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It is difficult to convert the PDEs defined on the whole space with higher order space derivatives into initial boundary value problems by proposing boundary conditions. In this paper, we use an absorbing boundary condition method to solve the Cauchy problem for one-dimensional Euler-Bernoulli beam with fast convolution boundary condition which is derived through the Padé approximation for the square root function ⋅ . We also introduce a constant damping term to control the error between the resulting approximation of the Euler-Bernoulli system and the original one. Numerical examples verify the theoretical results and demonstrate the performance for the fast numerical method. [ABSTRACT FROM AUTHOR]

The paper considers a boundary value problem for stationary equations of complex heat transfer with an undetermined boundary condition for the radiation intensity on a part of the boundary and an overdetermined condition on another part of the boundary. An optimization method for solving this problem is proposed, and an analysis of the corresponding problem of boundary optimal control is presented. It is shown that the sequence of solutions of extremum problems converges to the solution of a problem with Cauchy-type conditions. The efficiency of the algorithm is illustrated by numerical examples. [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]

In this paper, we present the formulation and a method for solving the problem of freezing living biological tissue with a flat circular cryoapplicator. The model is a two-dimensional boundary-value problem of Stefan type with nonlinear heat sources of a special type that provide the actually observed spatial localization of the temperature field. Some results of computer simulation are presented. [ABSTRACT FROM AUTHOR]

The paper presents an accelerating of solving potential boundary value problems (BVPs) with curvilinear boundaries by modified parametric integral equations system (PIES). The fast multipole method (FMM) known from the literature was included into modified PIES. To consider complex curvilinear shapes of a boundary, the modification of a binary tree used by the FMM is proposed. The FMM combined with the PIES, called the fast PIES, also allows a significant reduction of random access memory (RAM) utilization. Therefore, it is possible to solve complex engineering problems on a standard personal computer (PC). The proposed algorithm is based on the modified PIES and allows for obtaining accurate solutions of complex BVPs described by the curvilinear boundary at a reasonable time on the PC. [ABSTRACT FROM AUTHOR]

Surface comparison and matching is a challenging problem in computer vision. While elastic Riemannian metrics provide meaningful shape distances and point correspondences via the geodesic boundary value problem, solving this problem numerically tends to be difficult. Square root normal fields considerably simplify the computation of certain distances between parametrized surfaces. Yet they leave open the issue of finding optimal reparametrizations, which induce corresponding distances between unparametrized surfaces. This issue has concentrated much effort in recent years and led to the development of several numerical frameworks. In this paper, we take an alternative approach which bypasses the direct estimation of reparametrizations: we relax the geodesic boundary constraint using an auxiliary parametrization-blind varifold fidelity metric. This reformulation has several notable benefits. By avoiding altogether the need for reparametrizations, it provides the flexibility to deal with simplicial meshes of arbitrary topologies and sampling patterns. Moreover, the problem lends itself to a coarse-to-fine multi-resolution implementation, which makes the algorithm scalable to large meshes. Furthermore, this approach extends readily to higher-order feature maps such as square root curvature fields and is also able to include surface textures in the matching problem. We demonstrate these advantages on several examples, synthetic and real. [ABSTRACT FROM AUTHOR]

The paper presents an approach for set-membership estimation of the state of a heterogeneous population in which an infectious disease is spreading. The population state may consist of susceptible, infected, recovered, etc. groups, where the individuals are heterogeneous with respect to traits, relevant to the particular disease. Set-membership estimations in this context are reasonable, since only vague information about the distribution of the population along the space of heterogeneity is available in practice. The presented approach comprises adapted versions of methods which are known in estimation and control theory, and involve solving parametrized families of optimization problems. Since the models of disease spreading in heterogeneous populations involve distributed systems (with non-local dynamics and endogenous boundary conditions), these problems are non-standard. The paper develops the needed theoretical instruments and a solution scheme. SI and SIR models of epidemic diseases are considered as case studies and the results reveal qualitative properties that may be of interest. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the problem where $${\Omega}$$ is a open bounded subset of $${{\mathbb R}^N}$$ with C boundary ( $${N \ge 2}$$ ), $${\Gamma = \partial\Omega}$$ , $${(\Gamma_{0},\Gamma_{1})}$$ is a measurable partition of $${\Gamma}$$ , $${\Delta_{\Gamma}}$$ denotes the Laplace-Beltrami operator on $${\Gamma}$$ , $${\nu}$$ is the outward normal to $${\Omega}$$ , and the terms P and Q represent nonlinear damping terms, while f and g are nonlinear subcritical perturbations. In the paper a local Hadamard well-posedness result for initial data in the natural energy space associated to the problem is given. Moreover, when $${\Omega}$$ is C and $${\overline{\Gamma_{0}} \cap \overline{\Gamma_{1}} = \emptyset}$$ , the regularity of solutions is studied. Next a blow-up theorem is given when P and Q are linear and f and g are superlinear sources. Finally a dynamical system is generated when the source parts of f and g are at most linear at infinity, or they are dominated by the damping terms. [ABSTRACT FROM AUTHOR]

