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

Gauge-invariant quantum fields are constructed in an Abelian power-counting renormalizable gauge theory with both scalar, vector and fermionic matter content. This extends previous results already obtained for the gauge-invariant description of the Higgs mode via a propagating gauge-invariant field. The renormalization of the model is studied in the Algebraic Renormalization approach. The decomposition of Slavnov–Taylor identities into separately invariant sectors is analyzed. We also comment on some non-renormalizable extensions of the model whose 1-PI Green's functions are the flows of certain differential equations of the homogeneous Euler type, exactly resumming the dependence on a certain set of dim. 6 and dim. 8 derivative operators. The latter are identified uniquely by the condition that they span the mass and kinetic terms in the gauge-invariant dynamical fields. The construction can be extended to non-Abelian gauge groups. [ABSTRACT FROM AUTHOR]

In the paper we use recently proposed concepts of granular derivative and horizontal membership function to introduce and study the equivalence relation between granular differential equations and a class of second-order linear fuzzy differential equations. It is well-known that in the study of boundary value problem (BVP) of differential equations, the equivalence of the analytical solution for BVP plays an extremely important role. The BVP of second-order linear fuzzy differential equation is solved by the inverse operator method of the second-order fuzzy differential operator and the Green function is defined. It is usually necessary to transform the BVP under consideration into its equivalent integral equations, and the key to the transformation is how to construct Green function for BVP of differential equation. The Green function method is mainly used to solve a class of second-order linear fuzzy differential equation semi-homogeneous BVP here. Moreover, the existence and uniqueness of Green function for semi-homogeneous BVP of self-conjugate equations with parameter λ is proposed. At the same time, several examples are also shown to illustrate the solution of BVP of fuzzy differential equations with Green function method. [ABSTRACT FROM AUTHOR]

The main purpose of this investigation is to revisit solvability process of the Cauchy problems of Caputo fractional two-term initial value problems. To this aim, the Green function technique has chosen to make a bridge between the operator and the fixed point theories. The appeared Green functions in this paper are constructed by the Fox-Wright functions. Our solvability tools include the existence and uniqueness criteria as novel refinements of the Banach contraction principal and Schauder fixed point theorem. This investigation will be finalized by presenting some numerical applications that illustrate proposed solvability criteria. [ABSTRACT FROM AUTHOR]

Due to the restrictive growth and/or monotonicity requirements inherent in their employment, classical iterative fixed-point theorems are rarely used to approximate solutions to an integral operator with Green's function kernel whose fixed points are solutions of a boundary value problem. In this paper, we show how one can decompose a fixed-point problem into multiple fixed-point problems that one can easily iterate to approximate a solution of a differential equation satisfying one boundary condition, then apply a bisection method in an intermediate value theorem argument to meet a second boundary condition. Error estimates on the iterates are also established. The technique will be illustrated on a second-order right focal boundary value problem, with an example provided showing how to apply the results. [ABSTRACT FROM AUTHOR]

In this work, we use techniques from Stieltjes calculus and fixed point index theory to show the existence and multiplicity of solution of a first order non-linear boundary value problem with linear boundary conditions that extend the periodic case. We also provide the Green’s function associated to the problem as well as an example of application. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study the existence and uniqueness of a nonlinear implicit ψ-Caputo fractional order integro-differential boundary value problem using Schauder's and Banach's fixed point theorems. Besides, we study its stability using Ulam-Hyers-Rassias stability type. Finally, we demonstrate our main findings, with a particular case example included to show the significance of our results. [ABSTRACT FROM AUTHOR]

This paper is associated with Sturm–Liouville type boundary value problems and periodic boundary value problems for quaternion-valued differential equations (QDEs). Employing the theory of quaternionic matrices, we prove the conditions for the solvability of the linear boundary valued problem and find Green's function. [ABSTRACT FROM AUTHOR]

