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In this paper, we derive some error estimates of Simpson's second type quadrature formula for functions of bounded variation and Lipschitzian mappings. Also, similar error estimations for absolutely continuous functions whose first derivatives belong to Lp[γ, δ] with (p < 1 ≤ ∞) are established. Finally, with the help of the results given in this work, some Simpson's type inequalities involving special means are presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider a pseudo‐parabolic equation with the Caputo fractional derivative. We study the existence and uniqueness of a class of mild solutions of these equations. For a nonlinear problem, we first investigate the global solution under the initial data u0 ∈ L2. In the case of initial data u0 ∈ Lq, q ≠ 2, we obtain the local existence result. Our main tool here is using fundamental tools, namely, Banach fixed point theorem and Sobolev embeddings. In addition, we give an example to illustrate the effectiveness of the method has been proposed. [ABSTRACT FROM AUTHOR]

This paper attempts to present an iterative‐computational method for solving weakly singular nonlinear Volterra integral equations (WSNVIEs) of the second type based on rationalized Haar wavelets (RHWs). The proposed algorithm does not need to solve any linear or nonlinear system for evaluating the wavelet coefficients. After a brief introduction of rationalized Haar functions (RHFs), the computational matrices of integration and fractional integral are applied to reduce the approximation of integral equations to some matrix algebraic equations. Next, the error analysis of the problem by using the two‐dimensional iterated projection operator is offered. We will prove that the rate of convergence of the proposed algorithm is O(βh), where h is the iteration number and β is the contraction constance. The method for any WSNVIE of the second kind with 0 ≤ β < 1 is convergent. The proposed method is computationally attractive, and comparing the results obtained of the known technique is more efficient. [ABSTRACT FROM AUTHOR]

Zika is a flavivirus that is transmitted to humans either through the bites of infected Aedes mosquitoes or through sexual transmission. Zika has been associated with congenital anomalies, like microcephalus. We developed and analyzed the fractional‐order Zika virus model in this paper, considering the vector transmission route with human influence. The model consists of four compartments: susceptible individuals are x1(t), infected individuals are x2(t), x3(t) shows susceptible mosquitos, and x4(t) shows the infected mosquitos. The fractional parameter is used to develop the system of complex nonlinear differential equations by using Caputo and Atangana–Baleanu derivative. The stability analysis as well as qualitative analysis of the fractional‐order model has been made and verify the non‐negative unique solution. Finally, numerical simulations of the model with Caputo and Atangana Baleanu are discussed to present the graphical results for different fractional‐order values as well as for the classical model. A comparison has been made to check the accuracy and effectiveness of the developed technique for our obtained results. This investigative research leads to the latest information sector included in the evolution of the Zika virus with the application of fractional analysis in population dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Cauchy problem for a Caputo‐type time delay linear system of fractional differential equations with permutable matrices. First, we provide a new representation of solutions to linear homogeneous fractional differential equations using the Laplace integral transform and variation of constants formula via a newly defined delayed Mittag‐Leffler type matrix function introduced through a three‐parameter Mittag‐Leffler function. Second, with the help of a delayed perturbation of a Mittag‐Leffler type matrix function, we attain an explicit formula for solutions to a linear nonhomogeneous time delay fractional order system using the superposition principle. Furthermore, we prove the existence and uniqueness of solutions to nonlinear fractional delay differential equations using the contraction mapping principle. Finally, we present an example to illustrate the applicability of our results. [ABSTRACT FROM AUTHOR]

In this paper, we study systems of Fredholm integral equations with a singular matrix multiplying the leading part. We address the solvability issues of such systems with separable kernels and, similarly to differential‐algebraic and integral algebraic equations, introduce the notion of index. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the rotational periodic boundary value problem for a fractional‐order nonlinear differential equation. Applying topology‐degree theory and the Leray‐Schauder fixed‐point theorem, we prove the existence and uniqueness of solution for the fractional‐order differential system. Furthermore, the existence of solution for a nonlinear differential system with a multivalued perturbation term is investigated by using set‐valued theory and techniques of functional analysis. Two examples of applications are given at the end. [ABSTRACT FROM AUTHOR]

