's advantages for the optimal control of delay fractional dynamical equations.

The discretization of initial and boundary value problems and their existence behaviors are of great significance in various fields. This paper explores the existence of a class of self-adjoint delta fractional difference equations. The study begins by demonstrating the uniqueness of an initial value problem of delta Riemann–Liouville fractional operator type. Based on this result, the uniqueness of the self-adjoint equation will be examined and determined. Next, we define the Cauchy function based on the delta Riemann–Liouville fractional differences. Accordingly, the solution of the self-adjoint equation will be investigated according to the delta Cauchy function. Furthermore, the research investigates the uniqueness of the self-adjoint equation including the component of Green’s functions of and examines how this equation has only a trivial solution. To validate the theoretical analysis, specific examples are conducted to support and verify our results

The study is being applied to a model involving silane and on cyclopentasilane graph. We consider a graph with labeled vertices by 0 or 1 inspired by the molecular structure of cyclopentasilane. In this paper, we first study the existence of solutions to fractional conformable boundary value problem on the cyclopentasilane graph by applying Scheafer and Krasnoselskii fixed point theorems. Furthermore, we investigate different kinds of Ulam stability such as Ulam-Hyers stable, generalized Ulam-Hyers stable, Ulam-Hyers-Rassias stable and generalized Ulam-Hyers-Rassias stable for the given problem. Finally, we give an example to support our important results.

The progression of the still ongoing COVID-19 epidemic must be studied in the world of differential operators other than those specified with integer-order temporal derivatives, according to ongoing scientific studies in the fields of fractional calculus, mathematical modeling, and epidemiology. Infectious diseases leave behind a historical footprint because of their long memory. With this in mind, the article below makes an effort to probe an epidemiological model using a Caputo differential operator with a singular kernel of power-law type. The ability of the Caputo operator to capture the evolution of complicated phenomena has been demonstrated in a number of studies, prompting us to conduct the analysis presented here. The analysis contains solid reasons while using the fractional operator for the COVID-19 epidemiological paradigm and presents the fixed point concept for the existence and uniqueness of its solutions. Hyers–Ulam–Rassias stability aids in finding model equilibrium, and the nonlinear least squares method yields the unknown parameters that also include the model’s fractional order. The actual cases of the infection support the superiority of the Caputo concept with evidence of smaller residuals. The numerical simulations are run to see how varying important parameters affect the disease’s spread.

In this paper, we present a dynamic model of syphilis. The goal of this work is to provide a mathematical model for syphilis disease that contains information about the suspect, infected, recovered, and exposed. Boundedness, positivity, and unique solutions at equilibrium points are used in the qualitative and quantitative study of a proposed model using the power law kernel. The Routh-Hurwitz stability criteria are used to address the model's local and global stability. Chaos control will use the regulate for linear responses approach to bring the system to stabilize according to its points of equilibrium. Using Lyapunov function tests, the global stability of free and endemic equilibrium points is confirmed. The Lipschitz condition and fixed point theory are utilized to satisfy the requirements for the existence and uniqueness of the exact solution. The generalized form of the power law kernel is solved using a two-step Lagrange polynomial method to investigate the impact of the fractional operator using numerical simulations that illustrate the effects of various parameters on the illness.

In this paper, we presented and proved a general Lyapunov's inequality for a class of fractional boundary problems (FBPs) involving a new fractional derivative, named $ \lambda $-Hilfer. We proved a criterion of existence which extended that of Lyapunov concerning the ordinary case. We used this criterion to solve the fractional differential equation (FDE) subject to the Dirichlet boundary conditions. In order to do so, we invoked some properties and essential results of $ \lambda $-Hilfer fractional boundary value problem (HFBVP). This result also retrieved all previous Lyapunov-type inequalities for different types of boundary conditions as mixed. The order that we considered here only focused on $ 1 < r\leq 2 $. General Hartman-Wintner-type inequalities were also investigated. We presented an example in order to provide an application of this result.