This paper considers two variants of problems that arise in developing optical computers. The first variant is related to the mathematical analysis of problems of optical bistability in the case of multibeam interaction of laser radiation in nonlinear media. The existence of optical bistability is confirmed by the results of solving the boundary value problem for a system of nonlinear ordinary differential equations. In the general case of an arbitrary nonstationary process, the problem is reduced to solving a system of two nonlinear integral equations with respect to complex amplitudes describing interference patterns. The second variant of problems is devoted to studying the absorption and scattering of light by nanomaterials. As a result, a multidimensional integral equation was obtained for the complex amplitude of the electric field. A fundamentally important feature of this equation is its singularity inside a nanoparticle. [ABSTRACT FROM AUTHOR]

The block element method is used to study a static boundary value problem for semi-infinite lithospheric plates interacting with a deformable basement along Conrad boundary. It is assumed that the lithospheric plates have straight line boundaries parallel to each other and are considered in two positions. In the first case, the distance between the ends of the plates does not vanish, whereas in the second case the distance is absent, although the plates do not interact. It is assumed that horizontal action on the plates, which are known to move extremely slowly, is so strong that vertical components of contact stresses can be neglected. Only shift stresses remain in the contact zone. The paper addresses the comparison of numerical simulation and block element approach to investigate this problem. In the first case, appearance of a concentration of contact stresses in the zone of contact of lithospheric plates is found, while in the second case the stress concentration turns out to be singular and leads to destruction of the base or edges of the lithospheric plates. In the second case, it is possible to determine the influence of various parameters of the problem on the magnitude of the coefficients for singularities in the contact stress concentrations. The numerical method does not have this capability. The obtained result allows one to predict the starting earthquakes based on monitoring of horizontal motions of the lithospheric plates. [ABSTRACT FROM AUTHOR]

The paper deals with energy (weak) solutions u (t; x) of the class of equations with the model representativeup−1ut−Δpu=0,tx∈0T×Ω,Ω∈ℝn,n≥1,p>0,and with the following blow-up condition for the energy:εt≔∫Ωutxp+1dx+∫0t∫Ω∇xuτxp+1dxdτ→∞ast→T,where Ω is a smooth bounded domain. In the case of flat peaking, namely, under the conditionεt≤Fαtω0T−t−α∀t0,α>1p+1,a sharp estimate of the profile of a solution has been obtained in a neighborhood of the blow-up time t = T. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and multiplicity of periodic solutions for a class of parametric nonlocal equations with critical and supercritical growth. It is well known that these equations can be realized as local degenerate elliptic problems in a half-cylinder of R+N+1 together with a nonlinear Neumann boundary condition, through the extension technique in periodic setting. Exploiting this fact, and by combining the Moser iteration scheme in the nonlocal framework with an abstract multiplicity result valid for differentiable functionals due to Ricceri, we show that the problem under consideration admits at least three periodic solutions with the property that their Sobolev norms are bounded by a suitable constant. Finally, we provide a concrete estimate of the range of these parameters by using some properties of the fractional calculus on a specific family of test functions. This estimate turns out to be deeply related to the geometry of the domain. [ABSTRACT FROM AUTHOR]