This article is a tribute to Anatolii Mykhailovych Samoilenko, a renowned Ukrainian mathematician who was born in 1938 and passed away in 2020. Samoilenko made significant contributions to the fields of differential equations, nonlinear mechanics, and the theory of functions. He was known for his work on impulsive differential equations and almost periodic impulsive systems. Samoilenko also developed theories on perturbations and stability of toroidal manifolds, as well as the theory of accelerated convergence. He authored numerous works, including monographs and textbooks, and mentored many young scientists throughout his career. Samoilenko received several awards and honors for his contributions to mathematics. His legacy as a brilliant mathematician, educator, and science organizer will be remembered by those who knew him. [Extracted from the article]

Iterative differential equations provide a new idea to study functional differential equations. The study of iterative equations can provide new methods for the study of differential equations with state-dependent delays. In this paper, we are concerned with proving the existence of infinitely many positive solutions to nonlinear first-order iterative systems of singular BVPs on time scales by using Krasnoselskii's cone fixed point theorem in a Banach space. It is worth pointing out that in this paper, we can use the symmetry of the iterative process and Green's function to transform the considered differential equation into an equivalent integral equation, which plays a key role in the proof of the theorem in this paper. [ABSTRACT FROM AUTHOR]

In this manuscript, we give sufficient conditions for a sequence to be Cauchy in the context of controlled fuzzy metric space. Furthermore, we generalize the concept of Banach's contraction principle by utilizing several new contraction conditions and prove several fixed point results. Furthermore, we provide a number of non-trivial examples to validate the superiority of main results in the existing literature. At the end, we discuss an important application to the transformation of solar energy to electric power by utilizing differential equations. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the issue of the Green function matrices that belongs to some three-point boundary- and eigenvalue problems. A detailed definition is given for the Green function matrices provided that the considered boundary value problems are governed by a class of ordinary differential equation systems associated with homogeneous boundary and continuity conditions. The definition is a constructive one, i.e., it provides the means needed for calculating the Green function matrices. The fundamental properties of the Green function matrices—existence, symmetry properties, etc.—are also clarified. Making use of these Green functions, a class of three-point eigenvalue problems can be reduced to eigenvalue problems governed by homogeneous Fredholm integral equation systems. The applicability of the novel findings is demonstrated through a Timoshenko beam with three supports. [ABSTRACT FROM AUTHOR]

Symmetry is presented in many works involving differential and integral equations. Whenever a human is involved in the design of an integral equation, they naturally tend to opt for symmetric features. The most common examples are the Green functions and linguistic kernels that are often designed symmetrically and regularly distributed over the universe of discourse. In the current study, the authors report a study on boundary value problem (BVP) for a nonlinear integro Volterra–Fredholm integral equation with variable coefficients and show the existence of solution by applying some fixed-point theorems. The authors employ various numerical common approaches as the homotopy analysis methodology established by Liao and the modified Adomain decomposition technique to produce a numerical approximate solution, then graphical depiction reveals that both methods are most effective and convenient. In this regard, the authors address the requirements that ensure the existence and uniqueness of the solution for various variations of nonlinearity power. The authors also show numerical examples of how to apply our primary theorems and test the convergence and validity of our suggested approach. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a nonlinear neutral differential equation of first order with iterative terms and constant time delays. We obtain new results on the existence and uniqueness of positive periodic solutions of the nonlinear neutral differential equation such that the solution depends on the functions of that the nonlinear neutral differential equation. The proof of the new results depends on some fixed point theorems and the Green's functions. We also provide a numerical example to demonstrate the conditions of the new results can be satisfied and applied. [ABSTRACT FROM AUTHOR]

In this paper, based on the properties of Green function and the eigenvalue of a corresponding linear operator, the existence of positive solutions is investigated by spectral analysis for a infinite-points singular p-Laplacian Hadamard fractional differential equation boundary value problem, and an example is given to demonstrate the validity of our main results. [ABSTRACT FROM AUTHOR]

