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The main objective of the paper is to study the final model for the Kirchhoff-type parabolic system. Such type problems have many applications in physical and biological phenomena. Under some smoothness of the final Cauchy data, we prove that the problem has a unique mild solution. The main tool is Banach's fixed point theorem. We also consider the non-well-posed problem in the Hadamard sense. Finally, we apply truncation method to regularize our problem. The paper is motivated by the work of Tuan, Nam, and Nhat [Comput. Math. Appl. 77(1):15–33, 2019]. [ABSTRACT FROM AUTHOR]

In this paper we review some of the previous work done by the earlier authors (Singh et al. in J. Inequal. Appl. 2017:101, 2017; Lal and Shireen in Bull. Math. Anal. Appl. 5(4):1–13, 2013), etc., on error approximation of a function g in the generalized Zygmund space and resolve the issue of these works. We also determine the best error approximation of the functions g and g ′ , where g ′ is a derived function of a 2π-periodic function g, in the generalized Zygmund class X z (η) , z ≥ 1 , using matrix-Cesàro (T C δ) means of its Fourier series and its derived Fourier series, respectively. Theorem 2.1 of the present paper generalizes eight earlier results, which become its particular cases. Thus, the results of (Dhakal in Int. Math. Forum 5(35):1729–1735, 2010; Dhakal in Int. J. Eng. Technol. 2(3):1–15, 2013; Nigam in Surv. Math. Appl. 5:113–122, 2010; Nigam in Commun. Appl. Anal. 14(4):607–614, 2010; Nigam and Sharma in Kyungpook Math. J. 50:545–556, 2010; Nigam and Sharma in Int. J. Pure Appl. Math. 70(6):775–784, 2011; Kushwaha and Dhakal in Nepal J. Sci. Technol. 14(2):117–122, 2013; Shrivastava et al. in IOSR J. Math. 10(1 Ver. I):39–41, 2014) become particular cases of our Theorem 2.1. Several corollaries are also deduced from our Theorem 2.1. [ABSTRACT FROM AUTHOR]

In this paper, we establish inequalities of Hermite–Hadamard type for harmonically convex functions using a generalized fractional integral. The results of our paper are an extension of previously obtained results (İşcan in Hacet. J. Math. Stat. 43(6):935–942, 2014 and İşcan and Wu in Appl. Math. Comput. 238:237–244, 2014). We also discuss some special cases for our main results and obtain new inequalities of Hermite–Hadamard type. [ABSTRACT FROM AUTHOR]

Abstract The main objective of the paper is to study the final model for the Kirchhoff-type parabolic system. Such type problems have many applications in physical and biological phenomena. Under some smoothness of the final Cauchy data, we prove that the problem has a unique mild solution. The main tool is Banach’s fixed point theorem. We also consider the non-well-posed problem in the Hadamard sense. Finally, we apply truncation method to regularize our problem. The paper is motivated by the work of Tuan, Nam, and Nhat [Comput. Math. Appl. 77(1):15–33, 2019].

Abstract The purpose of this paper is to provide sufficient conditions for the local and global existence of solutions for the general nonlinear distributed-order fractional differential equations in the time domain. Also, we provide sufficient conditions for the uniqueness of the solutions. Furthermore, we use operational matrices for the fractional integral operator of the second kind Chebyshev wavelets and shifted fractional-order Jacobi polynomials via Gauss–Legendre quadrature formula and collocation methods to reduce the proposed equations into systems of nonlinear equations. Also, error bounds and convergence of the presented methods are investigated. In addition, the presented methods are implemented for two test problems and some famous distributed-order models, such as the model that describes the motion of the oscillator, the distributed-order fractional relaxation equation, and the Bagley–Torvik equation, to demonstrate the desired efficiency and accuracy of the proposed approaches. Comparisons between the methods proposed in this paper and the existing methods are given, which show that our numerical schemes exhibit better performances than the existing ones.

Abstract In this paper, we introduce multi-Lah numbers and multi-Stirling numbers of the first kind and recall multi-Bernoulli numbers, all of whose generating functions are given with the help of multiple logarithm. The aim of this paper is to study several relations among those three kinds of numbers. In more detail, we represent the multi-Bernoulli numbers in terms of the multi-Stirling numbers of the first kind and vice versa, and the multi-Lah numbers in terms of multi-Stirling numbers. In addition, we deduce a recurrence relation for multi-Lah numbers.

Abstract The aim of this paper is to introduce the degenerate generalized Laguerre polynomials as the degenerate version of the generalized Laguerre polynomials and to derive some properties related to those polynomials and Lah numbers, including an explicit expression, a Rodrigues type formula, and expressions for the derivatives. The novelty of the present paper is that it is the first paper on degenerate versions of orthogonal polynomials.

