Search

Soil reinforcement using geosynthetics is an efficient and cost-effective solution for a variety of geotechnical structures. Along with the increasing use of geosynthetics, there is a need to expand and enhance the design methodologies for these elements, which are still frequently based on conservative limit equilibrium approaches. In this paper, a bibliometric analysis was conducted on geosynthetic-reinforced soil structures (GRS), identifying the state of the art, research trends, and other indicators. The data were obtained from the Scopus platform and processed by VOSViewer v1.6 software. The initial search comprised 552 papers and the screening process selected 516 relevant papers from 1992 to October 2023. The study analyzed the occurrence of publications by year, keyword trends, authors, citations/co-citations, and bibliographic coupling. Then, a focus was given to 3D modeling research on geosynthetics, highlighting the dominant modeling techniques, material properties, and design challenges in GRS. The bibliometric analysis provided a crucial guideline in the identification of relevant papers and research trends, and a series of conclusions were presented regarding the 3D modeling techniques, choice of material properties, and boundary conditions. [ABSTRACT FROM AUTHOR]

This document is an editorial for a special issue of the journal Mathematics titled "Differential Equations of Mathematical Physics and Related Problems of Mechanics." The special issue covers a range of topics related to differential equations in mathematical physics and mechanics, including wave equations, spectral theory, scattering, and inverse problems. The editorial provides a summary of the published papers in the special issue, highlighting their contributions to the field. The document emphasizes the importance of the special issue in covering both applied and fundamental aspects of mathematics, physics, and their applications in various fields. The author expresses gratitude to the authors, reviewers, assistants, associate editors, and editors for their contributions to the special issue. The report does not provide specific details about the content of the papers or the nature of the special issue. [Extracted from the article]

In this paper, we consider the boundary value problem for the Sturm-Liouville type equation with the fixed delay π/2 and a non zero initial function under transmission conditions at the delay point. We study the case when all parameters within the transmission conditions are known and the potential function is known on the interval (0, π/2). We will prove the uniqueness theorem from two spectra, first with Neumann boundary conditions and second with Cauchy boundary condition. Additionally, we will present an algorithm for the construction of the potential function over the interval (π/2, π). [ABSTRACT FROM AUTHOR]

In this paper, Caputo boundary value problems of order 3 < ζ ≤ 4 are investigated on the interval [0, 1]. By Guo-Krasnoselskii fixed point theorem, some criteria of existence and multiplicity of positive and decreasing solutions are established. The main novelty of the paper lies in its capability to achieve positive solutions while the corresponding Green’s function changes sign. Finally, two examples are provided to illustrate the application of these results. [ABSTRACT FROM AUTHOR]

In this paper, we develop some new multiple fixed point theorems for the sum of two operators T + S where I - T is Lipschitz invertible and S is a k-set contraction on translate of a cone in a Banach space. New existence criteria for multiple positive solutions of a singular generalized Sturm-Liouville multipoint boundary value problem are established. The article ends with an illustrative example. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of nontrivial solutions for an anisotropic discrete nonlinear problem with variable exponent. The analysis makes use of variational methods and critical point theory. [ABSTRACT FROM AUTHOR]

The limit load is an important input parameter of analytical flaw assessment procedures. The accuracy of limit load solutions affects the accuracy of these procedures. The present paper evaluates the influence of a class of the work functions involved in the upper bound theorem on the upper bound limit load of a round highly undermatched tensile bar. It is assumed that the weld contains a crack. The boundary value problem is axisymmetric. The work functions considered cover the entire range of physically reasonable yield criteria, assuming the material is isotropic and incompressible. The kinematically admissible velocity field chosen accounts for some features of the real velocity field. In particular, the kinematically admissible velocity field is singular near the interface between the base material and weld. As a result, the new solution predicts a more accurate limit load than available solutions for the von Mises and Tresca yield criteria. Moreover, the effect of the generalized yield criterion associated with the work functions above on the limit load is shown. [ABSTRACT FROM AUTHOR]

