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The purpose of the present note is to point out mathematical mistakes in some published papers on boundary value problems to prevent usage of mistaken results and methods by mathematicians. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we provide an extension result for the existence and uniqueness of solutions for a coupled system of fractional Langevin differential equations with antiperiodic boundary conditions. The system involves two different Caputo fractional derivatives defined on different intervals and associated with boundary conditions described by sequential fractional derivatives. As a conclusion for our main result, we deduce the results of H. Fazli, J.J. Nieto, Fractional Langevin equation with anti‐periodic boundary conditions, Chaos Soliton Fract, 114 (2018),332–337 under less restrictive conditions. The consistency of the main results is demonstrated by two numerical examples. For the sake of completeness, we end the paper by a concluding discussion. [ABSTRACT FROM AUTHOR]

In the present paper, we construct the existence of nontrivial solutions to a new kind of Hadamard fractional boundary value problem on an unbounded domain. With the contribution of some fixed point theorems in cone and the corresponding Green function, we ensure sufficient conditions for the Hadamard fractional boundary value problem. Also, the paper concludes with two examples to demonstrate our results. [ABSTRACT FROM AUTHOR]

This paper deals with non-constant positive steady-state solutions of a predator-prey system with non-monotonic functional response, also called Holling type-IV interaction terms, and diffusion under the homogeneous Neumann boundary condition. We first establish positive upper and lower bounds for such solutions, and then study their non-existence, global existence and bifurcation. [ABSTRACT FROM AUTHOR]

The present paper explores a two‐dimensional mixed bio‐convective unsteady viscous hydro‐magnetic Casson Williamson nanofluid flow model with heat and mass transport incorporating motile microorganisms towards a stretchy spinning disc. The flow concept is accomplished by rotating a stretched disc with a time‐varying angular velocity. By applying a magnetic field normal to the axial direction, a magnetic interaction is taken into consideration. The Casson Williamson nanofluid contains nanosized particles suspended with swimming motile microorganisms and the rotation of the disc is exhibited by buoyancy forces, thermophoresis, suction/injection, zero mass flux conditions, variable thermal conductivity, Joule heating and so forth. The obtained flow narrating differential equations of the model are transformed into ordinary differential system. This is accomplished by simulating boundary value problems using the shooting technique using the ‘ND‐Solve’ approach included in the Mathematica software (Mathematica 12). The implications of the engaged parameters such as Williamson fluid parameter, Casson fluid (CF) parameter, thermophoresis parameter, Brownian motion parameter and so forth, on both axial and radial velocities, temperature, concentration of nanoparticles and microorganisms are explained by means of graphical and tabular constructions. This paper's validity has been confirmed and its findings align with those of other previously published papers. Furthermore, it is found that both the axial and radial velocity profiles are seen to be diminishing functions of the CF parameter. The identified observation may have theoretical implications for a number of engineering procedures, solar energy systems, biofuel cells and extrusion system improvement. Moreover, this work finds application in micro‐fabrication techniques and the chemical industry. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the boundary value problem of the wave equation with nonlocal damping ‖ut‖2qut$$ {\left\Vert {u}_t\right\Vert}_2^q{u}_t $$ and nonlinear source term. The main purpose of this paper is to provide a systematic research on the dynamic behavior of the solutions with three different energy levels. More precisely, we prove the global existence, energy decay estimate and blow‐up of solution at both subcritical ( E(0)

The paper concerns boundary value problems for an elastic body with a thin elastic inclusion for a noncoercive case. The inclusion is assumed to be delaminated thus forming a crack between the inclusion and the surrounding elastic body. To provide a mutual nonpenetration between crack faces, we consider inequality type boundary conditions with unknown set of a contact. In the paper, a solution existence of the equilibrium problems is proved for the cases when one or two given points of the inclusion are fixed as well as and for the case without these restrictions. [ABSTRACT FROM AUTHOR]