This paper deals with the boundary value problem involving the differential equation -y" + qy = λy, subject to the parameter dependent boundary conditions with finite number of transmission conditions. The potential function q ∈ L² (0, π) is real and λ is a spectral parameter. We develop the Hochstadt's results based on the transformation operator for inverse Sturm-Liouville problem when there are finite number of transmission and parameter dependent boundary conditions. Furthermore, we establish a formula for q(x) - q(x) in the finite interval [0, π], where q(x) and q(x) are analogous functions. [ABSTRACT FROM AUTHOR]

We study semilinear problems in bounded $ C^{1, 1} $ domains for non-local operators with a boundary condition. The operators cover and extend the case of the spectral fractional Laplacian. We also study harmonic functions with respect to the non-local operator and boundary behaviour of Green and Poisson potentials. [ABSTRACT FROM AUTHOR]

We prove the existence of the solutions for the new mixed differential equations, which is characteristic of the right-sided Caputo and the left-sided Riemann–Liouville fractional derivatives. There are four major ingredients. The first is composed of some basic definitions and lemmas. The second is the Green's function of the new mixed fractional differential equations. We calculate the corresponding Green's functions as well as their properties. The third, which is the main new ingredient of this paper, is demonstration of the existence of the solutions for fractional equations by the fixed-point theorem in cone expansion and compression of norm type. The fourth, as applications, is the example provided to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this work, we consider a class of second order iterative differential equations. Using Schauder's fixed point theorem and the Green's functions method, the existence of periodic solutions is proved after establishing the equivalence of our problem with a certain integral equation. Finally, we end this article with a simple conclusion recapitulating the guiding idea of our approach. Obtained findings complement some previous publications in the literature. [ABSTRACT FROM AUTHOR]

We present a comprehensive study for common second order PDE's in two dimensional disc-like systems and show how their solution can be approximated by finding the Green function of an effective one dimensional system. After elaborating on the formalism, we propose to secure an exact solution via a Fourier expansion of the Green function, which entails solving an infinitely countable system of differential equations for the Green–Fourier modes that in the simplest case yields the source-free Green distribution. We present results on non separable systems—or such whose solution cannot be obtained by the usual variable separation technique—on both annulus and disc geometries, and show how the resulting one dimensional Fourier modes potentially generate a near-exact solution. Numerical solutions will be obtained via finite differentiation using Finite Difference Method (FDM) or Finite Element Method (FEM) with the three-point stencil approximation to derivatives. Comparing to known exact solutions, our results achieve an estimated numerical relative error below 10 − 6 . Solutions show the well-known presence of peaks when r = r ′ and a smooth behavior otherwise, for differential equations involving well-behaved functions. We also verified how the Green functions are symmetric under the presence of a "weight function", which is guaranteed to exist in the presence of a curl-free vector field. Solutions of non-homogeneous differential equations are also shown using the Green formalism and showing consistent results. [ABSTRACT FROM AUTHOR]

We develop a method for constructing semiclassical asymptotic solutions of inhomogeneous partial differential and pseudodifferential equations with localized right-hand sides. These problems are related to the asymptotics of Green's function for this type of operators, in particular, for the Helmholtz equation, which has been studied in numerous papers. The method is based on a constructive description of the corresponding Lagrangian manifolds and on the recently proposed new representations of the Maslov canonical operator in a neighborhood of Lagrangian singularities (caustics and caustic sets). The method underlies an analytic-numerical algorithm for constructing efficient asymptotic solutions to problems of the above-mentioned type in various fields of physics and continuum mechanics. [ABSTRACT FROM AUTHOR]

We extend the study of banana diagrams in coordinate representation to the case of curved space-times. If the space is harmonic, the Green functions continue to depend on a single variable – the geodesic distance. But now this dependence can be somewhat non-trivial. We demonstrate that, like in the flat case, the coordinate differential equations for powers of Green functions can still be expressed as determinants of certain operators. Therefore, not-surprisingly, the coordinate equations remain straightforward – while their reformulation in terms of momentum integrals and Picard-Fuchs equations can seem problematic. However we show that the Feynman parameter representation can also be generalized, at least for banana diagrams in simple harmonic spaces, so that the Picard-Fuchs equations retain their Euclidean form with just a minor modification. A separate story is the transfer to the case when the Green function essentially depends on several rather than a single argument. In this case, we provide just one example, that the equations are still there, but conceptual issues in the more general case will be discussed elsewhere. [ABSTRACT FROM AUTHOR]

