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In this paper, we deal with an inverse boundary value problem for the Maxwell equations with boundary data assumed known only in accessible part Γ of the boundary. We aim to prove uniqueness results using the Dirichlet to Neumann data with measurements limited to an open part of the boundary and we seek to reconstruct the complex refractive index n in the interior of a body. Further, using the impedance map restricted to Γ , we may identify locations of small volume fraction perturbations of the refractive index. [ABSTRACT FROM AUTHOR]

The paper deals with the solvability analysis of boundary value problems for degenerate higher-order quasi-hyperbolic equations. The problems in question have the specific feature that the manifolds on which the equations characteristically degenerate are not freed from carrying boundary data. The aim of this paper is to prove the existence and uniqueness of regular solutions of the problems under study, that is, solutions all of whose generalized derivatives occurring in the corresponding equations exist as generalized derivatives in the sense of Sobolev. [ABSTRACT FROM AUTHOR]

Some gaps in two of my publications dated 1958 and 1959 years are indicated, and the way of closing them is given. Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

In this paper, we survey selected software packages for the numerical solution of boundary value ODEs (BVODEs), time-dependent PDEs in one spatial dimension (1DPDEs), and initial value ODEs (IVODEs). A unifying theme of this paper is our focus on software packages for these problem classes that compute error-controlled, continuous numerical solutions. A continuous numerical solution can be accessed by the user at any point in the domain. We focus on error-control software; this means that the software adapts the computation until it obtains a continuous approximate solution with a corresponding error estimate that satisfies the user tolerance. The second section of the paper will provide an overview of recent work on the development of COLNEWSC, an updated version of the widely used collocation BVODE solver, COLNEW, that returns an error-controlled continuous approximate solution based on the use of a superconvergent interpolant to the underlying collocation solution. The third section of the paper gives a brief review of recent work on the development of a new 1DPDE solver, BACOLIKR, that provides time- and space-dependent event detection for an error-controlled continuous numerical solution. In the fourth section of the paper, we briefly review the state of the art in IVODE software for the computation of error-controlled continuous numerical solutions. [ABSTRACT FROM AUTHOR]

This document is a correction notice for an article titled "Impulsive problems on the half-line with infinite impulse moments" by Feliz Minhós. The correction aims to align with the method used in the original paper and ensure appropriate conditions for the existence of solutions to a second-order impulsive boundary value problem. The correction adds a condition to the main result and provides revised steps in the proof of Theorem 2. The author acknowledges the referee for their careful reading of the manuscript, and the corrigendum was published after a positive report from the referee. [Extracted from the article]

In this paper, we are concerned with the existence and multiplicity of positive solutions of the boundary value problem for the fourth-order semipositone nonlinear Euler–Bernoulli beam equation where and are constants, is a parameter, and is a function satisfying for some positive constant ; here . The paper is concentrated on applications of the Green's function of the above problem to the derivation of the existence and multiplicity results for the positive solutions. One example is also given to demonstrate the results. [ABSTRACT FROM AUTHOR]

We study global attractors A = A f of semiflows generated by semilinear partial parabolic differential equations of the form u t = u xx + f (x , u , u x) , 0 < x < 1 , satisfying Neumann boundary conditions. The equilibria v ∈ E ⊂ A of the semiflow are the stationary solutions of the PDE, hence they are solutions of the corresponding second order ODE boundary value problem. Assuming hyperbolicity of all equilibria, the dynamic decomposition of A into unstable manifolds of equilibria provides a geometric and topological characterization of Sturm global attractors A as finite regular signed CW-complexes, the Sturm complexes, with cells given by the unstable manifolds of equilibria. Concurrently, the permutation σ = σ f derived from the ODE boundary value problem by ordering the equilibria according to their values at the boundaries x = 0 , 1 , respectively, completely determines the Sturm global attractor A . Equivalently, we use a planar curve, the meander M = M f , associated to the the ODE boundary value problem by shooting. In the previous paper (Fiedler and Rocha in J Dyn Differ Equ, 2020. https://doi.org/10.1007/s10884-020-09836-5), we set up to determine the boundary neighbors of any specific unstable equilibrium O , based exclusively on the information on the corresponding signed hemisphere complex. In addition, a certain minimax property of the boundary neighbors was established. In the signed hemisphere decomposition of the cell boundary of O , this property identifies the equilibria which are closest to, or most distant from, O at the boundaries x = 0 , 1 , in each hemisphere. The main objective of the present paper is to derive this minimax property directly from the Sturm permutation σ , or equivalently from the Sturm meander M , based on the Sturm nodal properties of the solutions of the ODE boundary value problem. This minimax result simplifies the task of identifying the equilibria on the cell boundary of each unstable equilibrium, directly from the Sturm meander M . We emphasize the local aspect of this result by an example for which the identification of the equilibria is obtained from the knowledge of only a segment of the Sturm meander M . [ABSTRACT FROM AUTHOR]