Abstract An approach to the generalized Bessel–Maitland function is proposed in the present paper. It is denoted by J ν , λ μ $\mathcal{J}_{\nu , \lambda }^{\mu }$ , where μ > 0 $\mu >0$ and λ , ν ∈ C $\lambda ,\nu \in \mathbb{C\ }$ get increasing interest from both theoretical mathematicians and applied scientists. The main objective is to establish the integral representation of J ν , λ μ $\mathcal{J}_{\nu ,\lambda }^{\mu }$ by applying Gauss’s multiplication theorem and the representation for the beta function as well as Mellin–Barnes representation using the residue theorem. Moreover, the mth derivative of J ν , λ μ $\mathcal{J}_{\nu ,\lambda }^{\mu }$ is considered, and it turns out that it is expressed as the Fox–Wright function. In addition, the recurrence formulae and other identities involving the derivatives are derived. Finally, the monotonicity of the ratio between two modified Bessel–Maitland functions I ν , λ μ $\mathcal{I}_{\nu ,\lambda }^{\mu }$ defined by I ν , λ μ ( z ) = i − 2 λ − ν J ν , λ μ ( i z ) $\mathcal{I}_{\nu ,\lambda }^{\mu }(z)=i^{-2\lambda -\nu }\mathcal{J}_{ \nu ,\lambda }^{\mu }(iz)$ of a different order, the ratio between modified Bessel–Maitland and hyperbolic functions, and some monotonicity results for I ν , λ μ ( z ) $\mathcal{I}_{\nu ,\lambda }^{\mu }(z)$ are obtained where the main idea of the proofs comes from the monotonicity of the quotient of two Maclaurin series. As an application, some inequalities (like Turán-type inequalities and their reverse) are proved. Further investigations on this function are underway and will be reported in a forthcoming paper.

Abstract In this research paper, we intend to study the stability of solutions of some nonlinear initial value fractional differential problems. These equations are studied within the generalized fractional derivative of various orders. In order to study the solutions’ decay to zero as a power function, we establish sufficient conditions on the nonlinear terms. To this end, some versions of inequalities are combined and generalized via the so-called Bihari inequality. Moreover, we employ some properties of the generalized fractional derivative and appropriate regularization techniques. Finally, the paper involves examples to affirm the validity of the results.

Abstract This paper deals with boundary value problems for local and nonlocal Laplace operator in 2D with exponential nonlinearities, the so-called Liouville type equations. They include the mean field equation and other equations arising in the statistical mechanics. Existence results into an explicit form for the Dirichlet problem in the unit disc B 1 ⊂ R 2 $B_{1} \subset {\mathbf{R}}^{2} $ and in the participation of positive parameters in the right-hand sides are proved in Theorems 2 and 3. Theorem 2 is illustrated by several examples including an application to the differential geometry. In Theorem 4 global radial solution of the Cauchy problem with constant data at ∂ B 1 $\partial B_{1} $ and under appropriate conditions is constructed. It develops logarithmic singularities for r = 0 $r = 0 $ , r = ∞ $r = \infty $ . An illustrative example to Theorem 4 in the case of two exponents is given at the end of the paper.

Abstract The present paper deals with a fractional-order mathematical epidemic model of malaria transmission accompanied by temporary immunity and relapse. The model is revised by using Caputo fractional operator for the index of memory. We also recommend the utilization of temporary immunity and the possibility of relapse. The theory of locally bounded and Lipschitz is employed to inspect the existence and uniqueness of the solution of the malaria model. It is shown that temporary immunity has a great effect on the dynamical transmission of host and vector populations. The stability analysis of these equilibrium points for fractional-order derivative α and basic reproduction number R 0 $\mathcal{R}_{0}$ is discussed. The model will exhibit a Hopf-type bifurcation. The two control variables are introduced in this model to decrease the number of populations. Mandatory conditions for the control problem are produced. Two types of numerical method via Laplace Adomian decomposition and Runge–Kutta of fourth order for simulating the proposed model with fractional-order derivative are presented. To validate the mathematical results, numerical simulations, sensitivity analysis, convergence analysis, and other important studies are given. The paper is finished with some conclusions and discussion.

Abstract In this paper, we prove that the self-improving property of the weighted Gehring class G λ p $G_{\lambda }^{p}$ with a weight λ holds in the non-homogeneous spaces. The results give sharp bounds of exponents and will be used to obtain the self-improving property of the Muckenhoupt class A q $A^{q}$ . By using the rearrangement (nonincreasing rearrangement) of the functions and applying the Jensen inequality, we show that the results cover the cases of non-monotonic functions. For applications, we prove a higher integrability theorem and report that the solutions of partial differential equations can be solved in an extended space by using the self-improving property. Our approach in this paper is different from the ones used before and is based on proving some new inequalities of Hardy type designed for this purpose.

Abstract Exact solutions to nonlinear differential equations play an undeniable role in various branches of science. These solutions are often used as reliable tools in describing the various quantitative and qualitative features of nonlinear phenomena observed in many fields of mathematical physics and nonlinear sciences. In this paper, the generalized exponential rational function method and the extended sinh-Gordon equation expansion method are applied to obtain approximate analytical solutions to the space-time conformable coupled Cahn–Allen equation, the space-time conformable coupled Burgers equation, and the space-time conformable Fokas equation. Novel approximate exact solutions are obtained. The conformable derivative is considered to obtain the approximate analytical solutions under constraint conditions. Numerical simulations obtained by the proposed methods indicate that the approaches are very effective. Both techniques employed in this paper have the potential to be used in solving other models in mathematics and physics.