Recently, Fe-doped semiconductors have been attracting much attention as ferromagnetic semiconductors due to the possibility that they may exhibit high Curie temperatures and low power consumption and that they may be useful for high-speed spin devices. High Curie temperature ferromagnetism has been observed in Fe-doped InAs, from which both n- and p-type ferromagnetic semiconductors can be fabricated. In order to obtain a higher Curie temperature than that of (In, Fe)As, we have focused on GaSb and InSb as host semiconductors. We have investigated their electronic structures, magnetic properties, and structural stability by using the Korringa-Kohn-Rostoker Green's function method within density functional theory. We have found that (Ga, Fe)Sb and (In, Fe)Sb show complex magnetic properties, which are determined by the correlation between magnetic exchange coupling constants and chemical pair interactions. Isoelectronic Fe-doped GaSb and InSb have strong antiferromagnetic interactions due to the super-exchange mechanism. By shifting the Fermi level–i.e., by n- or p-type doping–(Ga, Fe)Sb and (In, Fe)Sb can be made to undergo a magnetic transition from antiferromagnetic to ferromagnetic ordering. This transition can be well understood in terms of the Alexander-Anderson-Moriya mechanism. Our calculations indicate the possibility of manipulating (Ga, Fe)Sb and (In, Fe)Sb to achieve high Curie temperatures. [ABSTRACT FROM AUTHOR]

Based on a Rashba triple-quantum-dot ring, we theoretically investigate spin thermoelectric properties by using the nonequilibrium Green's function method. Our results show that thermoelectric properties are strongly influenced by the Rashba effect. The underlying reason is the antiresonance effect from Rashba spin-orbit interaction. When the magnetic field and Rashba phase factor satisfy a certain relationship, the value of the spin- dependent Seebeck coefficient alternates between its maximum and zero, and then a pure spin-dependent thermoelectric generator can be obtained. Moreover, we also find that the on-site Coulomb interaction is helpful to obtain a large amplitude for the figure of merit and a high Seebeck coefficient. [ABSTRACT FROM AUTHOR]

In this paper, a system of an arbitrary number of singularly perturbed semilinear differential equations of reaction-diffusion type having discontinuous source term is examined, for the case in which the diffusion parameters associated with each equation of the considered system are assumed to be different in magnitude. In addition to the occurrence of the boundary layers, the solution exhibits the overlapping and interacting internal layers due to the existence of discontinuity in the data. A central finite difference scheme is used in conjunction with a suitable piecewise uniform Shishkin mesh and a Bakhvalov mesh, to construct the numerical method. Using discrete Green's function technique, the proposed numerical scheme has been proved to be an almost second-order of uniform convergent for the Shishkin mesh and second-order of uniform convergent for the Bakhvalov mesh, independent of the perturbation parameters. Numerical test examples are presented to demonstrate the performance of the numerical scheme. [ABSTRACT FROM AUTHOR]

The antiplane deformation of a wedge made of a functionally graded material (FGM) with finite radius has been investigated analytically in the present article. In relation to the boundary conditions imposed on the arc portion of the wedge, displacement or traction, two problems have been studied. In each of the problems three various kinds of boundary conditions (tractiondisplacement, displacement-displacement and traction-traction) have been applied to the radial edges of the wedge. The governing differential equations have been solved by employing finite Fourier transforms and Green's function method. The closed form solutions for stress and displacement distribution have been achieved for the whole domain. Explicit relations have been extracted for the order of stress singularity in all cases. These relations indicated the dependence of the order of stress singularity on the boundary conditions, material property and wedge angle. In fact, despite of an isotropic wedge, for which the order of stress singularity depends only the geometry of the wedge, in an FG wedge the order of stress singularity depends both the geometry as well as the material property. [ABSTRACT FROM AUTHOR]