We propose a tailored finite point method (TFPM) for solving time fractional convection dominated diffusion equations in this paper. The main idea of TFPM is to firstly approximate the diffusion, convection coefficient near each grid by a constant, and then determine the weights of the finite difference scheme by using the exact solution of the convection diffusion equation with constant coefficients. This adaptation perfectly captures the rapid transition of the solutions which contain sharp boundary layers even with coarse meshes. The accuracy and stability of the scheme are rigorously analyzed. Numerical examples are shown to verify the accuracy and reliability of the proposed scheme. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present existence and uniqueness results for the solution of a general system of Fredholm integral equations by applying the vectorial form of Maia's fixed‐point theorem. A Gronwall‐type lemma and comparison theorems are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the structure of the solutions for linear interval‐valued Volterra integro‐differential equations (VIDEs) based on gH‐difference. The considered VIDEs consist of six forms. Among those six forms, three forms are obtained by using the definition of generalized Hukuhara difference (or gH‐difference). We obtain solutions of that linear interval‐valued VIDEs and discuss their properties. Some examples illustrate the theory. [ABSTRACT FROM AUTHOR]

In this paper, we consider a coupled Lamé system only with viscoealstic dampings. By assuming a more general of relaxation functions and by using some properties of convex functions, we establish optimal explicit and general energy decay results to the system. This result improves previous results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global (in time) existence of small data solutions to the Cauchy problem for the following structurally damped σ‐evolution model with nonlinear memory term: utt+(−Δ)σu+μ(−Δ)σ2ut=∫0t(t−τ)−γ|ut(τ,·)|pdτ,with σ>0. In particular, for γ∈((n−σ)/n,1), we find the sharp critical exponent, under the assumption of small data in L1. Dropping the L1 smallness assumption of initial data, we show how the critical exponent is consequently modified for the problem. In particular, we obtain a new interplay between the fractional order of integration 1−γ in the nonlinear memory term and the assumption that initial data are small in Lm, for some m>1. [ABSTRACT FROM AUTHOR]

In this paper, we study the extremal solutions of nonlinear fractional p‐Laplacian differential system with the fractional conformable derivative by applying monotone iterative method and a half‐pair of upper and lower solutions. For the smooth running of our work, we develop a comparison principle about linear system, which play a very crucial role in this article. At last, an illustrative example is given for the main result. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of solutions for a p–Laplacian boundary value problem defined by semilinear fractional system that involves Caputo–Katugampola fractional derivatives. Our main results rely on the implementation of the Banach and Schauder fixed point theorems. An example is introduced to expose the applicability of the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, the well‐posedness of Cauchy's problem of fractional Schrödinger equations with a power‐type nonlinearity on n‐dimensional manifolds with nonnegative Ricci curvature is studied. Under suitable volume conditions, the local solution with initial data in H[n2]+1 will blow up in finite time no matter how small the initial data is, which follows from a new weight function and ODE inequalities. Moreover, the upper‐bound of the lifespan can be estimated. [ABSTRACT FROM AUTHOR]

In this paper, we study boundary value problem of system of generalized Sturm–Liouville and Langevin Hadamard fractional differential equations. Existence and uniqueness results are proved via Banach contraction principle and Leray–Schauder fixed point theorem. Besides, the Ulam–Hyers and Ulam–Hyers–Rassias stability results are addressed for the proposed problem. An example illustrating the effectiveness of the theoretical results is presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Bresse‐Timoshenko‐type system with distributed delay term. Under suitable assumptions, we establish the global well‐posedness of the initial and boundary value problem by using the Faedo‐Galerkin approximations and some energy estimates. By using the energy method, we show two exponential stability results for the system with delay in vertical displacement and in angular rotation, respectively. This extends earlier results in the literature. [ABSTRACT FROM AUTHOR]

It is a very challenging task to solve a nonlinear integral equation in multidimensions. The main purpose of this paper is to develop and analyze a spectral collocation method for a class of nonlinear Fredholm integral equations of the second kind in multidimensions. The proposed spectral collocation method is based on a multivariate Legendre approximation in the frequency space. It is shown by both theory and numerics that the proposed spectral collocation scheme not only significantly reduces the number of degrees of freedom but also produces very accurate approximations. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to obtain the existence of at least one solution to the following coupled system of nonlinear fractional order differential equations under the integral type boundary conditions by using topological degree theory 0.1Dδμ(ℓ)=F1(ℓ,μ(λℓ),ν(λℓ)),ℓ∈[0,1],Dϱν(ℓ)=F2(ℓ,μ(λℓ),ν(λℓ)),ℓ∈[0,1],μ(0)=r(μ),μ(1)=1Γ(δ)∫01(1−η)δ−1φ(η,μ(η))dη,ν(0)=h(ν),ν(1)=1Γ(ϱ)∫01(1−η)ϱ−1ψ(η,ν(η))dη, where δ,ϱ∈(1,2], 0<λ<1, D denotes the standard Caputo fractional derivative, F1,F2:[0,1]×ℜ×ℜ→ℜ, φ,ψ:[0,1]×ℜ→ℜ and r,h:[0,1]→ℜ are continuous functions. For this intention, some results for the existence of at least one solution are constructed. For the validity of our results, an appropriate example is presented. [ABSTRACT FROM AUTHOR]