Copyright of Bautechnik is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we consider a class of nonlinear second‐order functional differential equations with piecewise constant arguments with applications to a thermostat that is controlled by the introduction of functional terms in the temperature and the speed of change of the temperature at some fixed instants. We first prove some comparison results for boundary value problems associated to linear delay differential equations that allow to give a priori bounds for the derivative of the solutions, so that we can control not only the values of the solutions but also their rate of change. Then, we develop the method of upper and lower solutions and the monotone iterative technique in order to deduce the existence of solutions in a certain region (and find their approximations) for a class of boundary value problems, which include the periodic case. In the approximation process, since the sequences of the derivatives for the approximate solutions are, in general, not monotonic, we also give some estimates for these derivatives. We complete the paper with some examples and conclusions. [ABSTRACT FROM AUTHOR]

The numerical solution of dynamic problems for porous fluid‐saturated solids is often performed with the use of simplified equations known as the u‐p approximation. The simplification of the equations consists in neglecting some acceleration terms, which is justified for a certain class of problems related, in particular, to geomechanics and earthquake engineering. There exist two u‐p approximations depending on how many acceleration terms are neglected. All comparative studies of the exact and u‐p formulations are focused on the question of how well the u‐p solutions approximate those obtained with the exact equations. In this paper, the equations are compared from a different point of view, addressing the question of well‐posedness of boundary value problems. The exact equations must be hyperbolic and satisfy the corresponding hyperbolicity conditions for the boundary value problems to be well posed. The u‐p equations are not of the form to which the conventional definition of hyperbolicity applies. A slight extension of the approach makes it possible to derive hyperbolicity conditions as necessary conditions for well‐posedness for the u‐p approximations. The hyperbolicity conditions derived in this paper for the u‐p approximations are formulated in terms of the acoustic tensor of the skeleton. They differ essentially from the hyperbolicity conditions for the exact equations. [ABSTRACT FROM AUTHOR]

In this paper, the linear theory of elasticity for materials with a triple porosity (macro, meso and micro levels of porosities) structure based on the volume fraction concept is considered. The external boundary value problems (BVPs) of steady vibrations of this theory are investigated. Namely, the uniqueness theorem for classical solutions of these BVPs is established. Then, the basic properties of the surface (single‐layer and double‐layer) potentials are given. Finally, on the basis of the properties of potentials, the existence theorems for classical solutions of the above mentioned BVPs of steady vibrations are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations. [ABSTRACT FROM AUTHOR]

In this paper, we continue to study an initial boundary value problems to a model describing the evolution in time of diffusive phase interfaces in sea ice growth. In a previous paper, the global existence and the large‐time behavior of weak solutions in one space was studied under Dirichlet boundary conditions. Here, we show that the global existence of weak solutions and the large‐time behavior are also studied under Neumann boundary condition. In this paper, we study in space dimension lower than or equal to 3. [ABSTRACT FROM AUTHOR]

In this paper, we study the long‐time dynamical behavior of the non‐autonomous velocity–vorticity–Voigt model of the 3D Navier–Stokes equations with damping and memory. We first investigate the existence and uniqueness of weak solutions to the initial boundary value problem for above‐mentioned model. Next, we prove the existence of uniform attractor of this problem, where the time‐dependent forcing term f∈Lb2(ℝ;H−1(Ω))$$ f\in {L}_b^2\left(\mathbb{R};{H}^{-1}\left(\Omega \right)\right) $$ is only translation bounded instead of translation compact. The results in this paper will extend and improve some results in Yue and Wang (Comput. Math. Appl., 2020) in the case of non‐autonomous and contain memory kernels which have not been studied before. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of a Dirichlet boundary value problem for a system of two coupled anisotropic Darcy–Forchheimer–Brinkman equations on a bounded Lipschitz domain in ℝn(n=2,3)$$ {\mathrm{\mathbb{R}}}^n\left(n=2,3\right) $$. Using variational methods and fixed point techniques, we obtain a well‐posedness result in L2$$ {L}^2 $$‐based Sobolev spaces for sufficiently small data. As an application, we investigate numerically the lid‐driven flow problem in a square cavity saturated with a bidisperse porous medium and analyze the effect of various physical parameters on the fluid motion. [ABSTRACT FROM AUTHOR]