An important question to be addressed regarding system control on a time interval [0, T] is whether some particular target state in the configuration space is reachable from a given initial state. When the target of interest refers only to a portion of the spatial domain, we speak about regional analysis. Cellular automata approach have been recently promoted for the study of control problems on spatially extended systems for which the classical approaches cannot be used. An interesting problem concerns the situation where the subregion of interest is not interior to the domain but a portion of its boundary. In this paper we address the problem of regional controllability of cellular automata via boundary actions, i.e., we investigate the characteristics of a cellular automaton so that it can be controlled inside a given region only acting on the value of sites at its boundaries. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the investigation of a parabolic equation with moving boundaries arising in ductal carcinoma in situ (DCIS) model. Approximation solution of this problem is implemented by Ritz-Galerkin, which is a first attempt at tackling such problem. In process of dealing with this moving boundary condition, we use a trick of introducing two transformations to convert moving boundary to nonclassical boundary that can be handled with Ritz-Galerkin method. Also, existence and uniqueness are proved. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. [ABSTRACT FROM AUTHOR]

An example of solving a boundary-value problem for a homogeneous Monge-Ampère equation is given, which produces a Bellman function for an extremal problem on the space BMO. The paper contains a step-by-step instruction for calculation of this function. Cases of rather complicated foliations are considered. This illustrates the technique elaborated in a paper by Ivanishvili, Stolyarov, Vasyunin, and Zatitskiy. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]

The main result of this paper is that, for $$\kappa \in (0,4]$$ , whole-plane SLE $$_\kappa $$ satisfies reversibility, which means that the time-reversal of a whole-plane SLE $$_\kappa $$ trace is still a whole-plane SLE $$_\kappa $$ trace. In addition, we find that the time-reversal of a radial SLE $$_\kappa $$ trace for $$\kappa \in (0,4]$$ is a disc SLE $$_\kappa $$ trace with a marked boundary point. The main tool used in this paper is a stochastic coupling technique, which is used to couple two whole-plane SLE $$_\kappa $$ traces so that they overlap. Another tool used is the Feynman-Kac formula, which is used to solve a PDE. The solution of this PDE is then used to construct the above coupling. [ABSTRACT FROM AUTHOR]

In this paper, we consider the boundary blow-up solutions to the nonlocal elliptic systems of cooperative type. By introducing the boundary measure, the boundary blow-up problem becomes a Cauchy problem. Then using the super-subsolution method, we obtain the existence and nonexistence of positive solutions. Moreover, we study the stability of the minimal solution to the Cauchy problem. [ABSTRACT FROM AUTHOR]

Advanced building design is a rather new interdisciplinary research branch, combining knowledge from physics, engineering, art and social science; its support from both theoretical and computational mathematics is needed. This paper shows an example of such collaboration, introducing a model problem of optimal heating in a low-energy house. Since all particular function values, needed for optimization are obtained as numerical solutions of an initial and boundary value problem for a sparse system of parabolic partial differential equations of evolution with at least two types of physically motivated nonlinearities, the usual gradient-based methods must be replaced by the downhill simplex Nelder-Mead approach or its quasi-gradient modifications. One example of the real low-energy house in Moravian Karst is demonstrated with references to other practical applications. [ABSTRACT FROM AUTHOR]

In this paper, the tension B-spline collocation method is used for finding the solution of boundary value problems which arise from the problems of calculus of variations. The problems are reduced to an explicit system of algebraic equations by this approximation. We apply some numerical examples to illustrate the accuracy and implementation of the method. [ABSTRACT FROM AUTHOR]

In high-speed and high precision machining, cornering acceleration profiles of the feed motion are always not smooth enough and cause severe inertial vibrations in feed drive system, which severely affect machining quality and elongate machining time. In view of this problem, many experts and scholars have proposed kinematic corner smoothing algorithms based on the jerk limited acceleration profile from the perspective of kinematics that generate continuous acceleration transition profiles. But continuous acceleration profiles still have non-differentiable points. In order to further generate smooth and continuous acceleration transition profiles, an acceleration smoothing algorithm based on the jounce limited acceleration profile is proposed in this paper. Firstly, by adding different velocity, acceleration, and displacement boundary conditions to the jounce limited acceleration profile and combined with user-specified contour error, the fastest cornering velocity is deduced. Next, by the fastest cornering velocity and motion performance of the motor driver, the cornering duration is deduced. Finally, smooth acceleration and velocity transition profiles can be controlled analytically by accurately calculating cornering duration. The proposed algorithms are divided into interrupted acceleration smoothing algorithmand uninterrupted acceleration smoothing algorithm based on additional different acceleration boundary conditions. Through the experimental analysis and comparison, the proposed algorithms can reduce overall machining time around 6-7% and deliver curvature smoothing motion profiles, curvature continuous velocity profiles, and tangent continuous acceleration profiles. The proposed algorithms achieve smooth acceleration transitions and improve machining quality. [ABSTRACT FROM AUTHOR]