In this paper, we examine the existence, uniqueness, and stability of solutions for a Caputo variable order ϑ-initial value problem (ϑ-IVP) with multi-point initial conditions. The proofs for uniqueness and existence leverage Sadovski's and Banach's fixed point theorems, along with the Kuratowski measure of noncompactness. Furthermore, we explore the Ulam–Hyers–Rassias (UHR) stability of the solution. To validate our findings, we present a numerical example. [ABSTRACT FROM AUTHOR]

In this paper, we consider an initial and boundary value problem to the three-dimensional (3D) nonhomogeneous nematic liquid crystal flows with density-dependent viscosity and vacuum. Combining delicate energy method with the structure of the system under consideration, the global well-posedness of strong solutions is established, provided that ‖ ρ 0 ‖ L 1 + ‖ ∇ d 0 ‖ L 2 is suitably small. In particular, the initial velocity can be arbitrarily large. Moreover, the exponential decay rates of the strong solution are also obtained. [ABSTRACT FROM AUTHOR]

This paper deals with the following indirect chemotaxis-consumption model with signal-dependent degenerate diffusion and logistic source u t = Δ u v α + a u - b u l , x ∈ Ω , t > 0 , v t = Δ v - v w , x ∈ Ω , t > 0 , w t = - δ w + u , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n ( n ≥ 1 ). Here, the parameters a > 0 , b > 0 , α ≥ 1 , δ > 0 and l ≥ 2 . For all suitably regular initial data, if one of the following cases holds: l > 2 ; l = 2 , n ≤ 3 ; l = 2 , n ≥ 4 , and b is sufficiently large, then the corresponding initial boundary value problem possesses a global classical solution. [ABSTRACT FROM AUTHOR]

In mathematical physics, the gradient operator with nonconstant coefficients encompasses various models, including Fourier’s law for heat propagation and Fick’s first law, that relates the diffusive flux to the gradient of the concentration. Specifically, consider n ≥ 3 orthogonal unit vectors e 1 , … , e n ∈ R n , and let Ω ⊆ R n be some (in general unbounded) Lipschitz domain. This paper investigates the spectral properties of the gradient operator T = ∑ i = 1 n e i a i (x) ∂ ∂ x i with nonconstant positive coefficients a i : Ω ¯ → (0 , ∞) . Under certain regularity and growth conditions on the a i , we identify bisectorial or strip-type regions that belong to the S-resolvent set of T. Moreover, we obtain suitable estimates of the associated resolvent operator. Our focus lies in the spectral theory on the S-spectrum, designed to study the operators acting in Clifford modules V over the Clifford algebra R n , with vector operators being a specific crucial subclass. The spectral properties related to the S-spectrum of T are linked to the inversion of the operator Q s (T) : = T 2 - 2 s 0 T + | s | 2 , where s ∈ R n + 1 is a paravector, i.e., it is of the form s = s 0 + s 1 e 1 + ⋯ + s n e n . This spectral problem is substantially different from the complex one, since it allows to associate general boundary conditions to Q s (T) , i.e., to the squared operator T 2 . [ABSTRACT FROM AUTHOR]