In this paper, we develop a hypercomplex operator calculus to treat fully analytically boundary value problems for the homogeneous and inhomogeneous fractional Helmholtz equation where fractional derivatives in the sense of Caputo and Riemann–Liouville are applied. Our method extends the recently proposed fractional reduced differential transform method (FRDTM) by using fractional derivatives in all directions. For the special separable case in three dimensions, we obtain completely explicit representations for the fundamental solution. This allows us to interpret and to understand the appearance of spatial steady‐state solutions or spatial blow‐ups of the fractional Helmholtz equation in a better way. More precisely, we were able to present explicit conditions for the parameters in the representation formulas of the fundamental solutions under which we obtain bounded or spatial decreasing steady‐solutions and when spatial blow‐ups occur. We also illustrate this with some representative numerical examples. Furthermore, we show that it is possible to recover the recently studied cases as well as the classical cases as particular limit cases within our more general setting. Using the hypercomplex operator approach also allows us to factorize the fractional Helmholtz operator and obtain some interesting duality relations between left and right derivatives, Caputo and Riemann–Liouville derivatives, and eigensolutions of antipodal eigenvalues in terms of a generalized Borel–Pompeiu formula. This factorization, in turn, allows us to tackle inhomogeneous fractional Helmholtz problems. [ABSTRACT FROM AUTHOR]

The paper presents the execution of the Adomian Decomposition Method to treat thermo‐fluidics of Williamson fluid flow in a vertical parallel plates channel filled with Williamson fluid–saturated porous medium. The channel walls experience asymmetric cooling. The fully developed flow setup constitutes a non‐linear boundary value problem tackled by the Adomain Decomposition Method (ADM) with the help of MAPLE. The computational procedure yields velocity, temperature distributions, skin friction, and Nusselt number. The results thus obtained are compared with those obtained by the RK‐Fehlberg numerical method to ascertain the accuracy of ADM. The results for quantities of interest viz. Skin friction. Nusselt number, entropy, and global entropy are shown in tabular/pictorial forms. These quantities exhibit qualitative and quantitative responses to embedded parameters discussed in detail in results and discussions section. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the existence and uniqueness of solutions for a sixth‐order differential equation involving nonlocal and integral boundary conditions. Firstly, we obtain the properties of the relevant Green's functions. The existence result of at least one nontrivial solution is obtained by applying the Krasnoselskii–Zabreiko fixed point theorem. Moreover, we also establish the existence of unique solution to the considered problem via Hölder and Minkowski inequalities and Rus's theorem. Finally, two numerical examples are included to show the applicability of our main results. [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]

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, 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]

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]

This paper is devoted to the theoretical and numerical investigation of the initial boundary value problem for a system of equations used for the description of waves in coastal areas, namely, the Boussinesq–Abbott system in the presence of topography. We propose a procedure that allows one to handle very general linear or nonlinear boundary conditions. It consists in reducing the problem to a system of conservation laws with nonlocal fluxes and coupled to an ordinary differential equation. This reformulation is used to propose two hybrid finite volumes/finite differences schemes of first and second order, respectively. The possibility to use many kinds of boundary conditions is used to investigate numerically the asymptotic stability of the boundary conditions, which is an issue of practical relevance in coastal oceanography since asymptotically stable boundary conditions would allow one to reconstruct a wave field from the knowledge of the boundary data only, even if the initial data are not known. [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]

As a first stage to study the global large solutions of the radiation hydrodynamics model with viscosity and thermal conductivity in the high‐dimensional space, we study the problems in high dimensions with some symmetry, such as the spherically or cylindrically symmetric solutions. Specifically, we will study the global classical large solutions to the radiation hydrodynamics model with spherically or cylindrically symmetric initial data. The key point is to obtain the strict positive lower and upper bounds of the density ρ$\rho$ and the lower bound of the temperature θ$\theta$. Compared with the Navier–Stokes equations, these estimates in the present paper are more complicated due to the influence of the radiation. To overcome the difficulties caused by the radiation, we construct a pointwise estimate between the radiative heat flux q$q$ and the temperature θ$\theta$ by studying the boundary value problem of the corresponding ordinary differential equation. And we consider a general heat conductivity: κ(ρ,θ)⩾C(1+θβ)$\kappa (\rho,\theta)\geqslant C(1+\theta ^\beta)$ if ρ⩽ρ+$\rho \leqslant \rho _+$; κ(ρ,θ)⩽C(1+θβ)$\kappa (\rho,\theta)\leqslant C(1+\theta ^\beta)$ if ρ⩾ρ−>0$\rho \geqslant \rho _->0$. This can be viewed as the first result about the global classical large solutions of the radiation hydrodynamics model with some symmetry in the high‐dimensional space. [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]