Abstract In this paper, an uncertain SIR (spreader, ignorant, stifler) rumor spreading model driven by one Liu process is formulated to investigate the influence of perturbation in the transmission mechanism of rumor spreading. The deduced process of the uncertain SIR rumor spreading model is presented. Then an existence and uniqueness theorem concerning the solution is proved. Moreover, the stability of uncertain SIR rumor spreading differential equation is proved. In addition, the influence of different parameters on rumor spreading is analyzed through numerical simulation. This paper also presents a paradox of stochastic SIR rumor spreading model.

Abstract In this paper, we extend existing population growth models and propose a model based on a nonlinear cubic differential equation that reveals itself as a special subclass of Abel differential equations of first kind. We first summarize properties of the time-continuous problem formulation. We state the boundedness, global existence, and uniqueness of solutions for all times. Proofs of these properties are thoroughly given in the Appendix to this paper. Subsequently, we develop an explicit–implicit time-discrete numerical solution algorithm for our time-continuous population growth model and show that many properties of the time-continuous case transfer to our numerical explicit–implicit time-discrete solution scheme. We provide numerical examples to illustrate different behaviors of our proposed model. Furthermore, we compare our explicit–implicit discretization scheme to the classical Eulerian discretization. The latter violates the nonnegativity constraints on population sizes, whereas we prove and illustrate that our explicit–implicit discretization algorithm preserves this constraint. Finally, we describe a parameter estimation approach to apply our algorithm to two different real-world data sets.

Abstract In this paper, we focus on the synchronization of fractional-order coupled neural networks (FCNNs). First, by taking information on activation functions into account, we construct a convex Lur’e–Postnikov Lyapunov function. Based on the convex Lyapunov function and a general convex quadratic function, we derive a novel Mittag-Leffler synchronization criterion for the FCNNs with symmetrical coupled matrix in the form of linear matrix inequalities (LMIs). Then we present a robust Mittag-Leffler synchronization criterion for the FCNNs with uncertain parameters. These two Mittag-Leffler synchronization criteria can be solved easily by LMI tools in Matlab. Moreover, we present a novel Lyapunov synchronization criterion for the FCNNs with unsymmetrical coupled matrix in the form of LMIs, which can be easily solved by YALMIP tools in Matlab. The feasibilities of the criteria obtained in this paper are shown by four numerical examples.

Abstract This paper is devoted to the study of existence and uniqueness of a mild solution for a parabolic equation with conformable derivative. The nonlocal problem for parabolic equations appears in many various applications, such as physics, biology. The first part of this paper is to consider the well-posedness and regularity of the mild solution. The second one is to investigate the existence by using Banach fixed point theory.

Abstract In this paper, we offer a new quantum integral identity, the result is then used to obtain some new estimates of Hermite–Hadamard inequalities for quantum integrals. The results presented in this paper are generalizations of the comparable results in the literature on Hermite–Hadamard inequalities. Several inequalities, such as the midpoint-like integral inequality, the Simpson-like integral inequality, the averaged midpoint–trapezoid-like integral inequality, and the trapezoid-like integral inequality, are obtained as special cases of our main results.

Abstract Ablowitz–Kaup–Newell–Segur (AKNS) linear spectral problem gives birth to many important nonlinear mathematical physics equations including nonlocal ones. This paper derives two fractional order AKNS hierarchies which have not been reported in the literature by equipping the AKNS spectral problem and its adjoint equations with local fractional order partial derivative for the first time. One is the space-time fractional order isospectral AKNS (stfisAKNS) hierarchy, three reductions of which generate the fractional order local and nonlocal nonlinear Schrödinger (flnNLS) and modified Kortweg–de Vries (fmKdV) hierarchies as well as reverse-t NLS (frtNLS) hierarchy, and the other is the time-fractional order non-isospectral AKNS (tfnisAKNS) hierarchy. By transforming the stfisAKNS hierarchy into two fractional bilinear forms and reconstructing the potentials from fractional scattering data corresponding to the tfnisAKNS hierarchy, three pairs of uniform formulas of novel N-fractal solutions with Mittag-Leffler functions are obtained through the Hirota bilinear method (HBM) and the inverse scattering transform (IST). Restricted to the Cantor set, some obtained continuous everywhere but nondifferentiable one- and two-fractal solutions are shown by figures directly. More meaningfully, the problems worth exploring of constructing N-fractal solutions of soliton equation hierarchies by HBM and IST are solved, taking stfisAKNS and tfnisAKNS hierarchies as examples, from the point of view of local fractional order derivatives. Furthermore, this paper shows that HBM and IST can be used to construct some N-fractal solutions of other soliton equation hierarchies.