In studying boundary value problems and coupled systems of fractional order in (1 , 2 ] , involving Hilfer fractional derivative operators, a zero initial condition is necessary. The consequence of this fact is that boundary value problems and coupled systems of fractional order with non-zero initial conditions cannot be studied. For example, such boundary value problems and coupled systems of fractional order are those including separated, non-separated, or periodic boundary conditions. In this paper, we propose a method for studying a coupled system of fractional order in (1 , 2 ] , involving fractional derivative operators of Hilfer and Caputo with non-separated boundary conditions. More precisely, a sequential coupled system of fractional differential equations including Hilfer and Caputo fractional derivative operators and non-separated boundary conditions is studied in the present paper. As explained in the concluding section, the opposite combination of Caputo and Hilfer fractional derivative operators requires zero initial conditions. By using Banach's fixed point theorem, the uniqueness of the solution is established, while by applying the Leray–Schauder alternative, the existence of solution is obtained. Numerical examples are constructed illustrating the main results. [ABSTRACT FROM AUTHOR]

This paper introduces a topology optimization approach based on the virtual element method (VEM), incorporating uncertainties. The objective of this optimization process is to design an optimal material layout for problems governed by linear elasticity equations to minimize the volume while satisfying probabilistic compliance constraints. The VEM is used to solve the boundary value problem in reliability-based topology optimization (RBTO). In the comparison between the VEM and the standard finite element method (FEM), a key difference emerges in the absence of explicitly defined shape functions tied to discrete degrees of freedom in VEM. Unlike FEM, VEM directly constructs the discrete bilinear form and load linear form without the need for computing shape function derivatives within the elements. This flexibility accommodates meshes with intricate geometries and arbitrarily shaped elements. The paper discusses the computational efficiency of VEM RBTO and explores the geometric impact of tessellations on converged topologies, demonstrating reduced susceptibility to checkerboard patterns compared to conventional quadrilateral elements. Additionally, the single-loop approach is examined, showcasing comparable accuracy to the first-order/second-order reliability methods (FORM/SORM) of RBTO using VEM. Numerical results for several problems that demonstrate the feasibility of the proposed method are presented. [ABSTRACT FROM AUTHOR]

This paper concerns with a Lyapunov-type inequality for the Riesz-Caputo fractional boundary value problem with anti-periodic boundary conditions. As an application for the obtained inequality, a lower bound for the eigenvalues of anti-periodic fractional boundary problems of the Riesz-Caputo derivative has been obtained. [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]

This paper develops the mathematical apparatus of studying control problems for the stationary model of magnetic hydrodynamics of viscous heat-conducting fluid in the Boussinesq approximation. These problems are formulated as problems of conditional minimization of special cost functionals by weak solutions of the original boundary value problem. The model under consideration consists of the Navier–Stokes equations, the Maxwell equations without displacement currents, the generalized Ohm's law for a moving medium and the convection-diffusion equation for temperature. These relations are nonlinearly connected via the Lorentz force, buoyancy force in the Boussinesq approximation and convective heat transfer. Results concerning the existence and uniqueness of the solution of the original boundary value problem and of its generalized linear analog are presented. The global solvability of the control problem under study is proved and the optimality system is derived. Sufficient conditions on the data are established which ensure local uniqueness and stability of solutions of the control problems under study with respect to small perturbations of the cost functional to be minimized and one of the given functions. We stress that the unique stability estimates obtained in the paper have a clear mathematical structure and intrinsic beauty. [ABSTRACT FROM AUTHOR]

Addressing the boundary value problems of fractional-order differential equations hold significant importance due to their applications in various fields. The aim of this paper was to approximate solutions for a class of boundary value problems involving Caputo fractional-order differential equations employing the AA-iterative scheme. Moreover, the stability and data dependence results of the iterative scheme were given for a certain class of mappings. Finally, a numerical experiment was illustrated to support the results presented herein. The results presented in this paper extend and unify some well-known comparable results in the existing literature. [ABSTRACT FROM AUTHOR]