In this paper, we study the existence of positive solution to boundary value problem of fractional differential equations with p$$ p $$‐Laplacian operator. By using Avery–Peterson theorem, some new existence results of three increasing positive solutions are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence of multiplicity of positive solutions to a new generalized Riemann–Liouville type fractional Fisher‐like equation on a semi‐infinite interval equipped with nonlocal multipoint boundary conditions involving Riemann–Liouville fractional derivative and integral operators. The existence of at least two positive solutions for the given problem is established by using the concept of complete continuity and iterative positive solutions. We show the existence of at least three positive solutions to the problem at hand by applying the generalized Leggett–Williams fixed‐point theorem due to Bai and Ge [Z. Bai, B. Ge, Existence of three positive solutions for some second‐order boundary value problems, Comput. Math. Appl. 48 (2014) 699‐70]. Illustrative examples are constructed to demonstrate the effectiveness of the main results. It has also been indicated in Section 5 that some new results appear as special cases by choosing the parameters involved in the given problem appropriately. [ABSTRACT FROM AUTHOR]

Data‐driven mechanics offers great potential in engineering applications, where the efficient simulation of complex and, in particular, path‐dependent material behavior is often challenging. Within this approach, conventional material models are replaced by data sets containing snapshots of stress, strain and the history of both assumed to be sufficiently accurate representations of the underlying material behavior. Based on these snapshots and on physical admissibility, a distance function is built which is minimized to yield the boundary value problems' solution. The aim of this paper is to apply this framework to accurately simulate the complex behavior of shape memory alloy wires under cyclic loading, under which these materials exhibit a degradation of their features, denoted functional fatigue. The constructed synthetic data sets are enriched by real experimental data, showcasing that the data‐driven method is capable of combining both approaches of classic material modeling and modern methods which directly incorporate experimental data. [ABSTRACT FROM AUTHOR]

This paper investigates the mechanical properties of carbon‐based nanostructures, in particular graphene, using the FE2${\mathrm{FE}}^{2}$ method. The local boundary value problem is calculated using molecular mechanics and is therefore, in contrast to continuum mechanics, discrete. In the case of graphene as a microstructure, atomistic interactions occur with a range of up to three or more atoms. For the local boundary value problem, a representative volume element is developed that is compatible with the atomistic interactions. Boundary effects resulting from the molecular mechanical consideration are eliminated. Boundary conditions compatible with the Hill–Mandel condition are formulated for the local boundary value problem. The results are verified with fully atomistic simulations. [ABSTRACT FROM AUTHOR]

In this paper, we study a free‐boundary problem modeling the growth of spherically symmetric tumors with angiogenesis and a necrotic core, where the Robin boundary condition is imposed for the nutrient concentration. The existence of a global solution is established by first reducing the free‐boundary problem into an equivalent initial boundary value problem for a nonlinear strongly singular parabolic equation on a fixed domain, then proving that an approximation problem admits a unique solution by the Schauder fixed point theorem combined with the Lp$$ {L}&#x0005E;p $$ estimates for parabolic equations, and finally taking the limit. Compared with the Dirichlet boundary value condition problem, the Robin condition causes some new difficulties in making rigorous analysis of the model, particularly on the uniqueness of solutions to the approximation problem. [ABSTRACT FROM AUTHOR]

This paper presents a spatiotemporal reaction–diffusion model for epidemics to predict how the infection spreads in a given space. The model is based on a system of partial differential equations with the Neumann boundary conditions. First, we study the existence and uniqueness of the solution of the model using the semigroup theory and demonstrate the boundedness of solutions. Further, the proposed model's basic reproduction number is calculated using the eigenvalue problem. Moreover, the dynamic behavior of the disease‐free steady states of the model for R0<1$$ {\mathcal{R}}_0<1 $$ is investigated. The uniform persistence of the model is also discussed. In addition, the global asymptotic stability of the endemic steady state is examined. Finally, the numerical simulations validate the theoretical results. [ABSTRACT FROM AUTHOR]