The paper considers an axisymmetric stress state of a homogeneous elastic space with a circular disc-shaped crack, one of the edges of which is pressed into a cylindrical circular stamp with static friction. It is assumed that the contact zone is considered under the generalized law of dry friction, i.e. tangential contact stresses are proportional to normal contact pressure, while the proportionality coefficient depends on the radial coordinates of the points of the contacting surfaces and is directly proportional to them. Considering the fact that in this case the Abel images of contact stresses are also related in a similar way, the solution of the problem, with the help of rotation operators and theory of analytical functions, is reduced to an inhomogeneous Riemann problem for two functions and the closed solution in quadratures is constructed. A numerical analysis was carried out and regularities of changes in both normal and shear real contact stresses, as well as rigid displacement of the stamp depending on the physical and geometric parameters were revealed. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the presence of positive solutions for system of fractional q-differential inclusions involving sequential derivatives with respect to the p-Laplacian operator. By using fixed point technique we obtain a new solution for inclusion or boundary value problems with special integral and derivative conditions. At the end, we give an example to show the effect of the solution of this device. The conclusion is expressed to introduce future works. [ABSTRACT FROM AUTHOR]

The paper is concerned with error identities for a class of parabolic equations. One side of such an identity is a natural measure of the distance between a function in the corresponding energy class and the exact solution of the problem in question. Another side is either directly computable or serves as a source of fully computable error bounds. Particular forms of the identities can be viewed as analogs of the hypercircle identity well known for elliptic problems. It is shown that identities possess an important consistency property. Therefore, the identities and the corresponding error estimates can be used in quantitative analysis of direct and inverse problems associated with parabolic equations. The first part of the paper deals with linear parabolic equations. A class of nonlinear problems is considered in the second part. In particular, this class includes problems whose spatial parts are presented by the α-Laplacian operator. [ABSTRACT FROM AUTHOR]

In this paper, a (p , q) -fractional nonlinear difference equation of different orders is considered and discussed. With the help of (p , q) -calculus for integrals and derivatives properties, we convert the main integral boundary value problem (IBVP) to an equivalent solution in the form of an integral equation, we use the upper–lower solution technique to prove the existence of positive solutions. We present an example of the IBVP to apply and demonstrate the results of our method. [ABSTRACT FROM AUTHOR]

Integrating evolutionary partial differential equations (PDEs) is an essential ingredient for studying the dynamics of the solutions. Indeed, simulations are at the core of scientific computing, but their mathematical reliability is often difficult to quantify, especially when one is interested in the output of a given simulation, rather than in the asymptotic regime where the discretization parameter tends to zero. In this paper we present a computer-assisted proof methodology to perform rigorous time integration for scalar semilinear parabolic PDEs with periodic boundary conditions. We formulate an equivalent zero-finding problem based on a variation of constants formula in Fourier space. Using Chebyshev interpolation and domain decomposition, we then finish the proof with a Newton–Kantorovich type argument. The final output of this procedure is a proof of existence of an orbit, together with guaranteed error bounds between this orbit and a numerically computed approximation. We illustrate the versatility of the approach with results for the Fisher equation, the Swift–Hohenberg equation, the Ohta–Kawasaki equation and the Kuramoto–Sivashinsky equation. We expect that this rigorous integrator can form the basis for studying boundary value problems for connecting orbits in partial differential equations. [ABSTRACT FROM AUTHOR]

This paper presents typical examples of the appearance of singular points in the proximity of optimal trajectories in different interorbital low-thrust missions. As a rule, the occurrence of singular points is accompanied by the appearance of computational difficulties in solving boundary-value problems. [ABSTRACT FROM AUTHOR]

The paper studies the asymptotic behavior of the optimal control for the Poisson type boundary value problem in a domain perforated by holes of an arbitrary shape with Robin-type boundary conditions on the internal boundaries. The cost functional is assumed to be dependent on the gradient of the state and on the usual L2-norm of the control. We consider the so-called "critical" relation between the problem parameters and the period of the structure . Two "strange" terms arise in the limit. The paper extends, by first time in the literature, previous papers devoted to the homogenization of the control problem which always assumed the symmetry of the periodic holes. [ABSTRACT FROM AUTHOR]

