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Norovirus infection has been documented to have a significant economic impact in different regions of the world. Even though young children bear the greatest economic burden, older age groups in some locations have the highest costs per illness. Most of these costs result from lost production caused by acute illnesses. This viral infection causes inflammation of the intestines and stomach, also called stomach flu and food poisoning. Our research work constructs a new compartmental model of norovirus infection based on contaminated water and food contamination to conceptualize how norovirus spreads. The proposed dynamics of norovirus have been presented in the fractional Caputo framework. We present the Caputo operator's rudimentary results for the system's examination. By applying the fixed point theory to the system, the existence theory has been investigated. To inspect the solution pathways of norovirus infection, we introduce a novel numerical scheme to explore the dynamical behavior of the system. Finally, a numerical investigation has been done to show the impact of different factors of the system for the control of norovirus illness. We suggested the most critical factors of the system to the policymakers to control and prevent infection in the community. [ABSTRACT FROM AUTHOR]

Abstract Recently, the area devoted to fractional calculus has given much attention by researchers. The reason behind such huge attention is the significant applications of the mentioned area in various disciplines. Different problems of real world processes have been investigated by using the concepts of fractional calculus and important and applicable outcomes were obtained. Because, there has been a lot of interest in fractional differential equations. It is brought on by both the extensive development of fractional calculus theory and its applications. The use of linear and quadratic perturbations of nonlinear differential equations in mathematical models of a variety of real-world problems has received a lot of interest. Therefore, motivated by the mentioned importance, this research work is devoted to analyze in detailed, a class of fractal hybrid fractional differential equation under Atangana- Baleanu- Caputo ABC derivative. The qualitative theory of the problem is examined by using tools of non-linear functional analysis. The Ulam-Hyer’s (U-H) type stability criteria is also applied to the consider problem. Further, the numerical solution of the model is developed by using powerful numerical technique. Lastly, the Wazewska-Czyzewska and Lasota Model, a well-known biological model, verifies the results. Several graphical representations by using different fractals fractional orders values are presented. The detailed discussion and explanations are given at the end.

Recently, the area devoted to fractional calculus has given much attention by researchers. The reason behind such huge attention is the significant applications of the mentioned area in various disciplines. Different problems of real world processes have been investigated by using the concepts of fractional calculus and important and applicable outcomes were obtained. Because, there has been a lot of interest in fractional differential equations. It is brought on by both the extensive development of fractional calculus theory and its applications. The use of linear and quadratic perturbations of nonlinear differential equations in mathematical models of a variety of real-world problems has received a lot of interest. Therefore, motivated by the mentioned importance, this research work is devoted to analyze in detailed, a class of fractal hybrid fractional differential equation under Atangana- Baleanu- Caputo ABC derivative. The qualitative theory of the problem is examined by using tools of non-linear functional analysis. The Ulam-Hyer's (U-H) type stability criteria is also applied to the consider problem. Further, the numerical solution of the model is developed by using powerful numerical technique. Lastly, the Wazewska-Czyzewska and Lasota Model, a well-known biological model, verifies the results. Several graphical representations by using different fractals fractional orders values are presented. The detailed discussion and explanations are given at the end. [ABSTRACT FROM AUTHOR]

In this dissertation, we study the coupled system of q$$ q $$‐fractional Langevin differential equations involving q$$ q $$‐Caputo derivative having q$$ q $$‐fractional integral conditions. With the help of some adequate conditions, we investigate the uniqueness and existence of mild solution of the aforementioned system. We also analyze various kinds of Ulam's stability. Banach fixed point theorem and Leray–Schauder of cone type are used to illustrate the existence and uniqueness results. We also used non‐linear functional analysis methods to explore variety of stability types. An example is provided to clearly demonstrate our theoretical outcomes. [ABSTRACT FROM AUTHOR]

This manuscript investigates the existence, uniqueness, and different forms of Ulam stability for a system of three coupled differential equations involving the Riemann–Liouville (RL) fractional operator. The Leray–Schauder alternative is employed to confirm the existence of solutions, while the Banach contraction principle is used to establish their uniqueness. Stability conditions are derived utilizing classical nonlinear functional analysis techniques. Theoretical findings are illustrated with an example. The proposed system generalizes third-order ordinary differential equations (ODEs) with different boundary conditions (BCs). [ABSTRACT FROM AUTHOR]