A new class of nonlocal boundary value problems consisting of multi-term delay fractional differential equations and multipoint-integral boundary conditions is studied in this paper. We derive a more general form of the solution for the given problem by applying a fractional integral operator of an arbitrary order βξ instead of β1; for details, see Lemma 2.2. The given problem is converted into an equivalent fixed-point problem to apply the tools of fixed-point theory. The existence of solutions for the given problem is established through the use of a nonlinear alternative of the Leray-Schauder theorem, while the uniqueness of its solutions is shown with the aid of Banach's fixed-point theorem. We also discuss the stability criteria, icluding Ulam-Hyers, generalized Ulam-Hyers, Ulam-Hyers-Rassias, and generalized Ulam-Hyers-Rassias stability, for solutions of the problem at hand. For illustration of the abstract results, we present examples. Our results are new and useful for the discipline of multi-term fractional differential equations related to hydrodynamics. The paper concludes with some interesting observations. [ABSTRACT FROM AUTHOR]

This paper proposes a new highly effective method for determining the optimal control of the stress-strain state of spatially multi-connected composite bodies using a stationary temperature field. The proposed method is considered based on the example of a stationary axisymmetric thermoelastic problem for a space with a spherical inclusion and cavity. The proposed method is based on the generalized Fourier method and reduces the original problem to an equivalent problem of optimal control, in which the state of the object is determined by an infinite system of linear algebraic equations, the right-hand side of which parametrically depends on the control. At the same time, the functional of the cost of the initial problem is transformed into a quadratic functional, which depends on the state of the equivalent system and parametrically on the control. The limitation on the temperature distribution is replaced by the value of the control norm in the space of square summable sequenc es. In fact, this paper considers for the first time the problem of optimal control of an infinite system of linear algebraic equations and develops a method for its solution. The proposed method is based on presenting the solutions of infinite systems in a parametric form, which makes it possible to reduce equivalent problem to the problem of conditional extremum of a quadratic functional, which explicitly depends on the control. A further solution to this problem A further solution to this problem is found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. The method developed in this paper is strictly justified. For all infinite systems, the Fredholm property of their operators is proved. As an important result necessary for substantiation, for the first time, an estimate from below of the module of the multi-parameter determinant of the resolving system of the boundary value problem of conjugation – space with a spherical inclusion – was obtained when solving it using the Fourier method. The theorem that establishes the conditions for the existence and uniqueness of the solution of equivalent problem or optimal control problem without restrictions in the space of square summable sequences is proved. The numerical algorithm is based on a reduction method for infinite systems of linear algebraic equations. Estimates of the practical accuracy of the numerical algorithm demo nstrated the stability of the method and sufficiently high accuracy even with close location of the boundary surfaces. Graphs showing the optimal temperature distribution for various geometric parameters of the problem and their analysis are provided. The proposed method extends to boundary value problems with different geometries. [ABSTRACT FROM AUTHOR]

This paper discusses the existence of solutions for fractional differential equations with nonlocal boundary conditions (NFDEs) under essential assumptions. The boundary conditions incorporate a point 0 ≤ c < d and fixed points at the end of the interval [0, d]. For i = 0, 1, the boundary conditions are as follows: ai, bi > 0, a0p(c) = -b0p(d), a1p'(c) = -b1p'(d). Furthermore, the research aims to expand the usability and comprehension of these results to encompass not just NFDEs but also classical fractional differential equations (FDEs) by using the Krasnoselskii fixedpoint theorem and the contraction principle to improve the completeness and usefulness of the results in a wider context of fractional differential equations. We offer examples to demonstrate the results we have achieved. [ABSTRACT FROM AUTHOR]