Minimax criterion is used to carry out the theoretical analysis of normal displacement of points at the boundary of a circular disk weakened by arbitrarily placed rectilinear cracks. This paper presents a criterion and method for solving the problem of fracture of the circular disk with mixed conditions on its boundary. A closed system of algebraic equations is constructed, which allows for minimization of stress intensity factors. The normal displacement of points at the boundary of the circular disk, for which the bearing capacity of the disk increases. [ABSTRACT FROM AUTHOR]

This paper is considered with a class of nonlinear fractional differential coupled system with fractional differential boundary value conditions and impulses. By means of the Banach contraction principle and the Schauder fixed point theorem, some sufficient criteria are established to guarantee the existence of solutions. As applications, some interesting examples are given to illustrate the effectiveness of our main results. [ABSTRACT FROM AUTHOR]

The objective of the present paper is to provide an efficient method for finding steady planar ideal plastic flows of anisotropic materials. The method consists of determining two mappings between coordinate systems. One of these mappings is between principal lines-based and characteristics-based coordinate systems, and the other is between Cartesian- and characteristics-based coordinate systems. Thus, the mapping between the Cartesian- and principal lines-based coordinate systems is given in parametric form. It is shown that the boundary value problem of finding the mapping between the principal lines-based and characteristics-based coordinate systems can be reduced to the solution of a telegraph equation where two families of characteristics are curved and to the evaluation of ordinary integrals where one family of characteristics is straight. In either case, after solving this problem the problem of finding the mapping between the Cartesian- and characteristics- based coordinate systems can be reduced to the evaluation of ordinary integrals. [ABSTRACT FROM AUTHOR]

In this paper, we present a general methodology for solving buckling problems for inhomogeneous columns. Columns that are treated are functionally graded in axial direction. The buckling mode is postulated as the general order polynomial function that satisfies all boundary conditions. For specificity, we concentrate on the boundary conditions of simple support, and employ the second-order ordinary differential equation that governs the buckling behavior. A quadratic polynomial is adopted for the description of the column's flexural rigidity. Satisfaction of the governing differential equation leads to a set of nonlinear algebraic equations that are solved exactly. In addition to the recovery of the solutions previously found by Duncan and Elishakoff, several new solutions are arrived at. [ABSTRACT FROM AUTHOR]

The present work aims at saving computational cost of multiscale simulation on major crack/minor crack interaction problems. The multiscale extended finite element method (MsXFEM) used for the numerical simulation is developed on multiscale projection technique which enables different scale decomposition, and transition of field variables between different scales. Both macroscale and microscale problems are solved independently and alternatively, in the framework of XFEM. The improvement made in this paper is to employ corrected XFEM on the macroscale level, so that a more accurate boundary condition can be obtained for the microscale problem. The modification leads to a reduced necessary microscale domain size, meanwhile a solution of higher accuracy and enhanced convergence rate can be achieved. The numerical examples of minor cracks near a major one are studied, which show that the effect of minor cracks on major crack can be efficiently captured. [ABSTRACT FROM AUTHOR]

In this paper, we present a strong-form framework for solving the boundary value problems with geometric nonlinearity, in which an incremental theory is developed for the problem based on the Newton-Raphson scheme. Conventionally, the finite element methods (FEMs) or weak-form based meshfree methods have often been adopted to solve geometric nonlinear problems. However, issues, such as the mesh dependency, the numerical integration, and the boundary imposition, make these approaches computationally inefficient. Recently, strong-form collocation methods have been called on to solve the boundary value problems. The feasibility of the collocation method with the nodal discretization such as the radial basis collocation method (RBCM) motivates the present study. Due to the limited application to the nonlinear analysis in a strong form, we formulate the equation of equilibrium, along with the boundary conditions, in an incremental-iterative sense using the RBCM. The efficacy of the proposed framework is numerically demonstrated with the solution of two benchmark problems involving the geometric nonlinearity. Compared with the conventional weak-form formulation, the proposed framework is advantageous as no quadrature rule is needed in constructing the governing equation, and no mesh limitation exists with the deformed geometry in the incremental-iterative process. [ABSTRACT FROM AUTHOR]