In the Caputo sense, the goal of this paper is to develop a thorough analysis for the Nipah virus sickness under piecewise equations with fractional order derivative. Here, we take into account a SIRD-type model with classes for susceptible, infected, recovered, and dead individuals. We evaluate the solution's viability using the notion of fractional order derivative. The fundamental reproduction number is also calculated. Here, we offer sufficient requirements for the Ulam–Hyers (UH) stability and the existence theory of the solution. We also give some numerical simulations for several compartments using the RK4 approach. In order to deepen our investigation, we therefore investigate the aforementioned disease's transmission process using some novel fractional calculus notions. [ABSTRACT FROM AUTHOR]

Variable-order derivatives are the natural extension of ordinary as well as of fractional-order differentiations and integration, respectively. Numerous suggestions for fractional variable-order operators have been made in the literature over time. Therefore, this is the moment to shine a light on the variable-order fractional calculus, due to the fact that it accurately describes the mathematical underpinnings and emphasizing the modeling utility via using contemporary numerical techniques. This study focuses on investigating a fractional variable-order model of lymphatic filariasis infection using with Atangana–Beleanue–Caputo derivative. Our investigations have led to the development of newly refined results, focusing on both qualitative and numerical aspects of analysis. To achieve our research objectives, we employ the fixed point theorems of Banach and Krasnoselskii. These theorems serve as powerful tools, allowing us to establish results regarding the existence of solutions to the model. Additionally, for precise numerical simulations, we employ the fractional Euler’s method, a sophisticated computational technique that allows us to effectively simulate and interpret the results both numerically and graphically. These graphs illustrate distinct variable-orders, providing a comprehensive understanding of the model’s behavior under different conditions. Here, it should be kept in mind that we have select various continuous functions for variable to present our graphical illustration.

Recently, fractals and fractional calculus have received much attention from researchers of various fields of science and engineering. Because the said area has been found applicable in modeling various real-world processes and phenomena. Hybrid differential equations (HDEs) play significant roles in mathematical modeling of various processes because the aforesaid equations incorporate different dynamical systems as specific cases. For instance, it is possible to model and describe non-homogeneous physical phenomena on using the said equations. Therefore, this research work is concerned with studying a class of nonlinear hybrid fractal–fractional differential equations. We develop the existence result for the qualitative study using a hybrid fixed point theorem. For the mentioned goal, a fixed point theory for the product of two operators is applied to deduce appropriate conditions for the existence of exactly one solution. Additionally, the stability result based on Ulam–Hyers is also deduced. The said stability results play an important role in numerical investigations. In addition, a numerical method based on Euler procedure is utilized to approximate the solution of the proposed problems. Various computational test problems are given to demonstrate the results. Also, using various fractal–fractional order values, several graphical presentations are given for the examples. The concerned analysis will help in investigating many real-world problems modeled using HDEs with fractal–fractional orders in the near future.

This manuscript established a comprehensive analysis of a general class of fractional order delay differential equations with Caputo-Fabrizio fractional derivative (CFFD). Functional analysis was used to examine the existence and uniqueness of the suggested class and to generate sufficient requirements for Ulam-Hyers (UH) type stability. Further, a numerical method based on Lagrange interpolation is used to compute approximate solution. Then, some applications in physical dynamics including a houseflies model and a Cauchy type problem were discussed to illustrate the established analysis with graphical illustrations.

This manuscript established a comprehensive analysis of a general class of fractional order delay differential equations with Caputo-Fabrizio fractional derivative (CFFD). Functional analysis was used to examine the existence and uniqueness of the suggested class and to generate sufficient requirements for Ulam-Hyers (UH) type stability. Further, a numerical method based on Lagrange interpolation is used to compute approximate solution. Then, some applications in physical dynamics including a houseflies model and a Cauchy type problem were discussed to illustrate the established analysis with graphical illustrations. [ABSTRACT FROM AUTHOR]

We first show how the classical Marguerre–von Kármán equations modeling the deformation of a nonlinearly elastic shallow shell can be recast as equations whose sole unknowns are the bending moments and stress resultants inside the middle surface of the shell. Thus, these equations allow to compute the stresses inside the shell without having to compute first the displacement field. We then show that the boundary value problem formed by these new equations is well posed by establishing an existence theorem. [ABSTRACT FROM AUTHOR]