For various kinds of parabolic and elliptic partial differential and differential-difference equations, results on the stabilization of solutions are presented. For the Cauchy problem for parabolic equations, the stabilization is treated as the existence of a limit as the time unboundedly increases. For the half-space Dirichlet problem for parabolic equations, the stabilization is treated as the existence of a limit as the independent variable orthogonal to the boundary half-plane unboundedly increases. In the classical case of the heat equation, the necessary and sufficient condition of the stabilization consists of the existence of the limit of mean values of the initial-value (boundary-value) function over balls as the ball radius tends to infinity. For all linear problems considered in the present paper, this property is preserved (including elliptic equations and differential-difference equations). The Wiener Tauberian theorem is used to establish this property. To investigate the differential-difference case, we use the fact that translation operators are Fourier multipliers (as well as differential operators), which allows one to use a standard Gel'fand-Shilov operational scheme. For all quasilinear problems considered in the present paper, the mean value from the stabilization criterion is changed: It undergoes a monotonic map, which is explicitly constructed for each investigated nonlinear boundary-value problem. [ABSTRACT FROM AUTHOR]

Copyright of Military Technical Courier / Vojnotehnicki Glasnik is the property of Military Technical Courier / Vojnotehnicki Glasnik 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, a high order accuracy method is developed for finding the approximate solution of two-point boundary value problems. The present approach is based on a special algorithm, taken from Pascal’s triangle, for obtaining a generalized form of the parametric splines of degree (2k + 1), k = 1, 2, . . ., which has a lower computational cost and gives the better approximation. Some appropriate band matrices are used to obtain a matrix form for this algorithm. The approximate solution converges to the exact solution of order O(h4k ), where k is a quantity related to the degree of parametric splines and the number of matrix bands that are applied in this paper. Some examples are given to illustrate the applicability of the method, and we compare the computed results with other existing known methods. It is observed that our approach produced better results. [ABSTRACT FROM AUTHOR]

Faber series are used extensively in the application of complex variable methods to two-dimensional elasticity theory, for example, in the mechanical analysis of composite materials where Faber series representations of complex potentials lead to convenient expressions for the corresponding displacement and stress distributions. In many cases, the use of the Faber series is combined with conformal mapping techniques which "transfer" a boundary value problem defined in the elastic body (physical plane) to a simpler problem posed in an imaginary plane characterized by the conformal mapping. In several instances in the literature, however, little attention has been paid to the domain of definition of the Faber series in the imaginary plane leading often to misunderstandings and erroneous conclusions regarding the concept and feasibility of the use of the Faber series. In this paper, we present a thorough and rigorous examination of the representation of the Faber series in both the physical (occupied by the material) and imaginary (defined by the conformal mapping) plane. In addition, we show that replacing a truncated Faber series by a truncated Taylor series does not induce any additional errors in the numerical analysis of the corresponding boundary value problem. We anticipate that the discussion in this paper will help clarify any existing misinterpretations regarding the application of the Faber series and help further extend their use to a range of problems in composite mechanics. [ABSTRACT FROM AUTHOR]

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]

In the present paper, a Sturm-Liouville equation and two nonlocal boundary conditions which are generalized versions of Bitsadze-Samarskii-type conditions are considered. The main goal of this paper is to obtain the uniqueness and the recovering algorithm of the operator via nodal points. [ABSTRACT FROM AUTHOR]

In this paper, we were concerned with the global behavior of positive solutions for third-order semipositone problems with an integral boundary condition y ‴ + β y ″ + α y ′ + λ f (t , y) = 0 , t ∈ (0 , 1) , y (0) = y ′ (0) = 0 , y (1) = χ ∫ 0 1 y (s) d s , where α ∈ (0 , ∞) and β ∈ (− ∞ , ∞) are two constants, λ , χ are two positive parameters, and f ∈ C ([ 0 , 1 ] × [ 0 , ∞) , R) with f (t , 0) < 0 . Our analysis mainly relied on the bifurcation theory. In this paper, we were concerned with the global behavior of positive solutions for third-order semipositone problems with an integral boundary condition where and are two constants, are two positive parameters, and with . Our analysis mainly relied on the bifurcation theory. [ABSTRACT FROM AUTHOR]