Fractional calculus is used to examine and enhance the concept of calculus in diverse fields of science. In this paper, we establish Hermite‐Hadamard inequalities for composite (α,m)$$ \left(\alpha, m\right) $$‐convex function. The generalized identities are established for Riemann‐type fractional integrals. The explored identities are used to examine error estimates of Hermite‐Hadamard inequalities for (α,m)$$ \left(\alpha, m\right) $$‐convex function concerning a strictly monotone function. Some results that exist in literature are obtained as special cases to our general results. The conclusions of this article may be useful in determining the uniqueness of partial differential equations and fractional boundary value problems. [ABSTRACT FROM AUTHOR]

Copyright of Bautechnik is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The paper is concerned with the Bari basis property of a boundary value problem associated in L2([0,1];C2)$L^2([0,1]; \mathbb {C}^2)$ with the following 2 × 2 Dirac‐type equation for y=col(y1,y2)$y = \operatorname{col}(y_1, y_2)$: LU(Q)y=−iB−1y′+Q(x)y=λy,B=b100b2,b1<0

Summary: This paper develops a Müntz–Legendre wavelet method for solving a fractional optimal control problem with dynamic constraint as a fractional Sturm–Liouville problem. We first derive the necessary optimality conditions as a two‐point boundary value problem by using the calculus of variation and integration by part formula. The operational matrices for Müntz–Legendre scaling functions have been obtained, and by utilizing it, the two‐point boundary value problem has been converted into a system of algebraic equations. The L2$$ {L}_2 $$‐error estimates in the approximation of operations matrices and in the approximation of unknown variable by the Müntz–Legendre wavelet has been derived. In the last, illustrative examples have been taken to show the applicability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, the mixed (Dirichlet–Neumann) boundary value problem (BVP) for the linear second‐order scalar elliptic differential equation with variable coefficients in a bounded two‐dimensional domain is considered. The two‐operator approach and appropriate parametrix (Levi function) are used to reduce this BVP to four systems of two‐operator boundary‐domain integral equations (BDIEs). Although the theory of two‐operator BDIEs in 3D is well developed, the BDIEs in 2D need a special consideration due to their different equivalence properties. As a result, we need to set conditions on the domain or on the associated Sobolev spaces to ensure the invertibility of corresponding parametrix‐based integral layer potentials and hence the unique solvability of BDIEs. The equivalence of the two‐operator BDIE systems to the original BVP is shown. The invertibility of the associated operators is proved in the appropriate Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper we study a system of equations which appear in the modelling of many physical phenomena. Initially this system appeared in description of the large scale structure formation. Recently it is derived as a second order queueing model. We construct weakly asymptotic solutions of the initial boundary value problem for the system and interaction of waves in the quarter plane {(x,t):x>0,t>0}$\lbrace (x,t): x>0,t>0\rbrace$ with boundary Riemann solution centered at (0,0) and Riemann solution centered at a point (x0,0),x0>0$(x_0,0), x_0>0$. [ABSTRACT FROM AUTHOR]