In this paper, the coupled system of sequential ψ-Hilfer fractional boundary value problems with non-instantaneous impulses is investigated. The existence results of the system are proved by means of topological degree theory. An example is constructed to demonstrate our results. Additionally, a graphical analysis is performed to verify our results. [ABSTRACT FROM AUTHOR]

In this paper, we examine the existence and uniqueness of solutions for a system of the first-order q-difference equations with multi-point and q-integral boundary conditions using various fixed point (fp) theorems. Also, we give two examples to support our results. [ABSTRACT FROM AUTHOR]

In this paper, the solvability of a new class of nonlocal boundary value problems for the Poisson equation is studied. Nonlocal conditions are specified in the form of a connection between the values of the unknown function at different points of the boundary. In this case, the boundary operator is determined using matrices of involution-type mappings. Theorems on the existence and uniqueness of solutions to the studied problems are proved. Using Green's functions of the classical Dirichlet and Neumann boundary value problems, Green's functions of the studied problems are constructed and integral representations of solutions to these problems are obtained. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study a generalized Riemann–Liouville fractional differential equation and system with nonlocal boundary conditions. Firstly, some properties of the Green function are presented and then Lyapunov-type inequalities for a sequential ψ-Riemann–Liouville fractional boundary value problem are established. Also, the existence and uniqueness of solutions are proved by using Banach and Schauder fixed-point theorems. Furthermore, the existence and uniqueness of solutions to a sequential nonlinear differential system is established by means of Schauder's and Perov's fixed-point theorems. Examples are given to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we obtain conditions of the solvability of the Riemann boundary value problem for sectionally analytic functions in multiply connected domains in Besov spaces embedded into the class of continuous functions. We indicate a new class of Cauchy-type integrals, which are continuous on a closed domain with continuous (not Hölder) density in terms of Besov spaces, and for which the Sokhotski–Plemelj formulas are valid. [ABSTRACT FROM AUTHOR]

In this paper, we have studied the behaviour of two symmetric mode-III collinear cracks in a functionally graded material (FGM). The fundamental goal of this paper is to provide insight on the interaction of two cracks in FGMs with the strain gradient effect. To assess the influence of gradient elasticity, we have considered two key parameters ℓ and ℓ ′ , which describe the size scale effect caused by the underlying microstructure and are related to volumetric and surface strain energy, respectively. The crack boundary value problem have been solved by the approach involving Fourier transforms and the innovative hyper-singular integro-differential equation method, where the integral equation contains the two terms in integrals for the both cracks. A system of equations has been constructed by employing the Chebyshev polynomial expansion and then by choosing the suitable collocation points the system of equation have been solved. Our investigation involves the determination of stress intensity factors at both crack tips. These factors are vital for understanding the material's fracture behavior and structural integrity. Furthermore, we explore the variations in the displacement profile when the distance between the cracks is reduced to close proximity. This particular scenario is of significant interest as it provides insights into how the interaction between the cracks impacts the overall structural response. [ABSTRACT FROM AUTHOR]

This paper deals with the following chemotaxis system: u t = ∇ · (γ (v) ∇ u - u ξ (v) ∇ v) + μ u (1 - u) , x ∈ Ω , t > 0 , v t = Δ v - u v , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n , n ≥ 2 , with smooth boundary. Here, the positive function γ ∈ C 2 ([ 0 , + ∞)) satisfies γ ′ (s) < 0 and γ ′ ′ (s) ≥ 0 for all s ≥ 0 , also ξ (s) = - (1 - α) γ ′ (s) with α ∈ (0 , 1) . For the above system, we prove that the corresponding initial boundary value problem admits a unique global classical solution which is uniformly in time bounded. This result is obtained for small initial data without any restriction on μ. The obtained result improves a recent result by Li and Lu (J Math Anal Appl 521:126902, 2023), which asserts the global existence of bounded classical solutions, provided that (γ ′ (s)) 2 γ ′ ′ (s) ≤ n 2 (n + 1) 3 and some conditions on initial data and μ. We should mention that in the special cases γ (s) = (1 + s) - k (k > 0) and γ (s) = e - χ s (χ > 0) , the result in Li and Lu (2023) is obtained under conditions on k and χ. But, our result is without any restriction on k and χ. [ABSTRACT FROM AUTHOR]