We study in this paper a geometric inverse problem in fluid mechanics. The goal is to determine the location of an object in the fluid domain from boundary information. We consider the case of a nonstationary two‐dimensional Stokes flow. Our approach is based on the Kohn–Vogelius concept and the topological gradient method. We develop a new topological asymptotic expansion, which can be used as a basic step for developing accurate detection algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we study the time‐dependent vorticity–velocity–pressure formulation of Stokes problem in two‐ and three‐dimensional domains provided with nonstandard boundary conditions, related to the normal component of the velocity and the tangential components of the vorticity. This problem is discretized by implicit Euler's scheme in time and spectral method in space. We prove an optimal error estimate for the three unknowns. [ABSTRACT FROM AUTHOR]

In this work, the optimal perturbation iteration method is briefly presented and employed for solving nonlinear Volterra‐Fredholm integral equations. The classical form of the optimal perturbation iteration method is modified, and new algorithms are constructed for integral equations. Comparing our new algorithms with some earlier papers proved the excellent accuracy of the newly proposed technique. [ABSTRACT FROM AUTHOR]

Richards in [35] proposed a modification of the logistic model to model growth of biological populations. In this paper, we give a new representation (or characterization) of the solution to the Richards‐type fractional differential equation Dαy(t)=y(t)·(1+a(t)yβ(t))$$ {\mathcal{D}}&amp;#x0005E;{\alpha }y(t)&amp;#x0003D;y(t)\cdotp \left(1&amp;#x0002B;a(t){y}&amp;#x0005E;{\beta }(t)\right) $$ for t≥0$$ t\ge 0 $$, where a:[0,∞)→ℝ$$ a:\left[0,\infty \right)\to \mathrm{\mathbb{R}} $$ is a continuously differentiable function on [0,∞),α∈(0,1)$$ \left[0,\infty \right),\alpha \in \left(0,1\right) $$ and β$$ \beta $$ is a positive real constant. The obtained representation of the solution can be used effectively for computational and analytic purposes. This study improves and generalizes the results obtained on fractional logistic ordinary differential equation. [ABSTRACT FROM AUTHOR]

This paper investigates the problems of exponential stability for a class of quaternion‐valued memristor‐based neural networks. By using M‐matrix theory and fixed point theorem, the existence and uniqueness of the equilibrium point of quaternion‐valued neural network are proved, respectively. Then, by combining M‐matrix with exponential stability theory, a non‐factorization method is obtained by using some inequality techniques to give the effective conditions of global exponential stability of quaternion‐valued memristor‐based neural network with time‐varying delay. Finally, numerical examples are given to demonstrate the validity of the derived results. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in propagation dynamics of a nonlocal dispersal Zika transmission model with general incidence. When the threshold R$$ \mathcal{R} $$ is greater than one, we prove that there is a wave speed c∗>0$$ {c}&#x0005E;{\ast }>0 $$ such that the model has a traveling wave solution with speed c>c∗$$ c>{c}&#x0005E;{\ast } $$, and there is no traveling wave solution with speed less than c∗$$ {c}&#x0005E;{\ast } $$. When the threshold R$$ \mathcal{R} $$ is less than or equal to one, we show that there is no nontrivial traveling wave solution. The approaches we use here are Schauder's fixed point theorem with an explicit construction of a pair of upper and lower solutions, the contradictory approach, and the two‐sided Laplace transform. [ABSTRACT FROM AUTHOR]

In this paper, we concentrate on the Bayesian inversion of a dispersion‐dominated fractional Helmholtz (DDFH) equation, which has been introduced in studies concerning seismic exploration. To establish the inversion theory, we meticulously examine the DDFH equation. We transform it into a system comprising both fractional‐ and integer‐order elliptic equations, extending the conventional definition of the spectral fractional Laplace operator to accommodate non‐homogeneous boundary conditions. Subsequently, we establish the well‐posedness theory for scenarios involving both small and large wavenumbers. Our proof hinges upon the regularity attributes of select fractional elliptic equations and capitalizes fully on the structural peculiarities of the elliptic system, which distinguish it from classical cases. Thereafter, we focus on the inverse medium scattering problem pertinent to the DDFH equation, framed within the Bayesian statistical framework. We address two scenarios: one devoid of model reduction errors and another characterized by such errors—arising from the implementation of certain absorbing boundary conditions. More precisely, based on the properties of the forward operator, well‐posedness of the posterior measures have been proved in both cases, which provide an opportunity to quantify the uncertainties of this problem. [ABSTRACT FROM AUTHOR]