We first show that the minimization problem associated with the nonlinear shell model of W.T. Koiter becomes coercive over its natural space of admissible deformations when the third fundamental form is added to its functional. Then, under the assumption that the middle surface of the shell is a minimal surface, we approach this minimization problem by a new minimization problem that is well-posed over the same space of admissible deformations. [ABSTRACT FROM AUTHOR]

This manuscript investigates the existence, uniqueness, and different forms of Ulam stability for a system of three coupled differential equations involving the Riemann–Liouville (RL) fractional operator. The Leray–Schauder alternative is employed to confirm the existence of solutions, while the Banach contraction principle is used to establish their uniqueness. Stability conditions are derived utilizing classical nonlinear functional analysis techniques. Theoretical findings are illustrated with an example. The proposed system generalizes third-order ordinary differential equations (ODEs) with different boundary conditions (BCs).

In this work, an epidemic model of a susceptible, exposed, infected and recovered SEIR-type is established for the distinctive dynamic compartments and epidemic characteristics of COVID-19 as it spreads across a population with a heterogeneous rate. The proposed model is investigated using a novel approach of fractional calculus known as piecewise derivatives. The existence theory is demonstrated through the establishment of sufficient conditions. In addition, result related to Hyers-Ulam stability is also derived for the considered model. A numerical method based on modified Euler procedure is also constructed to simulate the approximate solutions of the proposed model by employing various values of fractional orders. We testified the numerical results by using real available data of Japan. In addition, some results for the SEIR-type model are also presented graphically using the stochastic process, and the obtained results are discussed.

We study the Cauchy-Dirichlet problem for a class of nonlinear parabolic equations driven by nonstandard \(p(x,t),q(x,t)\)-growth condition. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces, derive global and local in time \(L^{\infty}\) bounds for the weak solutions.

In this note we present some existence results, in the spirit of the celebrated paper by Brezis and Nirenberg (CPAM, 1983), for a perturbed critical problem driven by a mixed local and nonlocal linear operator. We develop an existence theory, both in the case of linear and superlinear perturbations; moreover, in the particular case of linear perturbations we also investigate the mixed Sobolev inequality associated with this problem, detecting the optimal constant, which we show that is never achieved.

We study the Cauchy–Dirichlet problem for a class of nonlinear parabolic equations driven by nonstandard p(x, t), q(x, t)-growth condition. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz–Sobolev spaces, derive global and local in time L∞ bounds for the weak solutions. [ABSTRACT FROM AUTHOR]

In this work, an epidemic model of a susceptible, exposed, infected and recovered SEIRtype is established for the distinctive dynamic compartments and epidemic characteristics of COVID-19 as it spreads across a population with a heterogeneous rate. The proposed model is investigated using a novel approach of fractional calculus known as piecewise derivatives. The existence theory is demonstrated through the establishment of sufficient conditions. In addition, result related to Hyers-Ulam stability is also derived for the considered model. A numerical method based on modified Euler procedure is also constructed to simulate the approximate solutions of the proposed model by employing various values of fractional orders. We testified the numerical results by using real available data of Japan. In addition, some results for the SEIR-type model are also presented graphically using the stochastic process, and the obtained results are discussed. [ABSTRACT FROM AUTHOR]

In this study, we explore the epidemic spread of the coronavirus using the Caputo fractional variable order derivative as variable order derivative provides a natural extension to classical as well as fractional order derivatives. Using the variable order derivatives in investigation of biological models of infectious diseases is an important area of research in the current time. Using the fixed point technique, we discuss the existence and uniqueness of solution to the corona virus infectious disease 2019 environmental transformation model. In order to demonstrate the existence and novelty of our findings, we examine the results numerically and graphically with the help of Euler’s method. There are several graphs provided that are related to different variable orders.

Multiple sclerosis (MS) is a common neurological disorder that affects the central nervous system (CNS) and can cause lesions that spread over space and time. Our study proposes a mathematical model that illustrates the progression of the disease and its likelihood of recurrence. We use Caputo fractional-order (FO) derivative operators to represent non-negative solutions and to establish a steady-state point and basic reproductive number. We also employ functional analysis to prove the existence of unique solutions and use the Ulam-Hyres (UH) notion to demonstrate the stability of the solution for the proposed model. Furthermore, we conduct numerical simulations using an Euler-type numerical technique to validate our theoretical results. Our findings are presented through graphs that depict various behaviors of the model for different parameter values.