From the literature it is known that orthogonal polynomials as the Jacobi polynomials can be expressed by hypergeometric series. In this paper, the authors derive several contiguous relations for terminating multivariate hypergeometric series. With these contiguous relations one can prove several recursion formulas of those series. This theoretical result allows to compute integrals over products of Jacobi polynomials in a very efficient recursive way. Moreover, the authors present an application to numerical analysis where it can be used in algorithms which compute the approximate solution of boundary value problem of partial differential equations by means of the finite elements method. With the aid of the contiguous relations, the approximate solution can be computed much faster than using numerical integration. A numerical example illustrates this effect. [ABSTRACT FROM AUTHOR]

This paper considers the iterative solution of finite element discretizations of second-order elliptic boundary value problems. Mesh independent estimations are given for the rate of superlinear convergence of preconditioned Krylov methods, involving the connection between the convergence rate and the Lebesgue exponent of the data. Numerical examples demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper first we formulate the mixed problem with initial and boundary values in the context of the porous Cosserat thermoelastic bodies micropolar material with voids. WE have included among the independent constitutive variables the derivative with respect to time of the voids (pores) function. Under the conditions in which we imposed average restrictions on the functions used, we formulated and demonstrated the uniqueness of the solution to the mentioned mixed problem and we did a short analysis on the dilatational waves in this kind of media. [ABSTRACT FROM AUTHOR]

A two-point boundary value problem whose highest-order derivative is a Riemann–Liouville–Caputo derivative of order α ∈ (1 , 2) is considered. A similar problem was considered in Gracia et al. (BIT 60:411–439, 2020) but under a simplifying assumption that excluded singular solutions. In the present paper, this assumption is not imposed; furthermore, the finite difference method of the BIT paper, which was proved to attain 1st-order convergence under a sign restriction on the convective term, is replaced by a piecewise polynomial collocation method which can give any desired integer order of convergence on a suitably graded mesh. An error analysis of the collocation method is given which removes the above sign restriction and numerical results are presented to support our theoretical conclusions. The tools devised for this analysis include new comparison principles for Caputo initial-value problems and weakly singular Volterra integral equations that are of independent interest. Numerical experiments demonstrate the sharpness of our theoretical results. [ABSTRACT FROM AUTHOR]

A hybrid system interacts with the discrete and continuous dynamics of a physical dynamical system. The notion of a hybrid system gives embedded control systems a great advantage. The Langevin differential equation can accurately depict many physical phenomena and help researchers effectively represent anomalous diffusion. This paper considers a fractional hybrid Langevin differential equation, including the ψ-Caputo fractional operator. Furthermore, some novel boundaries selected are considered to be a problem. We used the Schauder and Banach fixed-point theorems to prove the existence and uniqueness of solutions to the considered problem. Additionally, the Ulam-Hyer stability is evaluated. Finally, we present a representative example to verify the theoretical outcomes of our findings. [ABSTRACT FROM AUTHOR]

A novel local and parallel multigrid method is proposed in this study for solving the semilinear Neumann problem with nonlinear boundary condition. Instead of solving the semilinear Neumann problem directly in the fine finite element space, we transform it into a linear boundary value problem defined in each level of a multigrid sequence and a small-scale semilinear Neumann problem defined in a low-dimensional correction subspace. Furthermore, the linear boundary value problem can be efficiently solved using local and parallel methods. The proposed process derives an optimal error estimate with linear computational complexity. Additionally, compared with existing multigrid methods for semilinear Neumann problems that require bounded second order derivatives of nonlinear terms, ours only needs bounded first order derivatives. A rigorous theoretical analysis is proposed in this paper, which differs from the maturely developed theories for equations with Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