We present an efficient Green's function based analytical method for forward but particularly also for the inverse modeling of light scattering by quasi-periodic and aperiodic surface nanostructures. In the forward modeling, good agreement over an important texture amplitude range is achieved between the developed formalism and exact rigorous calculations on the one hand and angle resolved light scattering measurements of complex quasi-periodic SiO2-Au nanopatterned interfaces on the other hand. Exploiting our formalism, we demonstrate for the first time how the inverse problem of quasi-periodic surface textures for a desired multiresonant absorption response can be expressed in terms of coupled systems of multivariate polynomial equations of the height profile's Fourier amplitudes. A good estimate of the required surface profile can thus be obtained in a computationally cheap manner via solving the multivariate polynomial equations. In principle, the inverse modeling formalism introduced here can be implemented in conjunction with any scattering model that provides expressions of the coupling coefficients between different modes in terms of the surface texture height profile. [ABSTRACT FROM AUTHOR]

By using the Green's function method, we have investigated spin Hall conductance (SHC) in a four terminal quantum spin Hall insulator. The results show that the intrinsic spin orbit coupling in a HgTe/CdTe hetero-structure interface naturally leads to separate probability distributions for the two spins in coordinate space, which leads to the spin Hall effect in our proposed device. We also find that the SHC of bulk states exhibits an oscillatory behavior as a function of the device width and persists at a broad device width. In addition, we calculate the effects of disorder on the SHC of the bulk states of non-trivial and trivial topology. The results indicate that the spin up and spin down conductances show different degrees of suppression by disorder; thus the SHC could be significantly enhanced by the disorder. This kind of property has a great value to practical applications. [ABSTRACT FROM AUTHOR]

• The static response of a generally restrained stress-driven nonlocal Euler-Bernoulli nanobeam (GRNB), subjected to arbitrary loads is theoretically and numerically investigated. • It is shown that the gradient elasticity theory is a stress-driven nonlocal model with unsymmetrical exponential kernel normalized along the finite beam. • The stress-driven nonlocal and the strain gradient elasticity theories are ruled by a similar 6th order differential equation, with distinct higher-order boundary conditions. • The Green's function method is handled to formulate the problem in a systematic integral approach. • Both theories give very close results and are associated to a stiffening effect of the small-scale effect. This paper focuses on the static analysis of a generally restrained Euler-Bernoulli nanobeam (GRNB), which is subjected to arbitrary distributed or concentrated loads. The small length scale effect is introduced through a strongly nonlocal integral elastic law called the stress-driven nonlocal Euler-Bernoulli beam model, and a weakly nonlocal elastic law called the strain-gradient Euler-Bernoulli beam model. The stress-driven nonlocal theory and the strain gradient elasticity theory are both employed, to formulate the differential equation governing the Euler-Bernoulli nanobeams. Both theories are ruled by a similar 6th order differential equation, with distinct higher-order boundary conditions. The paper demonstrates that the strain gradient theory can be also reformulated as a stress-driven nonlocal theory with unsymmetrical kernel, in contrast to the initial stress-driven nonlocal theory which has symmetrical exponential kernel. Additionally, some theoretical backgrounds about both nonlocal elasticity theorems are provided. Exact solutions for both nanobeam models are presented, using Green's function method. The parameters of the Green's functions are derived for each beam theory, and then utilized to establish the static displacement function of both nanobeam theories. Exact solutions are provided for the bending of both nanobeams with various end boundary conditions, that account for the small length scale effects. The paper also compares the responses of the stress-driven Euler-Bernoulli beam and the strain gradient Euler-Bernoulli beam, showing close deflections and similar effects of increasing the nonlocal parameter on beam stiffness. A major difference highlighted between the two theories is that the strain gradient theory predicts a homogeneous strain state for homogeneous stress state, as opposed to the stress-driven theory. [ABSTRACT FROM AUTHOR]