Abstract In this paper, a general class of Clifford-valued neutral high-order neural network (HNN) with D-operator on time scales is investigated. In this model, time-varying delays and continuously distributed delays are taken into account. As an extension of the real-valued neural network, the Clifford-valued neural network, which includes a familiar complex-valued neural network and a quaternion-valued neural network as special cases, has been an active research field recently. By utilizing this novel method, which incorporates the differential inequality techniques and the fixed point theorem and time-scale theory of computation, we derive a few sufficient conditions to ensure the existence, uniqueness, and exponential stability of the pseudo almost periodic (PAP) solution of the considered model. The results in this paper are new, even if time scale T = R $\mathbb{T}=\mathbb{R}$ or T = Z $\mathbb{T}=\mathbb{Z}$ , and complementary to the previously existing works. Furthermore, an example and its numerical simulations are included to demonstrate the validity and advantage of the obtained results.

Abstract In this paper, we introduce the non-convex interval-valued functions for fuzzy-interval-valued functions, which are called -convex fuzzy-interval-valued functions, by means of fuzzy order relation. This fuzzy order relation is defined level-wise through Kulisch–Miranker order relation given on the interval space. By using the -convexity concept, we present fuzzy-interval Hermite–Hadamard inequalities for fuzzy-interval-valued functions. Several exceptional cases are debated, which can be viewed as useful applications. Interesting examples that verify the applicability of the theory developed in this study are presented. The results of this paper can be considered as extensions of previously established results.

Abstract This paper is about to formulate a design of predator–prey model with constant and time fractional variable order. The predator and prey act as agents in an ecosystem in this simulation. We focus on a time fractional order Atangana–Baleanu operator in the sense of Liouville–Caputo. Due to the nonlocality of the method, the predator–prey model is generated by using another FO derivative developed as a kernel based on the generalized Mittag-Leffler function. Two fractional-order systems are assumed, with and without delay. For the numerical solution of the models, we not only employ the Adams–Bashforth–Moulton method but also explore the existence and uniqueness of these schemes. We use the fixed point theorem which is useful in describing the existence of a new approach with a particular set of solutions. For the illustration, several numerical examples are added to the paper to show the effectiveness of the numerical method.

Abstract In this paper, we have mathematically analyzed a within-host model of SARS-CoV-2 which is used by Li et al. in the paper “The within-host viral kinetics of SARS-CoV-2” published in (Math. Biosci. Eng. 17(4):2853–2861, 2020). Important properties of the model, like nonnegativity of solutions and their boundedness, are established. Also, we have calculated the basic reproduction number which is an important parameter in the infection models. From stability analysis of the model, it is found that stability of the biologically feasible steady states are determined by the basic reproduction number ( χ 0 ) $(\chi _{0})$ . Numerical simulations are done in order to substantiate analytical results. A biological implication from this study is that a COVID-19 patient with less than one basic reproduction ratio can automatically recover from the infection.

Abstract This paper is concerned with the Liouville type theorem for stable weak solutions to the following weighted Kirchhoff equations: − M ( ∫ R N ξ ( z ) | ∇ G u | 2 d z ) div G ( ξ ( z ) ∇ G u ) = η ( z ) | u | p − 1 u , z = ( x , y ) ∈ R N = R N 1 × R N 2 , $$\begin{aligned}& -M \biggl( \int_{\mathbb{R}^{N}}\xi(z) \vert \nabla_{G}u \vert ^{2}\,dz \biggr){ \operatorname{div}}_{G} \bigl(\xi(z) \nabla_{G}u \bigr) \\& \quad=\eta(z) \vert u \vert ^{p-1}u,\quad z=(x,y) \in \mathbb{R}^{N}=\mathbb{R}^{N_{1}}\times\mathbb{R}^{N_{2}}, \end{aligned}$$ where M ( t ) = a + b t k $M(t)=a+bt^{k}$ , t ≥ 0 $t\geq0$ , with a , b , k ≥ 0 $a,b,k\geq0$ , a + b > 0 $a+b>0$ , k = 0 $k=0$ if and only if b = 0 $b=0$ . Let N = N 1 + N 2 ≥ 2 $N=N_{1}+N_{2}\geq2$ , p > 1 + 2 k $p>1+2k$ and ξ ( z ) , η ( z ) ∈ L loc 1 ( R N ) ∖ { 0 } $\xi(z),\eta(z)\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})\setminus\{ 0\}$ be nonnegative functions such that ξ ( z ) ≤ C ∥ z ∥ G θ $\xi(z)\leq C\|z\|_{G}^{\theta}$ and η ( z ) ≥ C ′ ∥ z ∥ G d $\eta(z)\geq C'\|z\|_{G}^{d}$ for large ∥ z ∥ G $\|z\|_{G}$ with d > θ − 2 $d>\theta-2$ . Here α ≥ 0 $\alpha\geq0$ and ∥ z ∥ G = ( | x | 2 ( 1 + α ) + | y | 2 ) 1 2 ( 1 + α ) $\|z\|_{G}=(|x|^{2(1+\alpha)}+|y|^{2})^{\frac{1}{2(1+\alpha)}}$ . div G $\operatorname{div}_{G}$ (resp., ∇ G $\nabla_{G}$ ) is Grushin divergence (resp., Grushin gradient). Under some appropriate assumptions on k, θ, d, and N α = N 1 + ( 1 + α ) N 2 $N_{\alpha}=N_{1}+(1+\alpha)N_{2}$ , the nonexistence of stable weak solutions to the problem is obtained. A distinguished feature of this paper is that the Kirchhoff function M could be zero, which implies that the above problem is degenerate.