The aim of this paper is to determine the eigenvalue intervals of μk; 1 < k < n for which an iterative systems of a class of fractional-order differential equations with parameterized integral boundary conditions (BCs) has at least one positive solution by means of standard fixed point theorem of cone type. To the best of our knowledge, this will be the first time that we attempt to reach such findings for the topic at hand in the literature. The obtained results in the paper are illustrated with an example for their feasibility. [ABSTRACT FROM AUTHOR]

Asymmetry plays a significant role in the transmission dynamics in novel discrete fractional calculus. Few studies have mathematically modeled such asymmetry properties, and none have developed discrete models that incorporate different symmetry developmental stages. This paper introduces a Taylor monomial falling function and presents some properties of this function in a delta fractional model with Green's function kernel. In the deterministic case, Green's function will be non-negative, and this shows that the function has an upper bound for its maximum point. More precisely, in this paper, based on the properties of the Taylor monomial falling function, we investigate Lyapunov-type inequalities for a delta fractional boundary value problem of Riemann–Liouville type. [ABSTRACT FROM AUTHOR]

We provide criteria for the existence of solutions of nonlinear discrete-time boundary value problems on infinite-time intervals. The problems are formulated as nonlinear operator equations on sequence spaces and the tools of nonlinear functional analysis are employed throughout the paper. [ABSTRACT FROM AUTHOR]

In this article, the symmetric operator L corresponding to the boundary value problem is represented as the difference of two commuting operators A and  B.  The uniqueness of the solution is guaranteed if the spectra of the operators A and B do not intersect and the domain of the operator B is given by non-degenerate boundary conditions. In contrast to the existing papers, the criterion for the uniqueness of the boundary value problem formulated in this paper is satisfied even when the system of root functions of the operator B does not form a basis in the corresponding space. At the same time, only the closedness of the linear operator A is required. [ABSTRACT FROM AUTHOR]

The rationale of this work is to introduce the notion of graphical symmetric spaces and some fixed-point results are proved for H - (ϑ , φ) -contractions in this setting. The idea of graphical symmetric spaces generalizes various spaces equipped with a function which characterizes the distance between two points of the space. Some topological properties of graphical symmetric spaces are discussed. Some fixed-point results for the mappings defined on graphical symmetric spaces are proved. The fixed-point results of this paper generalize and extend several fixed-point results in this new setting. The main results of this paper are applied to obtain the positive solutions of fractional periodic boundary value problems. [ABSTRACT FROM AUTHOR]

This paper proves the well-posedness of locally smooth solutions to the free boundary value problem for the 1D degenerate drift diffusion equation. At the free boundary, the drift diffusion equation becomes a degenerate hyperbolic-Poisson coupled equation. We apply the Hardy's inequality and weighted Sobolev spaces to construct the appropriate a priori estimates, overcome the degeneracy of the system and successfully establish the existence of solutions in the Lagrangian coordinates. [ABSTRACT FROM AUTHOR]

This paper focuses on the analysis of a coupled system governed by a Caputo-fractional derivative with q-integral-coupled boundary conditions. This system is particularly relevant in modeling multi-atomic systems, including scenarios involving adsorbed atoms or clusters on crystalline surfaces, surface–atom scattering, and atomic friction. To investigate this system, we introduce an operator that exhibits fixed points corresponding to the solutions of the problem, effectively transforming the system into an equivalent fixed-point problem. We established the necessary conditions for the existence and uniqueness of solutions using the Leray–Schauder nonlinear alternative and the Banach contraction mapping principle, respectively. Stability results in the Ulam sense for the coupled system are also discussed, along with a sensitivity analysis of the range parameters. To support the validity of their findings, we provide illustrative examples. Overall, the paper offers a thorough examination and analysis of the considered coupled system, making important contributions to the understanding of multi-atomic systems and their mathematical modeling. [ABSTRACT FROM AUTHOR]

In this paper, we establish the uniqueness and existence of the classical solution to higher-order boundary value problems. Then, we give a new Lyapunov-type inequalities for higher order boundary value problems. Our study is based on Green’s functions corresponding to the five different types of two-point boundary value problems. In addition, some applications of the obtained inequalities are given. [ABSTRACT FROM AUTHOR]