A novel, robust, and computationally efficient approach is proposed for the evaluation of the Michalski–Zheng’s mixed-potential Green’s functions of general shielded layered media. A high-order variant of the spectral differential equation approximation method (SDEAM) is used to cast the spectra of Green’s functions’ components into pole-residue forms. The latter allows for closed-form evaluation of the Sommerfeld integrals (SIs) defining spatial components of the mixed-potential Green’s functions required by Method-of-Moment (MoM) discretization of different types of integral equations. The method produces Green’s functions for all elevations of interest at a fixed cost, making it substantially faster than popular alternative approaches. Numerical results validate the developed high-order SDEAM scheme and demonstrate its extended range of validity compared to the discrete complex image method (DCIM). [ABSTRACT FROM AUTHOR]

In this paper, a recently introduced method has been implemented for the forced vibration analysis of multi-span Timoshenko beams subjected to a wide range of external loads. To this end, the time-weighted residual approach has been developed for the vibration analysis of Timoshenko beams for the first time. Through a simplifying assumption, the governing system of the differential equations is firstly converted to two decoupled equations in terms of the vertical displacement and the rotation of the beam section. Next, the displacement and the rotation fields are considered as a series of exponential basis functions. Storing the information of the solution at each time step on the coefficients of these series plays a vital role in the proposed method. With such a feature, the solution advances in time through a set of recursive relations. The innovative concept of source functions has also been introduced and employed to direct both shear and flexural waves, tailored to the vertical displacement and the rotation of the beam section, toward the beam's ends. A new dynamic index is also proposed to investigate the validity of Bernoulli assumption for different ranges of the moving object's velocities. Compared with the results of some available common methods, the accuracy and efficiency of the proposed method have been appraised in the solution of five sample problems of single- and multi-span beams subjected to stationary or moving load/mass, including an overhead crane-suspended payload system. [ABSTRACT FROM AUTHOR]

The seepage law for asymmetric fractures can be solved by the Green’s function method. According to the basic seepage theory, the point source mathematical model for asymmetric fractures was established. The dimensionless point source mathematical model differential equation in the Laplacian space was obtained through the dimensionless transformation and the Laplacian transformation. By means of the unknown Green’s function combined with the point source differential equation, and in view of the homogeneous boundary conditions for the point source differential equation and the characteristics of the point source differential equation, a general construction method for Green’s function was given to meet the homogeneous boundary conditions for the point source differential equation and the solution of the unknown objective function. According to the symmetry and continuity of spatial Green’s function, the Green form of the asymmetric fracture point source model was obtained. Finally, through the seepage mathematical model for the asymmetric-fracture vertical well, it was verified that the 2 forms of Green’s function are consistent with the results calculated in references and with the commercial well test analysis software Saphir. [ABSTRACT FROM AUTHOR]

The present work is devoted to the buckling study of non-homogeneous fixedfixed beams with intermediate spring support. The stability issue of these beams leads to three-point boundary value problems. If the Green functions of these boundary value problems are known, the differential equations of the stability problems that contain the critical load sought can be turned into eigenvalue problems given by homogeneous Fredholm integral equations. The kernel function of these equations can be calculated from the associated Green functions. The eigenvalue issues can be reduced to algebraic eigenvalue problems, which are subsequently solvable numerically with the use of an efficient algorithm from the boundary element method. Within this article, the critical load findings of these beams are compared to those obtained using commercial finite element software, and the results are in excellent correlation. [ABSTRACT FROM AUTHOR]

In this paper, the existence and the Ulam-Hyers stability of solutions for the implicit secondorder differential equations are investigated via fractional-orders integral boundary conditions by direct application of the Banach contraction principle. Finally, we present some particular cases and two examples to illustrate our results. [ABSTRACT FROM AUTHOR]