Abstract This paper is motivated by the series of research papers that consider parasitoids’ external input upon the host–parasitoid interactions. We explore a class of host–parasitoid models with variable release and constant release of parasitoids. We assume that the host population has a constant rate of increase, but we do not assume any density dependence regulation other than parasitism acting on the host population. We compare the obtained results for constant stocking with the results for proportional stocking. We observe that under a specific condition, the release of a constant number of parasitoids can eventually drive the host population (pests) to extinction. There is always a boundary equilibrium where the host population extinct occurs, and the parasitoid population is stabilized at the constant stocking level. The constant and variable stocking can decrease the host population level in the unique interior equilibrium point; on the other hand, the parasitoid population level stays constant and does not depend on stocking. We prove the existence of Neimark–Sacker bifurcation and compute the approximation of the closed invariant curve. Then we consider a few host–parasitoid models with proportional and constant stocking, where we choose well-known probability functions of parasitism. By using the software package Mathematica we provide numerical simulations to support our study.

Abstract This paper gives an overview of systematic and analytic approach of operational technique involves to study multi-variable special functions significant in both mathematical and applied framework and to introduce new families of special polynomials. Motivation of this paper is to construct a new class of generalized Fubini-type polynomials of the parametric kind via operational view point. The generating functions, differential equations, and other properties for these polynomials are established within the context of the monomiality principle. Using the generating functions, various interesting identities and relations related to the generalized Fubini-type polynomials are derived. Further, we obtain certain partial derivative formulas including the generalized Fubini-type polynomials. In addition, certain members belonging to the aforementioned general class of polynomials are considered. The numerical results to calculate the zeros and approximate solutions of these polynomials are given and their graphical representation are shown.

Abstract In this paper, we obtain Hermite–Hadamard-type inequalities of convex functions by applying the notion of q b $q^{b}$ -integral. We prove some new inequalities related with right-hand sides of q b $q^{b}$ -Hermite–Hadamard inequalities for differentiable functions with convex absolute values of second derivatives. The results presented in this paper are a unification and generalization of the comparable results in the literature on Hermite–Hadamard inequalities.

Abstract In this paper we review some of the previous work done by the earlier authors (Singh et al. in J. Inequal. Appl. 2017:101, 2017; Lal and Shireen in Bull. Math. Anal. Appl. 5(4):1–13, 2013), etc., on error approximation of a function g in the generalized Zygmund space and resolve the issue of these works. We also determine the best error approximation of the functions g and g ′ $g^{\prime }$ , where g ′ $g^{\prime }$ is a derived function of a 2π-periodic function g, in the generalized Zygmund class X z ( η ) $X_{z}^{(\eta )}$ , z ≥ 1 $z\geq 1$ , using matrix-Cesàro ( T C δ ) $(TC^{\delta })$ means of its Fourier series and its derived Fourier series, respectively. Theorem 2.1 of the present paper generalizes eight earlier results, which become its particular cases. Thus, the results of (Dhakal in Int. Math. Forum 5(35):1729–1735, 2010; Dhakal in Int. J. Eng. Technol. 2(3):1–15, 2013; Nigam in Surv. Math. Appl. 5:113–122, 2010; Nigam in Commun. Appl. Anal. 14(4):607–614, 2010; Nigam and Sharma in Kyungpook Math. J. 50:545–556, 2010; Nigam and Sharma in Int. J. Pure Appl. Math. 70(6):775–784, 2011; Kushwaha and Dhakal in Nepal J. Sci. Technol. 14(2):117–122, 2013; Shrivastava et al. in IOSR J. Math. 10(1 Ver. I):39–41, 2014) become particular cases of our Theorem 2.1. Several corollaries are also deduced from our Theorem 2.1.

Abstract In this paper, we obtain sufficient conditions for the existence and uniqueness results of the pantograph fractional differential equations (FDEs) with nonlocal conditions involving Atangana–Baleanu–Caputo (ABC) derivative operator with fractional orders. Our approach is based on the reduction of FDEs to fractional integral equations and on some fixed point theorems such as Banach’s contraction principle and the fixed point theorem of Krasnoselskii. Further, Gronwall’s inequality in the frame of the Atangana–Baleanu fractional integral operator is applied to develop adequate results for different kinds of Ulam–Hyers stabilities. Lastly, the paper includes an example to substantiate the validity of the results.

Abstract The primary objective of this present paper is to establish certain new weighted fractional Pólya–Szegö and Chebyshev type integral inequalities by employing the generalized weighted fractional integral involving another function Ψ in the kernel. The inequalities presented in this paper cover some new inequalities involving all other type weighted fractional integrals by applying certain conditions on ω ( θ ) $\omega (\theta )$ and Ψ ( θ ) $\Psi (\theta )$ . Also, the Pólya–Szegö and Chebyshev type integral inequalities for all other type fractional integrals, such as the Katugampola fractional integrals, generalized Riemann–Liouville fractional integral, conformable fractional integral, and Hadamard fractional integral, are the special cases of our main results with certain choices of ω ( θ ) $\omega (\theta )$ and Ψ ( θ ) $\Psi (\theta )$ . Additionally, examples of constructing bounded functions are also presented in the paper.