This paper consists of four parts. It begins by using the authors' generalized Schauder formula, [41] , and the algebraic multiplicity, χ , of Esquinas and López-Gómez [15,14,31] to package and sharpening all existing results in local and global bifurcation theory for Fredholm operators through the recent author's axiomatization of the Fitzpatrick–Pejsachowicz–Rabier degree, [42]. This facilitates reformulating and refining all existing results in a compact and unifying way. Then, the local structure of the solution set of analytic nonlinearities F (λ , u) = 0 at a simple degenerate eigenvalue is ascertained by means of some concepts and devices of Algebraic Geometry and Galois Theory, which establishes a bisociation between Bifurcation Theory and Algebraic Geometry. Finally, the unilateral theorems of [31,33] , as well as the refinement of Shi and Wang [53] , are substantially generalized. This paper also analyzes two important examples to illustrate and discuss the relevance of the abstract theory. The second one studies the regular positive solutions of a multidimensional quasilinear boundary value problem of mixed type related to the mean curvature operator. [ABSTRACT FROM AUTHOR]

In this paper we ascertain the singular point-wise behavior of the positive solutions of a semilinear elliptic boundary value problem (1) at the critical value of the parameter, λ , where it begins its metasolution regime. As the weight function m (x) changes sign in Ω, our result is a substantial extension of a previous, very recent, result of Li et al. [8] , where it was imposed the (very strong) condition that m ≥ 0 on a neighborhood of b − 1 ({ 0 }). In this paper, we are simply assuming that m (x 0) > 0 for some x 0 ∈ b − 1 ({ 0 }). • Theorem 1.1 proves that the behavior of the solutions proved by Li et al. [8] also occurs with much weaker hypotheses. • Theorem 3.1 is a substantial extension of Theorem 2.1 of López-Gómez and Sabina de Lis [12]. • Lemma 2.1 provides a useful estimate of eigenfunctions associated to an eigenvalue problem with sign changing weight. [ABSTRACT FROM AUTHOR]

The interior inverse elastic scattering problem is considered in this paper. The scattering in the cavity is illuminated by an time-harmonic elastic point source at fixed frequency. The boundary of the cavity is assumed to be a small and smooth perturbation of some circle. In this paper, we are interested in the reconstruction of the boundary according to the measured data on a closed curve inside the cavity. In implementation, we split the displacement field into compressional and shear waves by the Helmholtz decomposition. Then we use coordinate transformation, power series expansion based on perturbed parameters and Fourier expansion of corresponding power series coefficients to simplify the original boundary value problem to a successive sequence of one-dimensional boundary value problem. We linearize the inverse problem by eliminating the higher-order terms in the power series expansion; and design a reconstruction algorithm by the explicit relationship between the Fourier coefficients of the boundary function and the total field. In order to obtain better resolution, we also use a nonlinear correction method in the numerical experiments. The numerical results have shown the effectiveness and the stableness of the proposed method. [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]

In this paper we consider the vanishing viscosity limit of solutions to the initial boundary value problem for the compressible viscoelastic equations in the half space. When the initial deformation gradient does not degenerate and there is no vacuum initially, we establish the uniform regularity estimates of solutions to the initial-boundary value problem for the three-dimensional compressible viscoelastic equations in the Sobolev spaces. Then we justify the vanishing viscosity limit of solutions of the compressible viscoelastic equations based on the uniform regularity estimates and the compactness arguments. Both the no-slip boundary condition and the Navier-slip type boundary condition on velocity are addressed in this paper. On the one hand, for the corresponding vanishing viscosity limit of the compressible Navier-Stokes equations with the no-slip boundary condition, it is impossible to derive such uniform energy estimates of solutions due to the appearance of strong boundary layers. Consequently, our results show that the deformation gradient can prevent the formation of strong boundary layers. On the other hand, our results also provide two different kinds of boundary conditions suitable for the well-posedness of the initial-boundary value problem of the elastodynamic equations via the method of vanishing viscosity. Finally, it is worth noting that we take advantage of the Lagrangian coordinates to study the vanishing viscosity limit for the fixed boundary problem in this paper. [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]