In modern discrete functional semiconductor devices and structural elements of micro- and nanoelectronics, use is made of materials with anisotropy of electrical properties. In particular, such materials are crystalline thermoelectrics, layered graphite structures, and strained silicon. In the practical application of these semiconductors, it becomes necessary to measure their kinetic coefficients. However, the electrodynamics of these media differs from that of isotropic ones, which requires the correction of existing methods for measuring the conductivity and concentration of the majority charge carriers. The paper presents a technique for solving the Neumann problem with inhomogeneous boundary conditions for the electric field potential in a rectangular region in a relatively weak magnetic field in a linear approximation. The boundary value problem considered in the paper is encountered in the analysis of measurements of the Hall effect by probe methods. Using the perturbation theory and the Fourier method, an expression for the Hall field potential is obtained and presented in rectangular coordinates as a series of harmonic functions, which is convenient for further application. Practically important expressions for analyzing the results of Hall measurements by probe methods have been obtained for anisotropic samples with flat boundaries. Analysis of the obtained solution and computer simulation of the electric potential in anisotropic semiconductor wafers with flat boundaries are performed. An experimental verification of the obtained distributions of potentials and practical recommendations on the application of the obtained theoretical expressions are presented. [ABSTRACT FROM AUTHOR]

Three-dimensional models of suspension transport in coastal marine systems are considered. The associated processes have a number of characteristic features, such as high concentrations of suspensions (e.g., when soil is dumped on the bottom), much larger areas of suspension spread than the reservoir depth, complex granulometric (multifractional) content of suspensions, and mutual transitions between fractions. Suspension transport can be described using initial-boundary value diffusion–convection–reaction problems. According to the authors' idea, on a time grid constructed for the original continuous initial-boundary value problem, the right-hand sides are transformed with a "delay" so that the right-hand side concentrations of the components other than the underlying one (for which the initial-boundary value problem of diffusion–convection is formulated) are determined at the preceding time level. This approach simplifies the subsequent numerical implementation of each of the diffusion–convection equations. Additionally, if the number of fractions is three or more, the computation of each of the concentrations at every time step can be organized independently (in parallel). Previously, sufficient conditions for the existence and uniqueness of a solution to the initial-boundary value problem of suspension transport were determined, and a conservative stable difference scheme was constructed, studied, and numerically implemented for test and real-world problems. In this paper, the convergence of the solution of the delay-transformed problem to the solution of the original suspension transport problem is analyzed. It is proved that the differences between these solutions tends to zero at an O(τ) rate in the norm of the Hilbert space as the time step approaches zero. [ABSTRACT FROM AUTHOR]

The paper continues the construction of the -theory of elliptic Dirichlet and Neumann boundary value problems with discontinuous piecewise constant coefficients in divergent form for an unbounded domain with a piecewise smooth noncompact Lipschitz boundary and smooth discontinuity lines of the coefficients. An earlier constructed -theory is generalized to the case of different smallest eigenvalues corresponding to a finite and an infinite singular point, and the effect of their interaction is further studied in the class of functions with first derivatives from in the entire range of the exponent . [ABSTRACT FROM AUTHOR]

In this paper, we show the existence and uniqueness of local regular solutions to the initial-Neumann boundary value problem of the Schrödinger flow from a smooth bounded domain Ω ⊂ R 3 into S 2 (namely Landau–Lifshitz equation without dissipation). The proof is built on a parabolic perturbation method, an intrinsic geometric energy argument, the symmetric (algebraic) properties of S 2 and some observations on the behaviors of some geometric quantities on the boundary of the domain manifold.It is based on methods from Ding and Wang (one of the authors of this paper) for the Schrödinger flows of maps from a closed Riemannian manifold into a Kähler manifold as well as on methods by Carbou and Jizzini for solutions of the Landau–Lifshitz equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence, nonexistence, and multiplicity of positive solutions to a nonlinear impulsive Sturm–Liouville boundary value problem with a parameter. By using a variational method, we prove that the problem has at least two positive solutions for the parameter λ ∈ (0 , Λ) , one positive solution for λ = Λ , and no positive solution for λ > Λ , where Λ > 0 is a constant. [ABSTRACT FROM AUTHOR]