The inertial forward–backward splitting algorithm can be considered as a modified form of the forward–backward algorithm for variational inequality problems with monotone and Lipschitz continuous cost mappings. By using parallel and inertial techniques and the forward–backward splitting algorithm, in this paper, we propose a new parallel inertial forward–backward splitting algorithm for solving variational inequality problems, where the constraints are the intersection of common solution sets of a finite family of variational inclusion problems. Then, strong convergence of proposed iteration sequences is showed under standard assumptions imposed on cost mappings in a real Hilbert space. Finally, some numerical experiments demonstrate the reliability and benefits of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this review paper, we discuss some of our recent results concerning the rigorous analysis of initial boundary value problems (IBVPs) and newly discovered effects for certain evolution partial differential equations (PDEs). These equations arise in the applied sciences as models of phenomena and processes pertaining, for example, to continuum mechanics, heat‐mass transfer, solid–fluid dynamics, electron physics and radiation, chemical and petroleum engineering, and nanotechnology. The mathematical problems we address include certain well‐known classical variations of the traditional heat (diffusion) equation, including (i) the Sobolev–Barenblatt pseudoparabolic PDE (or modified heat or second‐order fluid equation), (ii) a fourth‐order heat equation and the associated Cahn–Hilliard (or Kuramoto–Sivashinsky) model, and (iii) the Rubinshtein–Aifantis double‐diffusion system. Our work is based on the synergy of (i) the celebrated Fokas unified transform method (UTM) and (ii) a new approach to the rigorous analysis of this method recently introduced by one of the authors. In recent works, we considered forced versions of the aforementioned PDEs posed in a spatiotemporal quarter‐plane with arbitrary, fully non‐homogeneous initial and boundary data, and we derived formally effective solution representations, for the first time in the history of the models, justifying a posteriori their validity. This included the reconstruction of the prescribed initial and boundary conditions, which required careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula was utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. Importantly, this analysis is indispensable for proving (non)uniqueness of solution. These works extend previous investigations. The usefulness of our closed‐form solutions will be demonstrated by studying their long‐time asymptotics. Specifically, we will briefly review some asymptotic results about Barenblatt's equation. [ABSTRACT FROM AUTHOR]

The Ciarlet–Raviart mixed method is popular for the biharmonic equations/eigenvalue problem. In this paper, we propose a multigrid discretization based on the shifted‐inverse iteration of Ciarlet–Raviart mixed discontinuous Galerkin method for the biharmonic eigenvalue problem. We prove the a priori error estimates of the approximate eigenpairs. We also give the a posteriori error estimates of the approximate eigenvalues and prove the reliability of the estimator and implement adaptive computation. Numerical experiments show that our method can efficiently compute biharmonic eigenvalues. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of weak solutions of the system that describes the two‐phase two‐component fluid flow in porous media. Both two‐phase and possible one‐phase flow regions are taken into account. Our research is based on a global pressure, an artificial variable that allows us to partially decouple the original equations. As a second primary unknown for the system, we choose the gas pseudo‐pressure, a persistent variable which coincides with the gas pressure in the two‐phase regions while it does not have physical meaning in one‐phase flow regions, when only the liquid phase is present. This allows us to introduce an another persistent variable that is an artificial variable in one‐phase flow regions and a physical variable in two‐phase flow regions—the capillary pseudopressure. We rewrite the system's equations in a fully equivalent form in terms of the global pressure and the gas‐pseudo pressure. In order to prove the existence of weak solutions of obtained system, we also use the capillary pseudo‐pressure. By using it, we can decouple obtained equations on the discrete level. This allows us to derive the existence result for weak solutions in more tractable way. [ABSTRACT FROM AUTHOR]