In this paper, Galerkin method has been introduced using Legendre polynomials as basis functions over the interval $$[-1, 1]$$ to solve the eighth-order linear boundary value problems with two-point boundary conditions. Legendre Galerkin method is an effective tool in numerically solving such problems. The performance and applicability of the method is illustrated through some examples that reveal the method presents much better results. The obtained numerical results are convincing and very close to the analytical ones. [ABSTRACT FROM AUTHOR]

In the present paper, the Bitsadze-Samarski boundary-value problem is considered for a quasi-linear differential equation of first order on the plane and the existence and uniqueness theorem for a generalized solution is proved; the necessary (in the linear case) and sufficient optimality conditions for optimal control problems are found. The optimal control problem is posed, where the behavior of control functions is described by elliptic-type equations with Bitsadze-Samarski nonlocal boundary conditions. The necessary and sufficient optimality conditions are obtained in the form of the Pontryagin maximum principle and the solution existence and uniqueness theorem is proved for the conjugate problem. Nonlocal boundary-value problems and conjugate problems are solved by the algorithm, which reduces nonlocal boundary value problems to a sequence of Dirichlet problems. The numerical method of solution of an optimal control problem by the Mathcad package is presented. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of classical solutions for a second-order impulsive differential equation with non-separated periodic boundary conditions. By using the variational method and critical point theory, we give some new criteria to guarantee that the impulsive problem has at least one solution under some different conditions. Our results extend and improve some recent results. [ABSTRACT FROM AUTHOR]

In the paper, for a Lamé type system the inverse problem of recovering the fast and slow wave velocities from the boundary dynamical data (the response operator) is solved. The velocities are determined in a near-boundary domain, the depth of determination being proportional to the observation time. The BC-method, which is an approach to inverse problems based on their connections with boundary control theory, is used. [ABSTRACT FROM AUTHOR]

In this paper, we study the exterior problem for the spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients. Under certain assumptions imposed on the initial data, we show that there exists a unique global strong solution to the exterior problem and obtain the regularity of the strong solution. Some ideas and more delicate estimates are introduced to prove these results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence, nonexistence, uniqueness, and multiplicity of positive solutions for a class of eigenvalue problems of nonlinear fractional differential equations with a nonlinear integral term and a disturbance parameter in the boundary conditions. By using fixed point index theory we give the critical curve of eigenvalue λ and disturbance parameter μ that divides the range of λ and μ for the existence of at least two, one, and no positive solutions for the eigenvalue problem. Furthermore, by using fixed point theorem for a sum operator with a parameter we establish the maximum eigenvalue interval for the existence of the unique positive solution for the eigenvalue problem and show that such a positive solution depends continuously on the parameter λ for given μ. In particular, we give estimates for the critical value of parameters. Two examples are given to illustrate our main results. [ABSTRACT FROM AUTHOR]

Matrices where all minors are nonnegative are said to be totally nonnegative (TN) matrices. In the case of banded TN matrices, which can be expressed by products of several bidiagonal TN matrices, Fukuda et al. (Annal. Mat. Pura Appl. 192, 423-445, ) discussed the eigenvalue problem from the viewpoint of the discrete hungry Toda (dhToda) equation. The dhToda equation is a discrete integrable system associated with box and ball systems. In this paper, we consider an inverse eigenvalue problem for such banded TN matrices by examining the properties of the dhToda equation. This problem is a real-valued nonnegative inverse eigenvalue problem. First, we show the determinant solution to the dhToda equation with suitable boundary conditions. Next, we clarify the relationship between the characteristic polynomials of the banded TN matrices and the determinant solution. Finally, taking this relationship into account, we design a finite-step procedure for constructing banded TN matrices with specified eigenvalues. We also present an example to demonstrate this procedure. [ABSTRACT FROM AUTHOR]