Mathematics has several uses for operators on bounded symmetric domains of Bergman spaces including complex geometry, functional analysis, harmonic analysis and operator theory. They offer instruments for examining the interaction between complex function theory and the underlying domain geometry. Here, we extend the Atangana-Baleanu fractional differential operator acting on a special type of class of analytic functions with the m-fold symmetry characteristic in a bounded symmetric domain (we suggest the open unit disk). We explore the most significant geometric properties, including convexity and star-likeness. The boundedness in the weighted Bergman and the convex Bergman spaces associated with a bounded symmetric domain is investigated. A dual relations exist in these spaces. The subordination and superordination inequalities are presented. Our method is based on Young's convolution inequality.

A computer network can detect potential viruses through the use of kill signals, thereby minimizing the risk of virus propagation. In the realm of computer security and defensive strategies, computer viruses play a significant role. Understanding of their spread and extension is a crucial component. To address this issue of computer virus spread, we employ a fractional epidemiological SIRA model by utilizing the Caputo derivative. To solve the fractional-order computer virus model, we employ a computational technique known as the Jacobi collocation operational matrix method. This operational matrix transforms the problem of arbitrary order into a system of nonlinear algebraic equations. To analyze this system of arbitrary order, we derive an approximate solution for the fractional computer virus model, also considering the Vieta Lucas polynomials. Numerical simulations are performed and graphical representations are provided to illustrate the impact of order of the fractional derivative on different profiles.

Abstract This study focuses on improving the accuracy of assessing liver damage and early detection for improved treatment strategies. In this study, we examine the human liver using a modified Atangana-Baleanu fractional derivative based on the mathematical model to understand and predict the behavior of the human liver. The iteration method and fixed-point theory are used to investigate the presence of a unique solution in the new model. Furthermore, the homotopy analysis transform method, whose convergence is also examined, implements the mathematical model. Finally, numerical testing is performed to demonstrate the findings better. According to real clinical data comparison, the new fractional model outperforms the classical integer-order model with coherent temporal derivatives.

Abstract In this manuscript, our primary objective is to delve into the intricacies of an extended nonlinear Schrödinger equation. To achieve this, we commence by deriving a dynamical system tightly linked to the equation through the Galilean transformation. We then employ principles from planar dynamical systems theory to explore the bifurcation phenomena exhibited within this derived system. To investigate the potential presence of chaotic behaviors, we introduce a perturbed term into the dynamical system and systematically analyze the extended nonlinear Schrödinger equation. This investigation is further enriched by the presentation of comprehensive two- and 3D phase portraits. Moreover, we conduct a meticulous sensitivity analysis of the dynamical system using the Runge–Kutta method. Through this analytical process, we confirm that minor fluctuations in initial conditions have only minimal effects on solution stability. Additionally, we utilize the complete discrimination system of the polynomial method to systematically construct single traveling wave solutions for the governing model.

We explore the dynamics of a three-species Lotka-Volterra model incorporating intraguild (IG) predation. The model encompasses interactions between a basal prey, intraguild prey and omnivorous top/intraguild predator. These interactions are characterized by linear functional responses, while considering intraspecific competition and cooperative hunting dynamics. The study involves a comprehensive stability of different steady states and bifurcation analysis. Bifurcation structures unveil shifts in equilibrium stability and the emergence of new equilibrium states. Investigation into dynamics around the coexistence equilibrium highlights diverse behaviors, including stable coexistence, oscillations and chaos. Furthermore, exploration of species' densities under parameter variations uncovers distinct patterns, ranging from stability to chaos. Incorporating the concept of hunting cooperation among IG predators and IG prey can lead to the emergence or suppression of chaotic oscillations, respectively. Additionally, we observe that lower consumption rate of IG predator and cooperation of IG predator helps the system to keep in a stable state position.

In this study for examining the fractional Michaelis-Menten enzymatic reaction (FMMER) model, we suggested a computational method by using an operational matrix of Jacobi polynomials (JPs) as its foundation. We obtain an operational matrix for the arbitrary order derivative in the Caputo sense. The fractional differential equations (FDEs) are then reduced to a set of algebraic equations by using attained operational matrix and the collocation method. The approach which utilized in this study is quicker and more effective compared to other schemes. We also compared the suggested method with the Vieta-Lukas collocation technique (VLCM) and we obtain excellent results. A comparison between numerical outcomes is shown by figures and tables. Error analysis of the recommended methods is also presented.