This paper fits in the theory of international agreements by studying the success of stable coalitions of agents seeking the preservation of a public good. Extending Baliga and Maskin, we consider a model of N$$ N $$ homogeneous agents with quasi‐linear utilities of the form uj(rj;r)=rα−rj$$ {u}_j\left({r}_j;r\right)&#x0003D;{r}&#x0005E;{\alpha }-{r}_j $$, where r$$ r $$ is the aggregate contribution and the exponent α$$ \alpha $$ is the elasticity of the gross utility. When the value of the elasticity α$$ \alpha $$ increases in its natural range (0,1)$$ \left(0,1\right) $$, we prove the following five main results in the formation of stable coalitions: (i) the gap of cooperation, characterized as the ratio of the welfare of the grand coalition to the welfare of the competitive singleton coalition grows to infinity, which we interpret as a measure of the urge or need to save the public good; (ii) the size of stable coalitions increases from 1 up to N$$ N $$; (iii) the ratio of the welfare of stable coalitions to the welfare of the competitive singleton coalition grows to infinity; (iv) the ratio of the welfare of stable coalitions to the welfare of the grand coalition “decreases” (a lot), up to when the number of members of the stable coalition is approximately N/e$$ N/e $$ and after that it “increases” (a lot); and (v) the growth of stable coalitions occurs with a much greater loss of the coalition members when compared with free‐riders. Result (v) has two major drawbacks: (a) A priori, it is difficult to “convince” agents to be members of the stable coalition and (b) together with results (i) and (iv), it explains and leads to the “pessimistic” Barrett's paradox of cooperation, even in a case not much considered in the literature: The ratio of the welfare of the stable coalitions against the welfare of the grand coalition is small, even in the extreme case where there are few (or a single) free‐riders and the gap of cooperation is large. “Optimistically,” result (iii) shows that stable coalitions do much better than the competitive singleton coalition. Furthermore, result (ii) proves that the paradox of cooperation is resolved for larger values of α$$ \alpha $$ so that the grand coalition is stabilized. [ABSTRACT FROM AUTHOR]

This paper considers a non‐standard Optimal Control problem that has an application in economics. The primary focus of this research is to solve the royalty problem, which has been categorized as a non‐standard Optimal Control problem, where the final state value and its functional performance index value are unknown. A new continuous necessary condition is investigated for the final state value so that it will convert the final costate value into a non‐zero value. The research analyzes the seven‐stage royalty piecewise function, which is then approximated to continuous form with the help of the hyperbolic tangent function and solves the problem by using a new modified shooting method. This modified shooting method applies Sufahani–Ahmad–Newton–Brent–Royalty Algorithm and Sufahani‐Ahmad‐Powell‐Brent‐Royalty Algorithm. For a validation process, the results are compared with the existing methods such as Euler, Runge–Kutta, Trapezoidal, and Hermite–Simpson approximations, and the results show that the proposed method yields an accurate terminal state value. [ABSTRACT FROM AUTHOR]

This paper investigates an optimal investment‐reinsurance problem for an insurer who possesses inside information regarding the future realizations of the claim process and risky asset process. The insurer sells insurance contracts, has access to proportional reinsurance business, and invests in a financial market consisting of three assets: one risk‐free asset, one bond, and one stock. Here, the nominal interest rate is characterized by the Vasicek model, and the stock price is driven by Heston's stochastic volatility model. Applying the enlargement of filtration techniques, we establish the optimal control problem in which an insurer maximizes the expected power utility of the terminal wealth. By using the dynamic programming principle, the problem can be changed to four‐dimensional Hamilton–Jacobi–Bellman equation. In addition, we adopt a deep neural network method by which the partial differential equation is converted to two backward stochastic differential equations and solved by a stochastic gradient descent‐type optimization procedure. Numerical results obtained using TensorFlow in Python and the economic behavior of the approximate optimal strategy and the approximate optimal utility of the insurer are analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the following fractional Kirchhoff‐type pseudo parabolic equation driven by a nonlocal integro‐differential operator ℒK$$ {\mathcal{L}}_K $$: ut|x|2s+M([u]s2)ℒKu+ℒKut=|u|p−2ulog|u|,$$ \frac{u_t}{{\left&#x0007C;x\right&#x0007C;}&#x0005E;{2s}}&#x0002B;M\left({\left[u\right]}_s&#x0005E;2\right){\mathcal{L}}_Ku&#x0002B;{\mathcal{L}}_K{u}_t&#x0003D;{\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-2}u\log \mid u\mid, $$ where [u]s$$ {\left[u\right]}_s $$ represents the Gagliardo seminorm of u$$ u $$. Instead of imposing specific assumptions on the Kirchhoff function, we introduce a more general sense to establish the local existence of weak solutions. Moreover, via the sharp fractional Hardy inequality, the decay estimates for global weak solutions, the blow‐up criterion, blow‐up rate, and the upper and lower bounds of the blow‐up time are derived. Lastly, we discuss the global existence and finite time blow‐up results with high initial energy. [ABSTRACT FROM AUTHOR]