The study of nonlinear problems related to heat transfer in a substance is of great practical important. Earlier, this paper's authors proposed an effective algorithm for determining the volumetric heat capacity and thermal conductivity of a substance based on experimental observations of the dynamics of the temperature field in the object. In this paper, the problem of simultaneous identification of temperature-dependent volumetric heat capacity and thermal conductivity of the substance under study from the heat flux at the boundary of the domain is investigated. The consideration is based on the first (Dirichlet) boundary value problem for a one-dimensional unsteady heat equation. The coefficient inverse problem under consideration is reduced to a variational problem, which is solved by gradient methods based on the application of fast automatic differentiation. The uniqueness of the solution of the inverse problem is investigated. [ABSTRACT FROM AUTHOR]

This paper deals with some existence and uniqueness results for a class of deformable fractional differential equations. These problems encompassed nonlinear implicit fractional differential equations involving boundary conditions and various types of delays, including finite, infinite, and state-dependent delays. Our approach to proving the existence and uniqueness of solutions relied on the application of the Banach contraction principle and Schauder's fixed-point theorem. In the last section, we provide different examples to illustrate our obtained results. [ABSTRACT FROM AUTHOR]

The title problem which has recently been addressed in this journal is revisited in the present paper under a new point of view. It is shown that the joint effect of the Berman suction or injection normal to the boundaries and the velocity slip along the boundaries is equivalent to the sole effect of an oblique suction or injection of the fluid. The solution of the corresponding boundary value problem is given by a Maclaurin series expansion of the similar stream function to powers of the scaled transverse coordinate y/h. Compared to the classical Berman problem, the existence of several new solution branches of the oblique suction/injection problem is reported. Subsequently, the physical and mathematical aspects of the mentioned equivalence are discussed in the paper in some detail. It is pointed out that the vanishing midplane velocity represents the crossover from the physically feasible unidirectional flows to the unfeasible bidirectional flow configurations, where in the neighborhood of the midplane of the channel reverse flows occur. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a mixed boundary value problem for a complete Sturm–Liouville equation, where the coefficients can also be negative. In particular, the existence of infinitely many distinct positive solutions to the given problem is obtained by using critical point theory. [ABSTRACT FROM AUTHOR]

Recent research indicates the need for improved models of physical phenomena with multiple shocks. One of the newest methods is to use differential inclusions instead of differential equations. In this work, we intend to investigate the existence of solutions for an m-dimensional system of quantum differential inclusions. To ensure the existence of the solution of inclusions, researchers typically rely on the Arzela–Ascoli and Nadler's fixed point theorems. However, we have taken a different approach and utilized the endpoint technique of the fixed point theory to guarantee the solution's existence. This sets us apart from other researchers who have used different methods. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables, and some figures. The paper ends with an example. [ABSTRACT FROM AUTHOR]

In this paper, we compute holographic Euclidean thermal correlators of the stress tensor and U(1) current from the AdS planar black hole. To this end, we set up perturbative boundary value problems for Einstein's gravity and Maxwell theory in the spirit of Gubser-Klebanov-Polyakov-Witten, with appropriate gauge fixing and regularity boundary conditions at the horizon of the black hole. The linearized Einstein equation and Maxwell equation in the black hole background are related to the Heun equation of degenerate local monodromy. Leveraging the connection relation of local solutions of the Heun equation, we partly solve the boundary value problem and obtain exact two-point thermal correlators for U(1) current and stress tensor in the scalar and shear channels. [ABSTRACT FROM AUTHOR]