In this paper, exact solutions for the primary field variables and their first derivatives are derived for the coupled magneto-mechanical problem of a circular inclusion embedded in an infinite surrounding. Both material domains possess linear and isotropic material properties. The assumed planarity of all fields enables and recommends the description with analytic functions depending on the complex variable z. The well-known technique of a complex stress potential satisfying the bipotential equation is used and adapted to coupled magneto-mechanical problems. The provided analytic expressions are of special interest for the validation of numeric solutions of coupled magneto-mechanical boundary value problems. In this contribution, they are applied to analyze the convergence behavior of an extended finite element formulation for coupled magneto-mechanical problems. Based on the analytic results, proper boundary conditions are prescribed to the numeric model. The convergence in terms of polynomial degree and mesh refinement of the implemented element formulation is proved by computing the L and the energy norm which requires the knowledge of the exact solution. The generality of the proposed formulas allows for the deduction of some other kind of problems, e.g., the pure mechanical displacement and stress field of a bimaterial setting or the stress concentration around a circular hole. [ABSTRACT FROM AUTHOR]

This paper is a review of studies carried out on the basis of two-dimensional boundary-value problems of filtration theory. The role of critical regimes determining the specifics of filtration flows with moving boundaries is noted. [ABSTRACT FROM AUTHOR]

We study the worst case complexity and the tractability of the model second-order problem $$-\Delta u+u=f$$ on $$I^d=(0,1)^d $$ with homogeneous Neumann boundary conditions. As is often the case, we study the variational formulation of this problem. Previous work on the tractability of problems such as this relied on the fact that such problems were reducible to the $$L_2(I^d)$$ -approximation problem, which allowed us to find necessary and sufficient conditions for the problem to have a given degree of tractability. However, such an approach can only yield sufficient conditions, and not necessary conditions, for the model second-order Neumann problem to have a given degree of tractability. In this paper, we remedy this gap and find necessary and sufficient conditions for this Neumann problem to exhibit a given degree of tractability. [ABSTRACT FROM AUTHOR]

In this paper, a new method based on neural network is developed for obtaining the solution of the Navier-Stokes equations in an analytical function form. The solution procedure is based upon forming a trial solution consisting of two parts. The first part directly satisfies the boundary conditions and therefore, contains no adjustable parameters. The second part is constructed such that the governing equation is satisfied inside the solution domain, while the boundary conditions remain untouched. This part involves a feed-forward neural network, containing adjustable parameters (the weights), which must be determined such that the resulting approximate error function is minimized. The details of the method are discussed, and the capabilities of the method are illustrated by solving Navier-Stokes problem with different boundary conditions. The performance of the method and the accuracy of the results are evaluated by comparing with the available numerical and analytical solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study optimal control problems whose behavior is described by second-order differential equations with nonlocal Bitsadze-Samarski boundary conditions. Necessary conditions of optimality are obtained in terms of the maximum principle; adjoint equations are constructed in the differential and integral form. Necessary and sufficient optimality conditions are obtained for a linear problem, a difference scheme is constructed and examined, and a numerical algorithm is proposed. [ABSTRACT FROM AUTHOR]

The present paper is devoted to optimal control problems whose behavior is described by quasilinear first-order differential equations on the plane with nonlocal Bitsadze-Samarski boundary conditions. A theorem on the existence and uniqueness of a generalized solution in the space $$ {C}_{\mu}\left(\overline{G}\right) $$ is proved for quasilinear differential equations; necessary optimality conditions are obtained in terms of the maximum principle; the Bitsadze-Samarski boundary-value problem is examined for a first-order linear differential equation; the existence of a solution in the space $$ {C}_{\mu}^p\left(\overline{G}\right) $$ is proved, and an a priori estimate is derived. A necessary and sufficient optimality condition is proved for a linear optimal control problem. [ABSTRACT FROM AUTHOR]

In this paper we study a class of singularly perturbed interface boundary value problems with discontinuous source terms. We first establish a lemma of lower-upper solutions by using the Schauder fixed point theorem. By the method of boundary functions and the lemma of lower-upper solutions we obtain the existence, asymptotic estimates, and uniqueness of the solution with boundary and interior layers for the proposed problem. [ABSTRACT FROM AUTHOR]