Chaos, comprehended characteristically, is the mathematical property of a dynamical system which is a deterministic mathematical model in which time can be either continuous or discrete as a variable. These respective models are investigated as mathematical objects or can be employed for describing a target system. As a long-term aperiodic and random-like behavior manifested by many nonlinear complex dynamic systems, chaos induces that the system itself is inherently unstable and disordered, which requires the revealing of representative and accessible paths towards affluence of complexity and experimental processes so that novelty, diversity and robustness can be generated. Hence, complexity theory focuses on non-deterministic systems, whereas chaos theory rests on deterministic systems. These entailments demonstrate that chaos and complexity theory provide a synthesis of emerging wholes of individual components rather than the orientation of analyzing systems in isolation. Therefore, mathematical modeling and scientific computing are among the chief tools to solve the challenges and problems related to complex and chaotic systems through innovative ways ascribed to data science with a precisely tailored approach which can examine the data applied. The complexity definitions need to be weighed over different data offering a highly extensive applicability spectrum with more practicality and convenience owing to the fact that the respective processes lie in the concrete mathematical foundations, which all may as well indicate that the methods are required to be examined thoroughly regarding their mathematical foundation along with the related methods to be applied. Furthermore, making use of chaos theory can be considered to be a way to better understand the internal machinations of neural networks, and the amalgamation of chaos theory as well as Artificial Intelligence (AI) can open up stimulating possibilities acting instrumental to tackle diverse challenges, with AI algorithms providing improvements in the predictive capabilities via the introduction of adaptability, enabling chaos theory to respond to even slight changes in the input data, which results in a higher level of predictive accuracy. Therefore, chaos-based algorithms are employed for the optimization of neural network architectures and training processes. Fractional mathematics, with the application of fractional calculus techniques geared towards the problems’ solutions, describes the existence characteristics of complex natural, applied sciences, scientific, engineering related and medical systems more accurately to reflect the actual state properties co-evolving entities and patterns of the systems concerning nonlinear dynamic systems and modeling complexity evolution with fractional chaotic and complex systems. Complexity entails holistic understanding of various processes through multi-stage integrative models across spanning scales for expounding complex systems while following actuality across evolutionary path. Moreover, Fractional Calculus (FC), related to the dynamics of complicated real-world problems, ensures emerging processes adopting fractional dynamics rather than the ordinary integer-ordered ones, which means the related differential equations feature non integer valued derivatives. Given that slight perturbation leads to a significantly divergent future concatenation of events, pinning down the state of different systems precisely can enable one to unveil uncertainty to some extent. Predicting the future evolution of chaotic systems can screen the direction towards distant horizons with extensive applications in order to understand the internal machinations of neural and chaotic complex systems. Even though many problems are solvable and have been solved, they remain to be open constantly under transient circumstances. Thus, fields with a broad range of spectrum range from mathematics, physics, biology, fluid mechanics, medicine, engineering, image analysis, based on differing perspectives in our special issue which presents a compilation of recent research elaborating on the related advances in foundations, theory, methodology and topic-based implementations regarding fractals, fractal methodology, fractal spline, non-differentiable fractal functions, fractional calculus, fractional mathematics, fractional differential equations, differential equations (PDEs, ODEs), chaos, bifurcation, Lie symmetry, stability, sensitivity, deep learning approaches, machine learning, and so forth through advanced fractional mathematics, fractional calculus, data intensive schemes, algorithms and machine learning applications surrounding complex chaotic systems.

In this paper, we derive a new generalized integral transform on Dzherbashian–Nersesian fractional operator and give some special cases. We make a generalization of the application of integral transformations to different fractional operators, where several previous results can be invoked from a single relation. We also use the new results obtained to solve some fractional differential equations involving the recent revival of Dzherbashian-Nersesian fractional operators.