In this paper, a singularly perturbed one-dimensional initial boundary value problem of a quasilinear Sobolev-type equation is presented. The nonlinear term of the problem is linearized by Newton's linearization method. Time derivatives are discretized by implicit Euler's method on nonuniform step size. A uniform trigonometric B-spline collocation method is used to treat the spatial variable. The convergence analysis of the scheme is proved, and the accuracy of the method is of order two in space and order one in time direction, respectively. To test the efficiency of the method, a model example is demonstrated. Results of the scheme are presented in tabular, and the figure indicates the scheme is uniformly convergent and has an initial layer at t = 0. [ABSTRACT FROM AUTHOR]

The purpose of this paper is the study of the eigenvalues of the second order fuzzy boundary value problem (FBVP). By using the (α-β)-level set of intuitionistic fuzzy numbers and Zadeh's extension principle, the FBVP is solved with the proposed method. Furthermore, a numerical example is illustrated and the advantages of the proposed approach are compared with other well-known methods such as the solutions based on the generalized Hukuhara derivative. [ABSTRACT FROM AUTHOR]

Time-delay fractional optimal control problems (OCPs) are an important research area for developing effective control and optimization strategies to address complex phenomena occurring in various natural sciences, such as physics, chemistry, biology, and engineering. By considering fractional OCPs with time delays, we can design control strategies that take into account the system's history and optimize its behavior over a given time horizon. However, applying the Pontryagin principle of maximization to solve these problems leads to a boundary value problem (BVP) that includes delay and advance terms, making analytical solutions difficult and demanding. To address this issue, this paper presents a precise finite difference formula to solve the aforementioned advance-delay BVP numerically. The suggested approximate method's error analysis and convergence properties are provided, and several illustrative examples demonstrate the applicability, validity, and accuracy of the proposed approach. Simulation results confirm the proposed technique's advantages for the optimal control of delay fractional dynamical equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive solutions to a Hadamard-type fractional integral boundary value problem using fixed point index. We construct a new linear operator and obtain our main results under some conditions concerning the spectral radius of this linear operator. Our method improves and generalizes some results in the literature. [ABSTRACT FROM AUTHOR]

The Blasius equation is a well-known third-order nonlinear ordinary differential equation that can be found in some fluid dynamics boundary layer problems. In this paper, we convert the nonlinear differential equation to an integral equation, this integral equation has a shifted kernel. Our goal is to propose an efficient modification of the standard Adomian decomposition method, combined with the Laplace transform, for solving the Blasius equation. The main impediment to solving the Blasius equation is the absence of the second derivative at zero. Once this derivative has been correctly evaluated, an analytical solution to the Blasius problem can be easily found; as a result, we use our approximate solution to estimate the value of y″(0), also known as the Blasius constant. Understanding the Blasius constant is essential for calculating shear stress at a plate. Furthermore, once this value is determined, we have the initial value problem, which can be solved numerically. [ABSTRACT FROM AUTHOR]

The technique to broaden the scope of fixed point theory is to extend the class of spaces that have stronger conceptual frameworks than metric spaces. Therefore, this paper explores the introduction of novel metric spaces, namely, Branciari suprametric spaces, and investigates some of its fundamental topological properties. An illustration is provided to validate the newly defined idea of Branciari suprametric spaces. Further two intriguing, fixed point results are proved, and a corollary is presented as an implication of our main result. The following is a specification of the analogue of the rectangle inequality in Branciari suprametric spaces dB (τ,ι) ≤ dB (τ,ν)+dB(ν,σ)+dB(σ,ι) + μdB (τ,ν)dB(ν,σ)dB(σ,ι) for all t≠v, v≠σ and σ≠1. Furthermore, by employing the results obtained, the present study intends to provide an appropriate solution for the nonlinear fractional differential equations of the RiemannLiouville type. [ABSTRACT FROM AUTHOR]