This paper is concerned with the class of quasilinear Schrödinger equations: 0.1 −Δu+V(x)u−Δ(1+u2)α2αu2(1+u2)(2−α)/2=h(x,u),x∈ℝN,$$ -\Delta u&#x0002B;V(x)u-\left[\Delta {\left(1&#x0002B;{u}&#x0005E;2\right)}&#x0005E;{\frac{\alpha }{2}}\right]\frac{\alpha u}{2{\left(1&#x0002B;{u}&#x0005E;2\right)}&#x0005E;{\left(2-\alpha \right)/2}}&#x0003D;h\left(x,u\right),x\in {\mathrm{\mathbb{R}}}&#x0005E;N, $$ which models the self‐channeling of a high‐power ultra short laser in matter provided α=1$$ \alpha &#x0003D;1 $$, where N>2,V(x)∈C1(ℝN),α≥1$$ N>2,V(x)\in {C}&#x0005E;1\left({\mathrm{\mathbb{R}}}&#x0005E;N\right),\alpha \ge 1 $$ is a parameter, and h(x,u)∈C(ℝN×ℝ)$$ h\left(x,u\right)\in C\left({\mathrm{\mathbb{R}}}&#x0005E;N\times \mathrm{\mathbb{R}}\right) $$. Under some appropriate assumptions on h(x,u)$$ h\left(x,u\right) $$, we establish the nonexistence of solutions for the above problem. [ABSTRACT FROM AUTHOR]

This paper presents two methods for solving two‐dimensional linear and nonlinear time‐fractional advection–diffusion equations with Caputo fractional derivatives. To effectively manage endpoint singularities, we propose an advanced space‐time Galerkin technique and a collocation spectral method, both employing generalized Bernstein‐like basis functions (GBFs). The properties and behaviors of these functions are examined, highlighting their practical applications. The space‐time spectral methods incorporate GBFs in the temporal domain and classical Bernstein polynomials in the spatial domain. Fractional equations frequently produce irregular solutions despite smooth input data due to their singular kernel. To address this, GBFs are applied to the time derivative and classical Bernstein polynomials to the spatial derivative. A thorough error analysis confirms the technique's accuracy and convergence, offering a robust theoretical basis. Numerical experiments validate the method, demonstrating its effectiveness in solving both linear and nonlinear time‐fractional advection–diffusion equations. [ABSTRACT FROM AUTHOR]

In this paper, our aim is to introduce a new definition of ( k,ℑ)$$ k,\mathfrak{\Im}\Big) $$‐incomplete Wright hypergeometric matrix functions ( k,ℑ$$ \left(k,\mathfrak{\Im}\right) $$‐IWHMFs) using the k$$ k $$‐incomplete Pochhammer matrix symbol. First, we define the k$$ k $$‐incomplete gamma matrix function and introduce the k$$ k $$‐incomplete Pochhammer matrix symbols. Furthermore, we present differential formulas and integral representation related to these k,ℑ$$ \left(k,\mathfrak{\Im}\right) $$‐IWHMFs. We have also obtained some results regarding the k$$ k $$‐fractional calculus operators of these k,ℑ$$ \left(k,\mathfrak{\Im}\right) $$‐IWHMFs. Finally, we investigate the solutions of fractional kinetic equations (FKEs) involving the k,ℑ$$ \left(k,\mathfrak{\Im}\right) $$‐IWHMFs. [ABSTRACT FROM AUTHOR]

In numerical analysis, the quadrature formulas serve as a pivotal tool for approximating definite integrals. In this paper, we introduce a family of quadrature formulas and analyze their associated error bounds for convex functions. The main advantage of these new error bounds is that from these error bounds, we can find the error bounds of different quadrature formulas. This work extends the traditional quadrature formulas such as the midpoint formula, trapezoidal formula, Simpson's formula, and Boole's formula. We also use the power mean and Hölder's integral inequalities to find more general and sharp bounds. Furthermore, we give numerical example and applications to quadrature formulas of the newly established inequalities. [ABSTRACT FROM AUTHOR]

In this paper, Dirac system with interface conditions and spectral parameter dependent boundary conditions is investigated. By introducing a new Hilbert space, the original problem is transformed into an operator problem. Then the continuity and differentiability of the eigenvalues with respect to the parameters in the problem are showed. In particular, the differential expressions of eigenvalues for each parameter are given. These results would provide theoretical support for the calculation of eigenvalues of the corresponding problems. [ABSTRACT FROM AUTHOR]