The study is devoted to demonstrating the hypothesis necessary for the existence, uniqueness and at least one solution of implicit Hadamard fractional differential equations with nonlocal semi-coupled fractional integro-multipoints boundary conditions. Using Banach contraction principle, the uniqueness result is obtained. Leray–Schauder fixed point theorem supports the results of at least one solution of the aforementioned system. Additionally, By two different approaches, Hyers–Ulam stabilities are discussed for the proposed system. For verification, an example is presented. Through Remarks, ordinary differential equation of second-order with periodic and initial conditions are given by fixing some parameters in the suggested system.

Multiple sclerosis (MS) is a common neurological disorder that affects the central nervous system (CNS) and can cause lesions that spread over space and time. Our study proposes a mathematical model that illustrates the progression of the disease and its likelihood of recurrence. We use Caputo fractional-order (FO) derivative operators to represent non-negative solutions and to establish a steady-state point and basic reproductive number. We also employ functional analysis to prove the existence of unique solutions and use the Ulam-Hyres (UH) notion to demonstrate the stability of the solution for the proposed model. Furthermore, we conduct numerical simulations using an Euler-type numerical technique to validate our theoretical results. Our findings are presented through graphs that depict various behaviors of the model for different parameter values. [ABSTRACT FROM AUTHOR]

This article outlines the necessary requirements for a coupled system of fractional order boundary value involving the Caputo fractional derivative, including its existence, uniqueness, and various forms of Ulam stability. We demonstrate the existence and uniqueness of the proposed coupled system by using the cone-type Leray–Schauder result and the Banach contraction principle. Based on the traditional method of nonlinear functional analysis, the stability is examined. An example is used to provide a clear illustration of our main results. [ABSTRACT FROM AUTHOR]

We study the homogeneous Dirichlet problem for a class of nonlinear anisotropic parabolic double phase equations with nonstandard growth conditions. We prove the existence of a unique nonnegative weak solution in suitable Orlicz-Sobolev spaces, and derive the global boundedness of the weak solution. [ABSTRACT FROM AUTHOR]

We provide a method for constructing (possibly non-trivial) measures on non-locally compact Polish subspaces of infinite-dimensional separable Banach spaces which, under suitable assumptions, are minimizers of causal variational principles in the non-locally compact setting. Moreover, for non-trivial minimizers the corresponding Euler–Lagrange equations are derived. The method is to exhaust the underlying Banach space by finite-dimensional subspaces and to prove existence of minimizers of the causal variational principle restricted to these finite-dimensional subsets of the Polish space under suitable assumptions on the Lagrangian. This gives rise to a corresponding sequence of minimizers. Restricting the resulting sequence to countably many compact subsets of the Polish space, by considering the resulting diagonal sequence, we are able to construct a regular measure on the Borel algebra over the whole topological space. For continuous Lagrangians of bounded range, it can be shown that, under suitable assumptions, the obtained measure is a (possibly non-trivial) minimizer under variations of compact support. Under additional assumptions, we prove that the constructed measure is a minimizer under variations of finite volume and solves the corresponding Euler–Lagrange equations. Afterwards, we extend our results to continuous Lagrangians vanishing in entropy. Finally, assuming that the obtained measure is locally finite, topological properties of spacetime are worked out and a connection to dimension theory is established. [ABSTRACT FROM AUTHOR]

The present research work is basically devoted to construction of a fractional order differential equation with time delay. Initially, integral representation is given to solution of the underline problem. Afterwards, operator form of solution is studied under some auxiliary hypothesis. Since uniqueness of solution is required, therefore we also provide results for exploring the uniqueness of solution for the underlying model. Using Lebesgue dominated convergence theorem and some other results from analysis, this work provides results devoted to existence of at least one solution. Also, for investigating the nature of solution for the proposed model, we study different kind of stability analysis. These stability related results show, how the solution behave with time. At the end of the article, we illustrate the obtained results via some examples. [ABSTRACT FROM AUTHOR]

This study uses variable order differentiation and integration to investigate the disease dynamical model of COVID-19. Here, we update the results of the qualitative and quantitative analysis. We obtain necessary conclusions for the existence theory of the solution to the suggested model in order to satisfy the aforementioned criteria using fixed point theories of Banach and Schauder. Additionally, we simulate the outcomes mathematically and graphically using the Euler modified technique for numerical purposes. There are several graphs provided that relate to various variable ordering. In addition, we compare our simulated results with the real data results also in case of infected class.