Abstract Variants of the Newton method are very popular for solving unconstrained optimization problems. The study on global convergence of the BFGS method has also made good progress. The q-gradient reduces to its classical version when q approaches 1. In this paper, we propose a quantum-Broyden–Fletcher–Goldfarb–Shanno algorithm where the Hessian is constructed using the q-gradient and descent direction is found at each iteration. The algorithm presented in this paper is implemented by applying the independent parameter q in the Armijo–Wolfe conditions to compute the step length which guarantees that the objective function value decreases. The global convergence is established without the convexity assumption on the objective function. Further, the proposed method is verified by the numerical test problems and the results are depicted through the performance profiles.

Abstract The main objective of this paper is to obtain a new κ-fractional analogue of Hermite–Hadamard’s inequality using the class of s-convex functions and χ κ $\chi _{{\kappa }}$ -Hilfer fractional integrals. In order to obtain other main results of the paper we derive two new fractional integral identities using the definitions of χ κ $\chi _{{\kappa }}$ -Hilfer fractional integrals. For the validity of these identities we also take some particular examples. Using these identities we then obtain some more new variants of Hermite–Hadamard’s inequality using s-convex functions.

Abstract Humans are always exposed to the threat of infectious diseases. It has been proven that there is a direct link between the strength or weakness of the immune system and the spread of infectious diseases such as tuberculosis, hepatitis, AIDS, and Covid-19 as soon as the immune system has no the power to fight infections and infectious diseases. Moreover, it has been proven that mathematical modeling is a great tool to accurately describe complex biological phenomena. In the recent literature, we can easily find that these effective tools provide important contributions to our understanding and analysis of such problems such as tumor growth. This is indeed one of the main reasons for the need to study computational models of how the immune system interacts with other factors involved. To this end, in this paper, we present some new approximate solutions to a computational formulation that models the interaction between tumor growth and the immune system with several fractional and fractal operators. The operators used in this model are the Liouville–Caputo, Caputo–Fabrizio, and Atangana–Baleanu–Caputo in both fractional and fractal-fractional senses. The existence and uniqueness of the solution in each of these cases is also verified. To complete our analysis, we include numerous numerical simulations to show the behavior of tumors. These diagrams help us explain mathematical results and better describe related biological concepts. In many cases the approximate results obtained have a chaotic structure, which justifies the complexity of unpredictable and uncontrollable behavior of cancerous tumors. As a result, the newly implemented operators certainly open new research windows in further computational models arising in the modeling of different diseases. It is confirmed that similar problems in the field can be also be modeled by the approaches employed in this paper.

Abstract This paper is devoted to Kirchhoff-type parabolic problem with nonlocal integral condition. Our problem has many applications in modeling physical and biological phenomena. The first part of our paper concerns the local existence of the mild solution in Hilbert scales. Our results can be studied into two cases: homogeneous case and inhomogeneous case. In order to overcome difficulties, we applied Banach fixed point theorem and some new techniques on Sobolev spaces. The second part of the paper is to derive the ill-posedness of the mild solution in the sense of Hadamard.

Abstract In this paper, we propose a mathematical model to predict the novel coronavirus. Due to the rapid spread of the novel coronavirus disease in the world, we add to the deterministic model of the coronavirus the terms of the stochastic perturbations. In other words, we consider in this paper a stochastic model to predict the novel coronavirus. The equilibrium points of the deterministic model have been determined, and the reproduction number of our deterministic model has been implemented. The asymptotic behaviors of the solutions of the stochastic model around the equilibrium points have been studied. The numerical investigations and the graphical representations obtained with the novel stochastic model are made using the classical stochastic numerical scheme.

Abstract In this paper, we assess an integral containing incomplete H-functions and utilize it to build up an expansion formula for the incomplete H-functions including the Bessel function. Next, we evaluate an integral containing incomplete H̅-functions and use it to develop an expansion formula for the incomplete H̅-functions including the Bessel function. The outcomes introduced in this paper are general in nature, and several particular cases can be acquired by giving specific values to the parameters engaged with the principle results. As particular cases, we derive expansions for the incomplete Meijer G ( Γ ) ${}^{(\Gamma )}G$ -function, Fox–Wright Ψ q ( Γ ) p ${}_{p}\Psi _{q}^{(\Gamma )}$ -function, and generalized hypergeometric Γ q p ${}_{p}\Gamma _{q}$ function.