In this article, we initially provided the relationship between the RL fractional integral and the Caputo fractional derivative of different orders. Additionally, it is clear from the literature that studies into boundary value problems involving multi‐term operators have been conducted recently, and the aforementioned idea is used in the formulation of several novel models. We offer a unique coupled system of fractional delay differential equations with proper respect for the role that multi‐term operators play in the research of fractional differential equations, taking into account the newly established solution for fractional integral and derivative. We also made the assumptions that connected integral boundary conditions would be added on top of n$$ n $$‐fractional differential derivatives. The requirements for the existence and uniqueness of solutions are also developed using fixed‐point theorems. While analyzing various sorts of Ulam's stability results, the qualitative elements of the underlying model will also be examined. In the paper's final section, an example is given for purposes of demonstration. [ABSTRACT FROM AUTHOR]

It is effective to consider discretized problems for partial differential equations that are difficult to analyze themselves. The global solvability of the initial value problem of the Kirchhoff equation is still unsolved except in cases where the initial value is a specially restricted class. In this paper, we prove the global solvability of the semidiscrete Kirchhoff equation, which is obtained by discretizing the Kirchhoff equation with respect to spatial variables. [ABSTRACT FROM AUTHOR]

In this article, we investigate the existence and uniqueness of solutions to a generalized nonlinear functional integral equation (G‐NLFIE) associated with certain epidemic models of infectious diseases, defined within the Banach space C[0,1]$$ C\left[0,1\right] $$. Our existence results include several specific cases of nonlinear functional integral equations that commonly occur in nonlinear sciences. We then introduce an iterative algorithm that combines Adomian's decomposition method (ADM) with the modified homotopy perturbation method (mHPM) to approximate solutions to the G‐NLFIE. The paper addresses the convergence properties and error analysis of this method. Finally, we present numerical examples to demonstrate the effectiveness and efficiency of our proposed approach. [ABSTRACT FROM AUTHOR]

This paper deals with the free boundary problem for a double‐layered tumor filled with quiescent cells and proliferating cells, where time delay τ>0$$ \tau >0 $$ in cell proliferation is taken into account. These two types of living cells exhibit different metabolic responses and consume nutrients σ$$ \sigma $$ at different rates λ1$$ {\lambda}_1 $$ and λ2$$ {\lambda}_2 $$ ( λ1⩽λ2$$ {\lambda}_1\leqslant {\lambda}_2 $$). Time delay happens between the time at which a cell commences mitosis and the time at which the daughter cells are produced. The problem is reduced to a delay differential equation on the tumor radius R(t)$$ R(t) $$ over time, and the difficulty arises from the jump discontinuity of the consumption rate function. We give rigorous analysis on this new model and study the dynamical behavior of the global solutions for any initial φ(t)$$ \varphi (t) $$. [ABSTRACT FROM AUTHOR]

In approximation theory, classical discrete operators, like generalized sampling, Szász‐Mirak'jan, Baskakov, and Bernstein operators, have been extensively studied for scalar functions. In this paper, we look at the approximation of curves by a class of discrete operators, and we exhibit graphical examples concerning several cases. The topic has useful implications about the computer graphics and the image processing: We discuss applications on the approximation and the reconstruction of curves in images. [ABSTRACT FROM AUTHOR]