In this paper, we establish existence of infinitely many weak solutions for a class of quasilinear stationary Kirchhoff‐type equations, which involves a general variable exponent elliptic operator with critical growth. Precisely, we study the following nonlocal problem: −M(풜(u))diva(|∇u|p(x))|∇u|p(x)−2∇u=λf(x,u)+|u|s(x)−2uinΩ,u=0on∂Ω,where Ω is a bounded smooth domain of ℝN, with homogeneous Dirichlet boundary conditions on ∂Ω, the nonlinearity f:Ω¯×ℝ→ℝ is a continuous function, a:ℝ+→ℝ+ is a function of the class C1,M:ℝ0+→ℝ+ is a continuous function, whose properties will be introduced later, λ is a positive parameter and p,s∈C(Ω¯). We assume that 풞={x∈Ω:s(x)=γ∗(x)}≠∅, where γ∗(x)=Nγ(x)/(N−γ(x)) is the critical Sobolev exponent. We will prove that the problem has infinitely many solutions and also we obtain the asymptotic behavior of the solution as λ → 0+. Furthermore, we emphasize that a difference with previous researches is that the conditions on a(·) are general overall enough to incorporate some interesting differential operators. Our work covers a feature of the Kirchhoff's problems, that is, the fact that the Kirchhoff's function M in zero is different from zero, it also covers a wide class of nonlocal problems for p(x) > 1, for all x∈Ω¯. The main tool to find critical points of the Euler–Lagrange functional associated with this problem is through a suitable truncation argument, concentration‐compactness principle for variable exponent found in Bonder and Silva (2010), and the genus theory introduced by Krasnoselskii. The result of this paper extends or complements, or else completes recent papers and is new in several directions for the stationary Kirchhoff equations involving the p(x)‐Laplacian type operators. [ABSTRACT FROM AUTHOR]

In this paper, an initial boundary value problem of the 2D incompressible Bénard system is concerned. The global existence of a unique strong solution is established by the energy method and a key logarithmic interpolation inequality. In particular, the exponential decay rates for the gradients of velocity and temperature field are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study existence of solutions to the scalar additive Jump problem and the Riemann boundary value problems in the context of vectorial Clifford analysis on domains with fractal boundaries. A reduction procedure is applied with great effectiveness to find the solution of the problems. [ABSTRACT FROM AUTHOR]

A parabolic–elliptic Keller–Segel model for multipopulations in a two‐dimensional unit ball will be considered in this paper. For the initial boundary value problem with mutually attractive populations, the global existence of bounded solutions has been proved through the logarithmic Hardy–Littlewood–Sobolev inequality for system when the initial masses satisfy the subcritical condition. Moreover, based on the moments technique, there exists a blow‐up solution when the initial masses satisfy the supercritical case. In a word, a critical mass phenomenon has been established in this multispecies chemotaxis system. [ABSTRACT FROM AUTHOR]

In this paper, the solvability of a fractional differential equation multipoint boundary value problem at resonance in ℝn$$ {\mathrm{\mathbb{R}}}&#x0005E;n $$ is investigated by utilizing the Mawhin's continuation theorem. In order to relax the assumptions of matrices in the boundary conditions, the Moore–Penrose pseudoinverse matrix is introduced to construct projectors. Furthermore, a ternary Carathéodory function that is nondecreasing in the last two variables and exhibits at most linear growth after integrating with respect to the first variable is used as a control function to constrain the nonlinear term. [ABSTRACT FROM AUTHOR]

In this paper, we study the phenomenon of increasing stability in the inverse boundary value problems for the biharmonic equation. The inverse problem is to determine the potential from DtN map. It is a kind of nonlinear inverse problem. By considering a linearized form, we obtain an increasing Lipschitz‐like stability when k$$ k $$ is large. Furthermore, we extend the discussion to the linearized inverse biharmonic potential problem with attenuation, where an exponential dependence of the attenuation constant is traced in the stability estimate. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a novel fixed point iterative scheme based on Green's function, called the Picard–Ishikawa–Green's iterative scheme and use it in approximating the solution of boundary value problems (BVPs). It is proved that Picard–Ishikawa–Green's scheme converges strongly for an integral operator which represents the solution of BVP and the scheme is stable. Moreover, we prove that the integral operator is a contraction. Furthermore, it is shown that the novel scheme converges faster than all of Ishikawa–Green's, Khan–Green's, and Mann–Green's schemes. Finally, numerical examples are given to substantiate the validity of our results for third‐order BVPs. Our results extend and generalize several other results in literature. [ABSTRACT FROM AUTHOR]

This paper addresses the stochastic linear quadratic (LQ) control problem with first‐ and second‐order moment constraints on the terminal state. The problem is a modified version of the optimal covariance control problem, where the terminal state is steered to a given probability distribution. Studying a multiplicative‐noise stochastic system rather than an additive‐noise system is a salient feature. Unlike the existing ideas in the optimal steering, by using the Lagrange multipliers method and establishing the stochastic maximum principle, our problem is converted into solving forward–backward stochastic difference equations (FBSDEs), which is a special stochastic two‐point boundary‐value problem (TPBVP). We provide the optimal closed‐loop controller and necessary and sufficient solvability conditions by developing a nonhomogeneous relationship between the optimal state and costate in FBSDEs. Finally, numerical examples are given to demonstrate our results. [ABSTRACT FROM AUTHOR]