Abstract This paper deals with a general SEIR model for the coronavirus disease 2019 (COVID-19) with the effect of time delay proposed. We get the stability theorems for the disease-free equilibrium and provide adequate situations of the COVID-19 transmission dynamics equilibrium of present and absent cases. A Hopf bifurcation parameter τ concerns the effects of time delay and we demonstrate that the locally asymptotic stability holds for the present equilibrium. The reproduction number is brief in less than or greater than one, and it effectively is controlling the COVID-19 infection outbreak and subsequently reveals insight into understanding the patterns of the flare-up. We have included eight parameters and the least square method allows us to estimate the initial values for the Indian COVID-19 pandemic from real-life data. It is one of India’s current pandemic models that have been studied for the time being. This Covid19 SEIR model can apply with or without delay to all country’s current pandemic region, after estimating parameter values from their data. The sensitivity of seven parameters has also been explored. The paper also examines the impact of immune response time delay and the importance of determining essential parameters such as the transmission rate using sensitivity indices analysis. The numerical experiment is calculated to illustrate the theoretical results.

Abstract In this paper, by making use of the familiar q-difference operators D q $D_{q}$ and D q − 1 $D_{q^{-1}}$ , we first introduce two homogeneous q-difference operators T ( a , b , c D q ) $\mathbb{T}(\mathbf{a},\mathbf{b},cD_{q})$ and E ( a , b , c D q − 1 ) $\mathbb{E}(\mathbf{a},\mathbf{b}, cD_{q^{-1}})$ , which turn out to be suitable for dealing with the families of the generalized Al-Salam–Carlitz q-polynomials ϕ n ( a , b ) ( x , y | q ) $\phi_{n}^{(\mathbf{a},\mathbf{b})}(x,y|q)$ and ψ n ( a , b ) ( x , y | q ) $\psi_{n}^{(\mathbf{a},\mathbf{b})}(x,y|q)$ . We then apply each of these two homogeneous q-difference operators in order to derive generating functions, Rogers type formulas, the extended Rogers type formulas, and the Srivastava–Agarwal type linear as well as bilinear generating functions involving each of these families of the generalized Al-Salam–Carlitz q-polynomials. We also show how the various results presented here are related to those in many earlier works on the topics which we study in this paper.

Abstract Fractal analysis is one of interesting research areas of computer science and engineering, which depicts a precise description of phenomena in modeling. Visual beauty and self-similarity has made it an attractive field of research. The fractal sets are the effective tools to describe the accuracy of the inequalities for convex functions. In this paper, we employ linear fractals R α $\mathbb{R}^{\alpha }$ to investigate the ( s , m ) $(s,m)$ -convexity and relate them to derive generalized Hermite–Hadamard (HH) type inequalities and several other associated variants depending on an auxiliary result. Under this novel approach, we aim at establishing an analog with the help of local fractional integration. Meanwhile, we establish generalized Simpson-type inequalities for ( s , m ) $(s,m)$ -convex functions. The results in the frame of local fractional showed that among all comparisons, we can only see the correlation between novel strategies and the earlier consequences in generalized s-convex, generalized m-convex, and generalized convex functions. We obtain application in probability density functions and generalized special means to confirm the relevance and computational effectiveness of the considered method. Similar results in this dynamic field can also be widely applied to other types of fractals and explored similarly to what has been done in this paper.

Abstract In this paper, we analyze the criteria for the stability of a method suited to the ordinary differential equations models. The relevant proof that the method satisfies the condition of stiff stability is also provided. The aim of this paper is therefore to construct an efficient two-point block method based on backward differentiation formula which is A-stable and converged. The new diagonally implicit scheme is formulated to approximate the solution of the pharmacokinetics models. By implementing the algorithm, the numerical solution to the models is compared with a few existing methods and established stiff solvers. It yields significant advantages when the diagonally implicit method with a lower triangular matrix and identical diagonal elements is considered. The formula is designed in such a way that it permits a maximum of one LU decomposition for each integration stage.

Abstract Anti-viral therapy is comparatively very effective for patients who get affected by the hepatitis B virus. It is of prime importance to understand the different relations among the viruses, immune responses and overall health of the liver. In this paper, mathematical modeling is done to analyze and understand the effect of antiviral therapy using LHAM which describes the possible relation to HBV and target liver cells. The numerical simulations and error analysis are done up to a sixth-order approximation with the help of Matlab. This paper analyzes how the number of infected cells largely gets reduced and also how the liver damage can be controlled. Therefore, the treatment is successful for HBV infected patients.

Abstract This paper aims to discuss a generalization of certain paraxial diffraction integral operator in a class of generalized functions. At the start of this paper, we propose a convolution formula and establish certain convolution theorem. Then, with the addition to the convolution theorem, we consider a set of approximating identities and substantially employ our results in generating sets of integrable and locally integrable Boehmians. The said generalized integral operator is tested and declared to be one-to-one and onto mapping. Continuity of the generalized operator with respect to the convergence of the Boehmian spaces is obtained. Over and above, an inversion formula and consistency results are also counted.

Abstract This paper proposes a modified adaptive sliding-mode control technique and investigates the reduced-order and increased-order synchronization between two different fractional-order chaotic systems using the master and slave system synchronization arrangement. The parameters of the master and slave systems are different and uncertain. These systems exhibit different chaotic behavior and topological properties. The dynamic behavior of the proposed synchronization schemes is more complex and unpredictable. These attributes of the proposed synchronization schemes enhance the security of the information signal in digital communication systems. The proposed switching law ensures the convergence of the error vectors to the switching surface and the feedback control signals guarantee the fast convergence of the error vectors to the origin. Lyapunov stability theory proves the asymptotic stability of the closed-loop. The paper also designs suitable parameters update laws the estimate the unknown parameters. Computer-based simulation results verify the theoretical findings.