In this paper, we aim to study the dynamical behaviour of the motion for a simple pendulum with a mass decreasing exponentially in time. To examine this interesting system, we firstly obtain the classical Lagrangian and the Euler-Lagrange equation of the motion accordingly. Later, the generalized Lagrangian is constructed via non-integer order derivative operators. The corresponding non-integer Euler-Lagrange equation is derived, and the calculated approximate results are simulated with respect to different non-integer orders. Simulation results show that the motion of the pendulum with time-dependent mass exhibits interesting dynamical behaviours, such as oscillatory and non-oscillatory motions, and the nature of the motion depends on the order of non-integer derivative; they also demonstrate that utilizing the fractional Lagrangian approach yields a model that is both valid and flexible, displaying various properties of the physical system under investigation. This approach provides a significant advantage in understanding complex phenomena, which cannot be achieved through classical Lagrangian methods. Indeed, the system characteristics, such as overshoot, settling time, and peak time, vary in the fractional case by changing the value of α. Also, the classical formulation is recovered by the corresponding fractional model when α tends to 1, while their output specifications are completely different. These successful achievements demonstrate diverse properties of physical systems, enhancing the adaptability and effectiveness of the proposed scheme for modelling complex dynamics.

In this study, we present a numerical method that utilizes trigonometric cubic B-spline functions to solve the time fractional gas dynamics equation, which is a key component in the study of physical phenomena such as explosions, combustion, detonation and condensation in a moving flow. The Caputo formula is used to define the fractional time derivative, which generalizes the framework for both singular and non-singular kernels. To discretize the unknown function and its derivatives in the spatial direction, we employ trigonometric cubic B-spline functions, while the usual finite difference formulation is used to approximate the Caputo time fractional derivative. A stability analysis of the scheme is provided to ensure that errors do not propagate over time, and a convergence analysis is conducted to measure the accuracy of the solution. To demonstrate the effectiveness of the proposed methodology, we solve various relevant examples and present graphical and tabular results to evaluate the outcomes of the strategy.

Abstract The theoretical approach of random impulsive stochastic integrodifferential equations (RISIDEs) with finite delay, noncompact semigroups, and resolvent operators in Hilbert space is investigated in this article. Initially, a random impulsive stochastic integrodifferential system is proposed and the existence of a mild solution for the system is established using the Mönch fixed-point theorem and contemplating Hausdorff measures of noncompactness. Then, the stability results including a continuous dependence of solutions on initial conditions, exponential stability, and Hyers–Ulam stability for the aforementioned system are investigated. Finally, an example is proposed to validate the obtained results.

A system of fractional differential equations (FDEs) with fractional derivatives of diverse orders is called an incommensurate system of FDEs. In this paper, the well-posedness of the initial value problem for incommensurate systems of FDEs is obtained on the space of continuous functions. Three different methods for this analysis are used and compared. The complexity of such analysis is reduced by new techniques. Strong existence results are obtained by weaker conditions. The uniqueness and the continuous dependency of the solution on initial values are investigated using the Gronwall inequality.

In the current manuscript, new alpha delta dynamic Hardy-Hilbert inequalities on time scales are discussed. These inequalities combine and expand a number of continuous inequalities and their corresponding discrete analogues in the literature. We shall illustrate our results using Hölder's inequality on time scales and a few algebraic inequalities.

Introduction: Recently, a new family of fractional derivatives called the piecewise fractional derivatives has been introduced, arguing that for some problems, each of the classical fractional derivatives may not be able to provide an accurate statement of the consideration problem alone. In defining this kind of derivatives, several types of fractional derivatives can be used simultaneously. Objectives: This study introduces a new kind of piecewise fractional derivative by employing the Caputo type distributed-order fractional derivative and ABC fractional derivative. The one- and two-dimensional piecewise fractional Galilei invariant advection–diffusion equations are defined using this piecewise fractional derivative. Methods: A new class of basis functions called the orthonormal piecewise Vieta-Lucas (VL) functions are defined. Fractional derivatives of these functions in the Caputo and ABC senses are computed. These functions are utilized to construct two numerical methods for solving the introduced problems under non-local boundary conditions. The proposed methods convert solving the original problems into solving systems of algebraic equations. Results: The accuracy and convergence order of the proposed methods are examined by solving several examples. The obtained results are investigated, numerically. Conclusion: This study introduces a kind of piecewise fractional derivative. This derivative is employed to define the one- and two-dimensional piecewise fractional Galilei invariant advection–diffusion equations. Two numerical methods based on the orthonormal VL polynomials and orthonormal piecewise VL functions are established for these problems. The numerical results obtained from solving several examples confirm the high accuracy of the proposed methods.