This paper introduces a novel model for a generalized thermoelastic medium with homogeneity and isotropy, by applying the Modified Green-Lindsay (MGL) theory of thermoelasticity. The focus is on analysing the deformation due to time-harmonic by considering the impact of non-local and two-temperature (TT) parameters. The governing equations are solved using dimensionless quantities and a potential function. To address the boundary value problem in the frequency domain, the Hankel transform is employed. Specific boundary conditions, such as a normal force or thermal source, are applied. Analytical expressions for components of displacement, stresses, conductive temperature, and temperature distribution are derived in the transformed domain A numerical inversion technique is utilised to translate the solution into the physical domain. The study delves into exploring the influence of non-local and two-temperature parameters, as well as different theories of thermoelasticity on stresses, temperature distribution and conductive temperature. These effects are visually represented through graphical illustrations. Additionally, special cases of interest are examined and discussed. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of at least one solution to the following higher order Riemann–Liouville fractional differential equation with Riemann–Stieltjes integral boundary condition at resonance: −(D0+αx)(t)=f(t,x(t),D0+α−1x(t)),n−1<α≤n,t∈[0,1],x(0)=x′(0)=...=x(n−2)(0)=0,x(k)(1)=∫01x(k)(t)dA(t),$$ {\displaystyle \begin{array}{cc}\hfill -\left({D}_{0+}^{\alpha }x\right)(t)=& f\left(t,x(t),{D}_{0+}^{\alpha -1}x(t)\right),n-1<\alpha \le n,t\in \left[0,1\right],\hfill \\ {}\hfill x(0)=& {x}^{\prime }(0)=\dots ={x}^{\left(n-2\right)}(0)=0,{x}^{(k)}(1)=\int_0^1{x}^{(k)}(t) dA(t),\hfill \end{array}} $$by using Mawhin's coincidence degree theory. Here, D0+α$$ {D}_{0+}^{\alpha } $$ is the standard Riemann–Liouville fractional derivative of order α,f:[0,1]×ℝ2→ℝ$$ \alpha, f:\left[0,1\right]\times {\mathbb{R}}^2\to \mathbb{R} $$, and ∫01x(k)(t)dA(t)$$ {\int}_0^1{x}^{(k)}(t) dA(t) $$ is the Riemann–Stieltjes integral of x(k)$$ {x}^{(k)} $$ with respect to A$$ A $$. Our choice of k$$ k $$ in the boundary condition can be any integer between 0 and n−1$$ n-1 $$, which supplements many boundary conditions assumed in the literature. Several examples are given to strengthen our result. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and the uniqueness of solutions for a class of nonlinear fractional differential equations with a mixed fractional boundary value, by using Banach fixed‐point theorem. Moreover, we compare obtained results with another two works considering similar problem. [ABSTRACT FROM AUTHOR]

This paper considers time‐inconsistent problems when control and stopping strategies are required to be made simultaneously (called stopping control problems by us). We first formulate the time‐inconsistent stopping control problems under general multidimensional controlled diffusion model and propose a formal definition of their equilibria. We show that an admissible pair (û,C)$(\hat{u},C)$ of control‐stopping policy is equilibrium if and only if the auxiliary function associated with it solves the extended HJB system, providing a methodology to verify or exclude equilibrium solutions. We provide several examples to illustrate applications to mathematical finance and control theory. For a problem whose reward function endogenously depends on the current wealth, the equilibrium is explicitly obtained. For another model with a nonexponential discount, we prove that any constant proportion strategy can not be equilibrium. We further show that general nonconstant equilibrium exists and is described by singular boundary value problems. This example shows that considering our combined problems is essentially different from investigating them separately. In the end, we also provide a two‐dimensional example with a hyperbolic discount. [ABSTRACT FROM AUTHOR]