Abstract Recently, some mathematicians have been studying a lot of degenerate versions of special polynomials and numbers in some arithmetic and combinatorial aspects. Our research is also interested in this field. In this paper, we introduce a new type of the degenerate poly-Genocchi polynomials and numbers, based on Kim and Kim’s (J. Math. Anal. Appl. 487(2):124017, 2020) modified polyexponential function. The paper is divided into two parts. In Sect. 2, we consider a new type of the degenerate poly-Genocchi polynomials and numbers constructed from the modified polyexponential function. We also show several combinatorial identities related to the degenerate poly-Genocchi polynomials and numbers. Some of them include the degenerate and other special polynomials and numbers such as the Stirling numbers of the first kind, the degenerate Stirling numbers of the second kind, degenerate Euler polynomials, degenerate Bernoulli polynomials and Bernoulli numbers of order α, etc. In Sect. 3, we also introduce the degenerate unipoly Genocchi polynomials attached to an arithmetic function by using the degenerate polylogarithm function. We give some new explicit expressions and identities related to degenerate unipoly Genocchi polynomials and special numbers and polynomials.

Abstract This paper investigates a deterministic and stochastic SIQS epidemic model with vertical transmission and Beddington–DeAngelis incidence. Firstly, for the corresponding deterministic system, the global asymptotic stability of disease-free equilibrium and the endemic equilibrium is proved through the stability theory. Secondly, for the stochastic system, the threshold conditions which decide the extinction or permanence of the disease are derived. By constructing suitable Lyapunov functions, we investigate the oscillation behavior of the stochastic system solution near the endemic equilibrium. The results of this paper show that there exists a great difference between the deterministic and stochastic systems, which implies that the large stochastic noise contributes to inhibiting the spread of disease. Finally, in order to validate the theoretical results, a series of numerical simulations are presented.

Abstract This paper is devoted to studying a stochastic patch structure Nicholson’s blowflies system with mixed delays which is a new model for the generalization of classic Nicholson’s blowflies system. We examine stochastically ultimate boundedness and global asymptotic stability for the considered model by stochastic analysis technique. Finally, numerical simulations verify theoretical results of the present paper.

Abstract This paper presents a fundamental solution method for nonlinear fractional regularized long-wave (RLW) models. Since analytical methods cannot be applied easily to solve such models, numerical or semianalytical methods have been extensively considered in the literature. In this paper, we suggest a solution method that is coupled with a kind of integral transformation, namely Elzaki transform (ET), and apply it to two different nonlinear regularized long wave equations. They play an important role to describe the propagation of unilateral weakly nonlinear and weakly distributer liquid waves. Therefore, these equations have been noticed by scientists who study waves their movements. Particularly, they have been used to model a large class of physical and engineering phenomena. In this context, this paper takes into consideration an up-to-date method and fractional operators, and aims to obtain satisfactory approximate solutions to nonlinear problems. We present this achievement, firstly, by defining the Elzaki transforms of Atangana–Baleanu fractional derivative (ABFD) and Caputo fractional derivative (CFD) and then applying them to the RLW equations. Finally, numerical outcomes giving us better approximations after only a few iterations can be easily obtained.

Abstract In this paper, we establish the generalized Riemann–Liouville (RL) fractional integrals in the sense of another increasing, positive, monotone, and measurable function Ψ. We determine certain new double-weighted type fractional integral inequalities by utilizing the said integrals. We also give some of the new particular inequalities of the main result. Note that we can form various types of new inequalities of fractional integrals by employing conditions on the function Ψ given in the paper. We present some corollaries as particular cases of the main results.

Abstract This paper is related to some dynamical aspects of a class of predator–prey interactions incorporating cannibalism and Allee effects for non-overlapping generations. Cannibalism has been frequently observed in natural populations, and it has an ability to alter the functional response concerning prey–predator interactions. On the other hand, from dynamical point of view cannibalism is considered as a procedure of stabilization or destabilization within predator–prey models. Taking into account the cannibalism in prey population and with addition of Allee effects, a new discrete-time system is proposed and studied in this paper. Moreover, existence of fixed points and their local dynamics are carried out. It is verified that the proposed model undergoes transcritical bifurcation about its trivial fixed point and period-doubling bifurcation around its boundary fixed point. Furthermore, it is also proved that the proposed system undergoes both period-doubling and Neimark–Sacker bifurcations (NSB) around its interior fixed point. Our study demonstrates that outbreaks of periodic nature may appear due to implementation of cannibalism in prey population, and these periodic oscillations are limited to prey density only without leaving an influence on predation. To restrain this periodic disturbance in prey population density, and other fluctuating and bifurcating behaviors of the model, various chaos control methods are applied. At the end, numerical simulations are presented to illustrate the effectiveness of our theoretical findings.