We develop a new self-consistent model for simulation of the electron–cyclotron maser interaction in cylindrical structures, where expansion of the fields in transverse eigenmodes cannot be directly applied. Instead of solving the nonhomogeneous equation for the fields as a differential equation, a different approach is followed. First, the Green's function for elementary azimuthal and radial RF current sources is analytically derived by expanding the fields in longitudinal modes. Then, the total generated field is calculated by representing the perturbed electron beam as a sum of elementary RF current sources along the axis with amplitude coefficients that are found from the kinematic quantities of the electrons. The self-consistent stationary solution is found by solving the equations of motion along with the field equation in an iterative procedure. The model is useful for the full-wave simulation of lossy structures, which are frequently found in gyro-devices, such as ceramic-loaded interaction circuits of gyro-traveling-wave tubes and beam tunnels of gyrotron oscillators. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the uniqueness and existence of the solution to a nonlinear coupled system of (FODEs) with p-Laplacian operator under integral boundary conditions (IBCs). Our problem is based on Caputo fractional derivative of orders σ, λ, where k - 1 ≤ σ, λ < k, k ≥ 3. For these aims, the nonlinear coupled system will be converted into an equivalent integral equations system by the help of Green function. After that, we use Leray-Schauder's and topological degree theorems to prove the existence and uniqueness of the solution. Further, we study certain conditions for the Hyers-Ulam stability of the solution to the suggested problem. We give a suitable and illustrative example as an application of the results. [ABSTRACT FROM AUTHOR]

When cracks start to surface in the surrounding areas of the structure, they create a local softness zone and influences on the dynamic response of the structure. The beams are more susceptible to shear and flexural cracks because of being subjected to shear and bending stress. In this study, the dynamic response of the single-span and multi-span damped beam under moving load with multi-crack and elastic boundary condition is studied based on Timoshenko's theory. The Green's function method is used to calculate the dynamic response of the cracked beam. In addition, the Green's function method provides a solution for the differential equations. Moreover, the effects of the crack on the essential characteristics of the multi-span beams, especially the natural frequencies, are investigated. In this study, crack by itself is modeled in different situations and its effect on the behavior of the beam is analyzed. Also, the elastically restrained beam is modeled and its effect on the behavior of the beam is assessed. Because of the fact that the Euler–Bernoulli theory is also used in most beams, in this study, the results of the numerical examples are compared with the Euler–Bernoulli theory. Several examples are analyzed for a better understanding of the Timoshenko cracked beam. [ABSTRACT FROM AUTHOR]

In this paper, a coupled system of m (≥ 2) second-order singularly perturbed differential equations of reaction–diffusion type with discontinuous source term subject to Dirichlet boundary conditions is studied, where the diffusive term of each equation is being multiplied by the small perturbation parameters having different magnitudes and coupled through their reactive term. A discontinuity in the source term causes the appearance of interior layers on either side of the point of discontinuity in the continuous solution in addition to the boundary layer at the end points of the domain. Unlike the case of a single equation, the considered system does not obey the maximum principle. To construct a numerical method, a classical finite difference scheme is defined in conjunction with a piecewise-uniform Shishkin mesh and a graded Bakhvalov mesh. Based on Green's function theory, it has been proved that the proposed numerical scheme leads to an almost second-order parameter-uniform convergence for the Shishkin mesh and second-order parameter-uniform convergence for the Bakhvalov mesh. Numerical experiments are presented to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

The purpose of the present article is to study weak solutions of viscous conservation laws in physics. We are interested in the well‐posedness theory and the propagation of singularity in the weak solutions for the initial value problem. Our approach is to convert the differential equations into integral equations on the level of weak solutions. This depends on exact analysis of the associated linear equations and their Green's functions. We carry out our approach for the Navier‐Stokes equations in gas dynamics. Local in time as well as time‐asymptotic behaviors of weak solutions, and the continuous dependence of the solutions on their initial data are established. © 2022 Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

This paper establishes the existence of positive solutions for 3nth order differential equations with p-Laplacian operator ... [ABSTRACT FROM AUTHOR]