The parabolic problem ut−Δu=λf(x)(1−u)2+P$$ {u}_t-\Delta u&#x0003D;\frac{\lambda f(x)}{{\left(1-u\right)}&#x0005E;2}&#x0002B;P $$ on a bounded domain Ω$$ \Omega $$ of Rn$$ {R}&#x0005E;n $$ with Dirichlet boundary condition models the microelectromechanical systems (MEMS) device with an external pressure term. In this paper, we classify the behavior of the solutions to this equation. We first show that under certain initial conditions, there exist critical constants P∗$$ {P}&#x0005E;{\ast } $$ and λP∗$$ {\lambda}_P&#x0005E;{\ast } $$ such that when 0≤P≤P∗,0<λ≤λP∗$$ 0\le P\le {P}&#x0005E;{\ast },0<\lambda \le {\lambda}_P&#x0005E;{\ast } $$, there exists a global solution, while for 0≤P≤P∗,λ>λP∗$$ 0\le P\le {P}&#x0005E;{\ast },\lambda >{\lambda}_P&#x0005E;{\ast } $$ or P>P∗$$ P>{P}&#x0005E;{\ast } $$, the solution quenches in finite time. The estimates of voltage λP∗$$ {\lambda}_P&#x0005E;{\ast } $$, quenching time T$$ T $$, and pressure term P∗$$ {P}&#x0005E;{\ast } $$ are investigated. The quenching set Σ$$ \varSigma $$ is proved to be a compact subset of Ω$$ \Omega $$ with an additional condition on f(x)$$ f(x) $$, provided Ω⊂Rn$$ \Omega \subset {R}&#x0005E;n $$ is a convex bounded set. In particular, if Ω$$ \Omega $$ is radially symmetric, then the origin is the only quenching point. Furthermore, we not only derive the two‐sided bound estimate for the quenching solution but also obtain its asymptotic behavior near the quenching time. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the minimal norm Hermitian solution, pure imaginary Hermitian solution and pure real Hermitian solution of the reduced biquaternion matrix equation. We introduce the new real representation of the reduced biquaternion matrix and the special properties of Vec(ΨPMQ)$$ Vec\left({\Psi}_{PMQ}\right) $$. We present the sufficient and necessary conditions of three solutions and the corresponding numerical algorithms for solving the three solutions. Finally, we show that our method is better than the complex representation method in terms of error and CPU time in numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider a diffusive predator–prey model with spatial memory of prey and gestation delay of predator. For the system without delays, we study the stability of the positive equilibrium in the case of diffusion and no diffusion, respectively. For the delayed model without diffusions, the existence of Hopf bifurcation is discussed. Further, we investigate the stability switches of the model with delays and diffusions when two delays change simultaneously by calculating the stability switching curves and obtain the existence of Hopf bifurcation. We also calculate the normal form of Hopf bifurcation to determine the direction of Hopf bifurcation and the stability of bifurcation periodic solutions. Finally, numerical simulations verify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse problem of determining the initial value of the distributed‐order time‐fractional diffusion equation from the final time observation data, which arises in some ultra‐slow diffusion phenomena in applied areas. Since the problem is ill‐posed, we propose an iterated regularization method based on the Nesterov acceleration strategy to deal with it. Convergence rates for the regularized approximation solution are given under both the a priori and a posteriori regularization parameter choice rules. It is shown that the proposed method can always yield the order optimal convergence rates as long as the iteration parameter which appears in the Nesterov acceleration strategy is chosen large enough. In numerical aspect, the main advantage of the proposed method lies in its simplicity. Specifically, due to the Nesterov acceleration strategy, only a few number of iteration steps are required to obtain the approximation solution, and at each iteration step, we only need to numerically solve the standard initial‐boundary value problem for the distributed‐order time‐fractional diffusion equation. Some numerical examples including one‐dimensional and two‐dimensional cases are presented to illustrate the validity and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The purpose of this article is to provide a number of Hermite–Hadamard and Fejér type integral inequalities for a class of h$$ h $$‐superquadratic functions. We then develop the fractional perspective of inequalities of Hermite–Hadamard and Fejér types by use of the Riemann–Liouville fractional integral operators and bring up with few particular cases. Numerical estimations based on specific relevant cases and graphical representations validate the results. Another motivating component of the study is that it is enriched with applications of modified Bessel function of first type, special means, and moment of random variables by defining some new functions in terms of modified Bessel function and considering uniform probability density function. The results in this paper have not been initiated before in the frame of h$$ h $$‐superquadraticity. We are optimistic that this effort will greatly stimulate and encourage additional research. [ABSTRACT FROM AUTHOR]

A dynamical system is considered such that, in this system, particles move on a toroidal lattice of the dimension N1×N2$$ {N}_1\times {N}_2 $$ according to a version of the rule of particle movement in Biham–Middleton–Levine traffic model. We introduce a stochastic case with direction choice for particles. Particles of the first type move along rows, and the particles of the second type move along columns. The goal is to find conditions of self‐organization system for any lattice dimension. We have proved that the BML model as a dynamical system is a special case of Buslaev nets. This equivalence allows us to use of Buslaev net analysis techniques to investigate the BML model. In Buslaev nets conception, the self‐organization property of the system corresponds to the existence of velocity single point spectrum equal to 1. In the paper, we consider the model version when one notable aspect is that a particle may change its type. Exactly, we assume a constant probability q$$ q $$ that a particle changes type at each step. In the case where q=0$$ q&#x0003D;0 $$, the system corresponds to the classical version of the BML model. We define a state of the system where all particles continue to move indefinitely, in both the present and the future, as a state of free movement. A sufficient condition for the system to result in a state of free movement from any initial state (condition for self‐organization) has been found. This condition is that the number of particles be not greater than half the greatest common divisor of the numbers N1,N2,…,Nn$$ {N}_1,{N}_2,\dots, {N}_n $$. It has been proved that, if n=2$$ n&#x0003D;2 $$, and whether 0