This paper explored a delayed diffusive prey-predator model with prey-taxis involving the volume-filling mechanism subject to homogeneous Neumann boundary condition. To figure out the impact on the dynamic of the prey-predator model due to prey-taxis and time delay, we treated the prey-tactic coefficient χ and time delay τ as the bifurcating parameters and did stability and bifurcation analysis. It showed that the time delay will induce Hopf bifurcations in the absence of prey-taxis, and the bifurcation periodic solution at the first critical value of τ was spatially homogeneous. Hopf bifurcations occurred in the model when the prey-taxis and time delay coexisted, and at the first critical value of τ , spatially homogeneous or nonhomogeneous periodic solutions might emerge. It was also discovered that the bifurcation curves will intersect, which implied that Hopf-Hopf bifurcations can occur. Finally, we did numerical simulations to validate our outcomes. [ABSTRACT FROM AUTHOR]

In this paper, we present a new computational methodology using the Residual Power Series Method (RPSM) to create a new model in the fuzzy domain suitable for solving a fourth-order of FBVP using the concept of generalized differentiation, and we present an algorithm for the numerical solutions of fourth-order fuzzy boundary value problems (FBVPs), which include ordinary differential equations, based on concepts from fuzzy set theory. The effectiveness of the suggested technique was tested to verify the accuracy of RPSM, and the series solutions were compared to the exact solution when the plane is equal to one. The results show that the algorithm is highly efficient and straightforward in implementation, compared to the other mentioned schemes, and fulfills the characteristics of fuzzy solution, thus it is a practically optimal solution for the most complex BVP programs in science. [ABSTRACT FROM AUTHOR]

This paper introduces a novel F fixed-point iteration method that leverages Green's function for solving the nonlinear Troesch problem in Banach spaces, which are symmetric spaces. The Troesch problem, characterized by its challenging boundary conditions and nonlinear nature, is significant in various physical and engineering applications. The proposed method integrates fixed-point theory with Green's function techniques to develop an iteration process that ensures convergence, stability, and accuracy. The numerical experiments demonstrate the method's efficiency and robustness, highlighting its potential for broader applications in solving nonlinear differential equations in Banach spaces. [ABSTRACT FROM AUTHOR]

After the initiation of Jachymski's contraction principle via digraph, the area of metric fixed point theory has attracted much attention. A number of outcomes on fixed points in the context of graph metric space employing various types of contractions have been investigated. The aim of this paper is to investigate some fixed point theorems for a class of nonlinear contractions in a metric space endued with a transitive digraph. The outcomes presented herewith improve, extend and enrich several existing results. Employing our findings, we describe the existence and uniqueness of a singular fractional boundary value problem. [ABSTRACT FROM AUTHOR]

This paper provides the process of incremental constitutive integration for the unified hardening model combined with the transformation stress method. The dimension-extending technique takes the hardening function of the hardening/softening model as the same position as the stress components, so that the constitutive integration of the plasticity can be reduced to an initial value problem of differential–complementarity equations, which is solved using the Gauss–Seidel algorithm-based Projection–Correction for the mixed complementarity problem. The Gauss–Seidel based Projection–Correction algorithm does not require the calculation of the Jacobean matrix of the potential function, making it relatively easy to implement in programming. The unified hardening model is proposed based on the modified Cam–Clay model and the sub-loading surface model, and the elastic properties are pressure-dependent. Two processing methods, backward Euler integration and exact elastic property, are used for the variable elasticity properties. The constitutive integration of the increased dimensional unified hardening model is reduced to a special mixed complementarity problem and solved by the proposed algorithm, which does not need to calculate the Jacobean matrix of the potential function, and greatly simplifies the derivation process. Several numerical examples are given to verify the feasibility of the incremental constitutive integration in the unified hardening model, including the single integral point and the boundary value problems. The research results have expanded the scope of use of the Gauss–Seidel based Projection–Correction algorithm. [ABSTRACT FROM AUTHOR]