In this paper, we are going to deal with fractional operators (FOs) with non-singular kernels which is not an easy task because of its restriction at the origin. In this work, we first show the boundedness of the extended form of the modified Atangana-Baleanu (A-B) Caputo fractional derivative operator. The generalized Laplace transform is evaluated for the introduced operator. By using the generalized Laplace transform, we solve some fractional differential equations. The corresponding form of the Atangana-Baleanu Caputo fractional integral operator is also established. This integral operator is proved bounded and obtained its Laplace transform. The existence and Hyers-Ulam stability is explored. In the last results, we studied the relation between our defined operators. The operators in the literature are obtained as special cases for these newly explored FOs.

In this manuscript, we study new solutions of generalized version of geophysical KdV equation which is called generalized perturbed KdV (gpKdV) under time–space conformable operator. We implement two methods to get some novel waves solution of the gpKdV equation. First, we use extended Tanh-method to extract new solutions of considered equations in the form of trigonometric hyperbolic functions. To achieve Sine and Cosine hyperbolic solutions, we use generalized Kudryashov (GK) technique with Riccati equation. We show the behaviour of solutions via 2D and 3D figures. Also, we analyze the Corioles effect on the evolution of waves solutions of the considered equation.

This research introduces novel concepts in sequence theory, including Bessel convergence, Bessel boundedness, Bessel statistical convergence, and Bessel statistical Cauchy sequences. These concepts establish new inclusion relations and related results within mathematical analysis. Additionally, we extend the first and second Korovkin-type approximation theorems by incorporating Bessel statistical convergence, providing a more robust and comprehensive framework than existing results. The practical implications of these theorems are demonstrated through examples involving the classical Bernstein operator and Fejér convolution operators. This work contributes to the foundational understanding of sequence behavior, with potential applications across various scientific disciplines.

This study delves deep into the complexities of the modified nonlinear Schrödinger equation. Through the Galilean transformation, we derive a dynamic system linked to the equation. Using planar dynamical systems theory, we investigate bifurcation phenomena and introduce perturbations to reveal chaotic behaviors. Phase portraits offer visual insights, while sensitivity analysis using the Runge–Kutta method emphasizes solution stability against initial condition variations. Leveraging the planar dynamical system method, we generate diverse solitons, including periodic, bright, and dark solitons. This work enhances our grasp of intricate dynamics and their broader implications.

In this study, steady and incompressible MHD movement of a viscous nanofluid past a non-linear stretching plate has been investigated mathematically by considering heat generation. Blood is employed as the base fluid. Mathematical modeling and analysis are attended in the presence of the permeable medium. Specifically, influences of porosity and heat generation are explored. Numerical solution of the enlisted mathematical model is carried out via an in-built bvp4c tool of MATLAB software. Impact of various parameters associated with the system of ordinary differential equations (ODEs) is reported through graphical depicturation and scrutinized thoroughly. Computed outcomes demonstrated that the increment in the porous media parameter caused a reduction in the velocity profile, while converse outcomes were sighted for the temperature and mass distribution. Temperature profile shows a boost with an increase in heat generation parameter whereas concentration profile indicates a non-monotonic behavior. Analogy between the results obtained in this study and already published work has been established. Thus, by adjusting the controlling parameters, the heating/cooling mechanism can be created. This will help the engineers in designing the application that needs high-temperature processing operations.

This paper investigates the existence, uniqueness and stability of solutions to the nonlinear Volterra–Fredholm integral equations (NVFIE) involving the Erdélyi–Kober (E–K) fractional integral operator. We use the Leray–Schauder alternative and Banach’s fixed point theorem to examine the existence and uniqueness of solutions, and we also explore Hyers–Ulam (H–U) and Hyers–Ulam–Rassias (H–U–R) stability in the space C([0,β],R). Furthermore, three solution sets Uσ,λ, Uθ,1 and U1,1 are constructed for σ>0, λ>0, and θ∈(0,1), and then we obtain local stability of the solutions with some ideal conditions and by using Schauder fixed point theorem on these three sets, respectively. Also, to achieve the goal, we choose the parameters for the NVFIE as δ∈(12,1), ρ∈(0,1), γ>0. Three examples are provided to clarify the results.