This research work is devoted to undertaking a dynamical system representing SARS-CoV-19 disease under the concept of piecewise fractional-order derivative using the Caputo concept since long-memory and short-memory terms are not well explained by ordinary fractional differential equations. It has been found that for such disruption, piecewise operators of fractional derivatives have been found useful in many cases. Therefore, we study a compartmental model of susceptible and infected individuals under the concept of piecewise derivative. We establish the existence theory of the considered model by using some Banach and Schauder fixed-point theorems. Keeping the importance of stability, a pertinent result related to the said area is also developed. The said concept of stability is based on the concept given by Ulam and Hyers. Further, to derive the numerical results, we use the Euler method to develop a numerical scheme for the considered model. Using real available data, we have presented various graphical presentations of two compartments against different fractional orders and various values of isolation parameters. The crossover behaviors in the dynamics can be clearly observed, which is explained by the piecewise operators, not the usual fractional-order derivative. [ABSTRACT FROM AUTHOR]

In the given manuscript, the fractional mathematical model for the current pandemic of COVID-19 is investigated. The model is composed of four agents of susceptible (S) , infectious (I) , quarantined (Q) and recovered (R) cases respectively. The fractional operator of Atangana-Baleanu-Caputo (ABC) is applied to the considered model for the fractional dynamics. The basic reproduction number is computed for the stability analysis. The techniques of existence and uniqueness of the solution are established with the help of fixed point theory. The concept of stability is also derived using the Ulam-Hyers stability technique. With the help of the fractional order numerical method of Adams-Bashforth, we find the approximate solution of the said model. The obtained scheme is simulated on different fractional orders along with the comparison of integer orders. Varying the numerical values for the contact rate ζ , different simulations are performed to check the effect of it on the dynamics of COVID-19. [ABSTRACT FROM AUTHOR]

In this research work, we establish some new results about piecewise equation involving Caputo Fabrizio derivative (CFD). The concerned class has been recently introduced and these results are fundamental for investigation of qualitative theory and numerical interpretation. We derive some necessary results for the existence, uniqueness and various form of Hyers-Ulam (H-U) type stability for the considered problem. For the required results, we need to utilize usual classical fixed point theorems due to Banach and Krasnoselskii's. Moreover, results devoted to H-U stability are derived by using classical tools of nonlinear functional analysis. Some pertinent test problems are given to demonstrate our results.

This manuscript is devoted to investigate a class of multi terms delay fractional order impulsive differential equations. Our investigation includes existence theory along with Ulam type stability. By using classical fixed point theorems, we establish sufficient conditions for existence and uniqueness of solution to the proposed problem. We develop some appropriate conditions for different kinds of Ulam-Hyers stability results by using tools of nonlinear functional analysis. We demonstrate our results by an example.

The fundamental purpose of this research is to investigate the existence theory as well as the uniqueness of solutions to a coupled system of fractional order differential equations with Caputo derivatives. In this regard, we utilize the definition and properties of a newly developed conception of complex valued fuzzy rectangular b-metric spaces to explore the fuzzy form of some significant fixed point and coupled fixed point results. We further present certain examples and a core lemma in the case of complex valued fuzzy rectangular b-metric spaces.

In this paper, we establish a qualitative theory for implicit fractional order differential equations (IFODEs) with nonlocal initial condition (NIC) with delay term. Because area related to investigate existence and uniqueness of solution is important field in recent times. Also researchers are using existence theory to derive some prior results about a dynamical problem weather it exists or not in reality. In literature, we have different tools to study qualitative nature of a problem. On the same line the exact solution of every problem is difficult to determined. Therefore, we use technique of numerical analysis to approximate the solutions, where stability analysis is an important aspect. Therefore, we use a tool from non-linear analysis known as topological degree theory to develop sufficient conditions for existence and uniqueness of solution to the considered problem. Further, we also develop sufficient conditions for Hyers- Ulam type stability for the considered problem. To justify our results, we also give an illustrative example.