This paper offers a number of examples showing that in the case of two independent variables the uniform ellipticity of a linear system of differential equations with partial derivatives of the second order, which fulfills condition (3), do not always cause the normal solvability of formulated exterior elliptic problems in the sense of Noether. Nevertheless, from the system of differential equations with partial derivatives of elliptic type it is possible to choose, under certain additional conditions, classes which are normally solvable in the sense of Noether. This paper also shows that for the so-called decomposed system of differential equations, with partial derivatives of an elliptic type in the case of exterior regions, the Noether theorems are valid. [ABSTRACT FROM AUTHOR]

In this paper we provide a way to handle some symmetric fractional boundary-value problems. Indeed, first, we consider some system of fractional equations. We introduce the existence and uniqueness of solutions of the systems of equations and we show that they are entire functions of the spectral parameter. In particular, we show that the solutions are at most of order 1/2. Moreover we share the integration by parts rule for vector-valued functions that enables us to obtain some symmetric equations. These symmetries allow us to handle 2 - sequential and 4 - sequential fractional boundary-value problems. We provide some expansion formulas for the bilinear forms of the solutions of 2 - sequential and 4 - sequential fractional equations which admit us to impose some unusual boundary conditions for the solutions of fractional differential equations. We show that the systems of eigenfunctions of 2 - sequential and 4 - sequential fractional boundary value problems are complete in both energy and mean. Furthermore, we study on the zeros of solutions of 2 - sequential fractional differential equations. At the end of the paper we show that 6 - sequential fractional differential equation can also be handled as a system of equations and hence almost all the results obtained in the paper can be carried for such boundary-value problems. [ABSTRACT FROM AUTHOR]

Purpose: The real-time dynamics of the single-link flexible manipulator is a challenging problem due to its inherent instability and in situ vibration. In order to add the criticalities to these real-time dynamics, run-time vibration does play a pivotal role in designing a robust control system for the flexible robotic manipulator. Methodology: Governing differential equations and the boundary conditions are usually considered exact of the single-link manipulator and the formulation leads to an Eigenvalue problem where the elements of the matrices are in exact form. Double parametric form has been used to solve the fuzzy differential equation. Results and Conclusions: In this paper, a new idea has been introduced in the above problem for a representative single-link robotic manipulator considering the uncertainty in the associated parameter(s) in the governing differential equation, which may mimic the actual scenario of the real environment. The uncertainty has been considered in terms of a novel fuzzy model which agrees with the crisp case model too. [ABSTRACT FROM AUTHOR]

In this paper, we consider an initial boundary value problem for the one-dimensional thermodiffusion equations of type III. By the semigroup approach and the energy method, we establish the global existence and exponential stability for the solutions in a bounded region. The main work of this paper is to extend the study of thermodiffusion equations to the model of type III, which can be used as a reference for the study of other types of partial differential equations. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the study of the asymptotic behavior of the optimal control for the boundary value problem in an ε-periodically perforated domain with linear Robin-type boundary condition, when the period of the structure tends to zero, and the problem parameters, diameter of perforations and adsorption coefficient, take critical values. [ABSTRACT FROM AUTHOR]

Porous solid structures have attracted much attention because they are widely used in national defense, military industry, machinery, civil engineering and other fields. In this paper, the problem of free vibration of a porous rectangular plate with variable thickness is investigated. Firstly, given the two distribution modes of the porous rectangular plate along the thickness direction, the dimensionless governing differential equations for the free vibration of the porous rectangular plate are derived using the classical plate theory and the Hamiltonian variational principle. Then, the dimensionless governing differential equations of motion and boundary conditions are derived by converting them into algebraic equations through the differential transformation method (DTM). The dimensionless natural frequencies of the porous rectangular plate are solved by iterative convergence method through MATLAB programming. Finally, numerical examples are given to analyze the influence of different parameters on the vibration frequency of porous rectangular plates with different porosity distributions under different boundary conditions. Numerical examples show that the method has fast convergence speed and high accuracy. In addition, some novel results are presented in this paper, which can be used for reference in the following research on the vibration mechanical behavior of graded porous rectangular plates with variable thickness. [ABSTRACT FROM AUTHOR]