The world is grappled with an unprecedented challenges due to Corona virus. We are all battling this epidemic together, but we have not been able to defeat this epidemic yet. A new variant of this virus, named ‘Omicron’ is spreading these days. The fractional differential equations are providing us with better tools to study the mathematical model with memory effects. In this paper, we will consider an extended SER mathematical model with quarantined and vaccinated compartment to speculate the Omicron variant. This extended Susceptible Exposed Infected Recovered SER model involves equations that associate with the group of individuals those are susceptible (S), exposed (E): this class includes the individuals who are infected but not yet infectious, infectious (W): this class includes the individuals who are infected but not yet Quarantined, quarantined (Q): this class includes those group of people who are infectious, confirmed and quarantined, recovered (R) this class includes the group of individuals who have recovered, and vaccinated (V): this class includes the group of individuals who have been vaccinated. The non-negativity and of the extended SER model is analysed, the equilibrium points and the basic reproduction number are also calculated. The proposed model is then extended to the mathematical model using AB derivative operator. Proof for the existence and the uniqueness for the solution of fractional mathematical model in sense of AB fractional derivative is detailed and a numerical method is detailed to obtain the numerical solutions. Further we have discussed the efficiency of the vaccine against the Omicron variant via graphical representation.

The present article introduces and studies the Fredholm-type integral equation with an incomplete I-function (IIF) and an incomplete $ \bar {I} $ -function (I $ \bar {I} $ F) in its kernel. First, using fractional calculus and the Mellin transform principle, we solve an integral problem involving IIF. The idea of the Mellin transform and fractional calculus is then used to analyse an integral equation using the incomplete $ \bar {I} $ -function. This is followed by the discovery and investigation of several important exceptional cases. This article's general discoveries may yield new integral equations and solutions. The desired outcomes seem to be very helpful in resolving many real-world problems whose solutions represent different physical phenomena. And also, findings help solve introdifferential, fractional differential, and extended integral equation problems.

Calcium is an essential element in our body and plays a vital role in moderating calcium signalling. Calcium is also called the second messenger. Calcium signalling depends on cytosolic calcium concentration. In this study, we focus on cellular calcium fluctuations with different buffers, including calcium-binding buffers, using the Hilfer fractional advection-diffusion equation for cellular calcium. Limits and start conditions are also set. By combining with intracellular free calcium ions, buffers reduce the cytosolic calcium concentration. The buffer depletes cellular calcium and protects against toxicity. Association, dissociation, diffusion, and buffer concentration are modelled. The solution of the Hilfer fractional calcium model is achieved through utilizing the integral transform technique. To investigate the influence of the buffer on the calcium concentration distribution, simulations are done in MATLAB 21. The results show that the modified calcium model is a function of time, position, and the Hilfer fractional derivative. Thus the modified Hilfer calcium model provides a richer physical explanation than the classical calcium model.

In this paper, under some conditions in the Banach space $ C ([0, \beta], \mathbb{R}) $, we establish the existence and uniqueness of the solution for the nonlinear integral equations involving the Riemann-Liouville fractional operator (RLFO). To establish the requirements for the existence and uniqueness of solutions, we apply the Leray-Schauder alternative and Banach's fixed point theorem. We analyze Hyers-Ulam-Rassias (H-U-R) and Hyers-Ulam (H-U) stability for the considered integral equations involving the RLFO in the space $ C([0, \beta], \mathbb{R}) $. Also, we propose an effective and efficient computational method based on Laguerre polynomials to get the approximate numerical solutions of integral equations involving the RLFO. Five examples are given to interpret the method.

This study is devoted to examine the existence and uniqueness behavior of a nonlinear integro-differential equation of Volterra-Fredholm integral type in continues space. Then, we examine its solution by modification of Adomian and homotopy analysis methods numerically. Initially, the proposed model is reformulated into an abstract space, and the existence and uniqueness of solution is constructed by employing Arzela-Ascoli and Krasnoselskii fixed point theorems. Furthermore, suitable conditions are developed to prove the proposed model's continues behavior which reflects the stable generation. At last, three test examples are presented to verify the established theoretical concepts.