After situating Andreas Reckwitz's The Society of Singularities within the broader context of the tradition of social theory, we discuss in detail the obvious strengths of this book, notably its impressive range and originality. Subsequently, we elaborate on two limitations of Reckwitz's argument. Firstly, we argue that Reckwitz's use of categories such as 'singularity' and 'universality' is too all-embracing, lacking the clarity and focus needed to sustain a productive line of inquiry. Secondly, and related to the previous point, we contend that Reckwitz's claims about the recent trend towards increasing singularity are so broad that they are difficult to refute empirically. Further, we discuss briefly contemporary political developments to demonstrate why the core societal issues at stake cannot be explained through all-inclusive categories such as singularity. Finally, we maintain that existence theory can provide an alternative fruitful perspective on some of the phenomena discussed in this book. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this research work, we establish some new results about piecewise equation involving Caputo Fabrizio derivative (CFD). The concerned class has been recently introduced and these results are fundamental for investigation of qualitative theory and numerical interpretation. We derive some necessary results for the existence, uniqueness and various form of Hyers-Ulam (H-U) type stability for the considered problem. For the required results, we need to utilize usual classical fixed point theorems due to Banach and Krasnoselskii's. Moreover, results devoted to H-U stability are derived by using classical tools of nonlinear functional analysis. Some pertinent test problems are given to demonstrate our results. [ABSTRACT FROM AUTHOR]

We prove the existence of minimizers of causal variational principles on second countable, locally compact Hausdorff spaces. Moreover, the corresponding Euler–Lagrange equations are derived. The method is to first prove the existence of minimizers of the causal variational principle restricted to compact subsets for a lower semi-continuous Lagrangian. Exhausting the underlying topological space by compact subsets and rescaling the corresponding minimizers, we obtain a sequence which converges vaguely to a regular Borel measure of possibly infinite total volume. It is shown that, for continuous Lagrangians of compact range, this measure solves the Euler–Lagrange equations. Furthermore, we prove that the constructed measure is a minimizer under variations of compact support. Under additional assumptions, it is proven that this measure is a minimizer under variations of finite volume. We finally extend our results to continuous Lagrangians decaying in entropy. [ABSTRACT FROM AUTHOR]

This manuscript is devoted to investigate a class of multi terms delay fractional order impulsive differential equations. Our investigation includes existence theory along with Ulam type stability. By using classical fixed point theorems, we establish sufficient conditions for existence and uniqueness of solution to the proposed problem. We develop some appropriate conditions for different kinds of Ulam-Hyers stability results by using tools of nonlinear functional analysis. We demonstrate our results by an example. [ABSTRACT FROM AUTHOR]

Abstract The major goal of this work is investigating sufficient conditions for the existence and uniqueness of solutions for implicit impulsive coupled system of φ-Hilfer fractional differential equations (FDEs) with instantaneous impulses and terminal conditions. First, we derive equivalent fractional integral equations of the proposed system. Next, by employing some standard fixed point theorems such as Leray–Schauder alternative and Banach, we obtain the existence and uniqueness of solutions. Further, by mathematical analysis technique we investigate the Ulam–Hyers (UH) and generalized UH (GUH) stability of solutions. Finally, we provide a pertinent example to corroborate the results obtained.

A fractional system for transmission dynamics behavior of influenza infection with two strains is investigated and applied to the qualitative assess of crossing immunities on the expansion of the disease. The generalized mathematical model is significantly more informative regarding the infection outbreak and gives a deep description of the infection’s dynamic process. We firstly take the integer order system given in current literature and then we globalize it by applying the Caputo operator of fractional derivatives. Moreover, we present the rudimentary results of the Caputo operator for the examination of the system, comprising the basic reproduction number R0 and equilibrium points stabilities with the help of the Routh–Hurwitz and the Lyapunov operators techniques. We also focus on the approximate root. For simulations, we apply the famous generalize predictor–corrector Adams–Bashforth Moulton method, which shows the impact of two strains of influenza disease.

This paper provides the effects of nutrients concentrations on the algae growth via solitary wave analysis and classical existence theory. There is a well known Diffusive Nutrient-Algae model based on the Sanyang wetland, is under consideration. The new families of exact solutions which represent nutrient concentration and algae density in different form are observed using the ϕ6-model expansion. Furthermore, the existence of these solutions are also discussed under different restrictive conditions and variables of nutrient concentration and algae density are represented in hyperbolic, trigonometric and rational forms. The graphics of these solutions are also sketched in 3D and 2D representations.

This article outlines the necessary requirements for a coupled system of fractional order boundary value involving the Caputo fractional derivative, including its existence, uniqueness, and various forms of Ulam stability. We demonstrate the existence and uniqueness of the proposed coupled system by using the cone-type Leray–Schauder result and the Banach contraction principle. Based on the traditional method of nonlinear functional analysis, the stability is examined. An example is used to provide a clear illustration of our main results.