This paper investigates some nonlinear third-order ordinary differential equations and inclusions with anti-periodic type integral boundary conditions and multi-strip boundary conditions. To establish the existence results for the given problems, we apply standard tools of fixed point theory for single-valued and multi-valued problems. The results are illustrated with the aid of examples. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of positive solutions to the singular elliptic boundary value problem involving p-Laplace operator ... where Ω⊂ RN (N = 1) is a bounded domain with smooth boundary ∂ Ω, h ∈ L¹(Ω), h(x) > 0 almost everywhere in Ω, k ∈ L∞(Ω) is nonnegative, p > 2, α > 1 and ß ∈ (0, p - 1). A compatibility condition on the couple (h(x),a) is given for the problem to have at least one solution. More precisely, it is shown that the problem admits a solution if and only if there exists u0 ∈ H¹0(Ω) such that ƒΩ hu01-a 0 dx <∞. [ABSTRACT FROM AUTHOR]

We shall consider the Sturm-Liouville boundary value problem y(m)(t) + λF(t,y(t), y'(t),..., y(q)(t)) = 0, t ∈ (0, 1), y(k)(0) = 0, 0 ≤ k ≤ m- 3, ζ y(m-2)(0) - θy(m-1)(0) = 0, py(m-2)(1) + δy(m-1)(1) = 0 where m ≥ 3, 1 ≤ q ≤ m - 2, and λ > 0. It is noted that the boundary value problem considered has a derivative-dependent nonlinear term, which makes the investigation much more challenging. In this paper we shall develop a new technique to characterize the eigenvalues? so that the boundary value problem has a positive solution. Explicit eigenvalue intervals are also established. Some examples are included to dwell upon the usefulness of the results obtained. [ABSTRACT FROM AUTHOR]

In this paper we will study the equation with $$N=3,$$ where $$ S_2(D^2u)(x)=\sum _{1\le i0$$ , and $$0

In this paper, a mid-knot cubic non-polynomial spline is applied to obtain the numerical solution of a system of second-order boundary value problems. The numerical method is proved to be uniquely solvable and it is of second-order accuracy. We further include three examples to illustrate the accuracy of our method and to compare with other methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider wave viscoelastic equation with dynamic boundary condition in a bounded domain, and we establish a general decay result of energy by exploiting the frequency domain method which consists in combining a contradiction argument and a special analysis for the resolvent of the operator of interest with assumptions on past history relaxation function. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of infinitely many solutions for a class of impulsive fractional boundary value problems with p-Laplacian. Based on a variant fountain theorem, the existence of infinitely many nontrivial high or small energy solutions is obtained. In addition, two examples are worked out to illustrate the effectiveness of the main results. [ABSTRACT FROM AUTHOR]

A prominent model for transportation networks is branched transport, which seeks the optimal transportation scheme to move material from a given initial to a given final distribution. The cost of the scheme encodes a higher transport efficiency the more mass is moved together, which automatically leads to optimal transportation networks with a hierarchical branching structure. The two major existing model formulations use either mass fluxes (vector-valued measures, Eulerian formulation) or patterns (probabilities on the space of particle paths, Lagrangian formulation). In the branched transport problem the transportation cost is a fractional power of the transported mass. In this paper we instead analyse the much more general class of transport problems in which the transportation cost is merely a nonnegative increasing and subadditive function (in a certain sense this is the broadest possible generalization of branched transport). In particular, we address the problem of the equivalence of the above-mentioned formulations in this wider context. However, the newly-introduced class of transportation costs lacks strict concavity which complicates the analysis considerably. New ideas are required, in particular, it turns out convenient to state the problem via 1-currents. Our analysis also includes the well-posedness, some network properties, as well as a metrization and a length space property of the model cost, which were previously only known for branched transport. Some already existing arguments in that field are given a more concise and simpler form. [ABSTRACT FROM AUTHOR]

The combined compact difference (CCD) scheme has better spectral resolution than many other existing compact or noncompact high-order schemes, and is widely used to solve many differential equations. However, due to its implicit nature, very little theoretical results on the CCD method are known. In this paper, we provide a rigorous theoretical proof for the unique solvability of the CCD scheme for solving the convection-diffusion equation with variable convection coefficients subject to periodic boundary conditions. [ABSTRACT FROM AUTHOR]