We revisit the convergence of loop-erased random walk, LERW, to SLE2 when the curves are parametrized by capacity. We construct a Markovian coupling of the driving processes and Loewner chains for the chordal version of LERW and chordal SLE2 based on the Green's function for LERW as martingale observable and using an elementary discrete-time Loewner "difference" equation. We keep track of error terms and obtain power-law decay. This coupling is different than the ones previously considered in this context, e.g., in that each of the processes has the domain Markov property at mesoscopic capacity time increments, given the sigma algebra of the coupling. At the end of the paper we discuss in some detail a version of Skorokhod embedding. Our recent work on the convergence of LERW parametrized by length to SLE2 parameterized by Minkowski content uses specific features of the coupling constructed here. [ABSTRACT FROM AUTHOR]

A new local boundary value problem for an equation parabolichyperbolic type with a fractional-order operator in the sense of Gerasimov-Caputo is formulated. An equation of the first kind, that is, the line of change of type is not a characteristic of the equation. This problem is analogous to the well-known Tricomi problem. In this paper, we prove the existence and uniqueness of a solution to the problem posed in the sense of a classical and strong solution. [ABSTRACT FROM AUTHOR]

This paper deals with designing a control force to create nodal point(s) having zero displacement and/or zero slope at selected locations in a vibrating beam structure excited by multiple harmonic forces. It is shown that the steady state vibrations at desired points can be eliminated using applied control forces. The control forces design method is implemented using dynamic Green’s functions that transform the equations of motion from differential to algebraic equations, in which the resulting solution is analytic and exact. The control problem is greatly simplified by utilizing the superposition principle that leads to calculating the control forces to create node(s) for each excitation frequency independently. The calculated control forces can be realized using passive elements such as masses and springs connected to the beam having reaction forces equal to the calculated control forces. The effectiveness of the proposed method is demonstrated on various cases using numerical examples. Through examples, it was shown that creating node(s) with zero deflection, as well as zero slope, not only results in isolated stationary points, but also suppresses the vibrations along a wide region of the beam. [ABSTRACT FROM AUTHOR]

We present a systematic comparison of the results from our non-equilibrium Green's function formalism with a large number of AlGaAs-GaAs terahertz quantum cascade lasers previously published in the literature. Employing identical material and simulation parameters for all samples, we observe that the discrepancies between measured and calculated peak currents are similar for samples from a given group. This suggests that the differences between experiment and theory are partly due to a lacking reproducibility for devices fabricated at different laboratories. Varying the interface roughness height for different devices, we find that the peak current under lasing operation hardly changes, so that differences in interface quality appear not to be the sole reason for the lacking reproducibility. [ABSTRACT FROM AUTHOR]

In this literature, we study the existence and stability of the solution of the boundary value problem of fractional differential equations with Φp-Laplacian operator. Our problem is based on Caputo fractional derivative of orders σ, ϵ, where k − 1 < σ, ϵ ≤ k, and k ≥ 3. By using the Schauder fixed point theory and properties of the Green function, some conditions are established which show the criterion of the existence and non-existence solution for the proposed problem. We also investigate some adequate conditions for the Hyers-Ulam stability of the solution. Illustrated examples are given as an application of our result. [ABSTRACT FROM AUTHOR]

The local lattice distortion (LLD) effects in the disordered alloys with the large atomic-size misfit between the constituent elements are well known to be essential for reproducing theoretically the observed phase diagrams including order-disorder critical temperatures, although the theoretical approach from first principles remains a long-standing problem. We propose the Kanzaki model combined with the full potential (FP) Korringa-Kohn-Rostoker (KKR) Green's function method, as an approximation for the direct FPKKR calculations, which is based on the harmonic approximation of the atomic displacements and enables us to study the long-ranged LLD effects in the disordered alloys. We first show that the present Kanzaki model may reproduce very accurately the LLD energies obtained by the direct FPKKR calculations with the restriction on the displacement of only the 1st-nearest neighboring atoms around a single impurity X in Al, although the discrepancy increases with the atomic-size misfit between Al and X atoms. Second we clarify the fundamental features of the LLD energies (over -10000 atoms) around a single impurity X (= H-Sn) in Al, corresponding to the 1-body part in our real space cluster expansion for the LLD energy in the Al-rich AlX disordered alloy; the 1-body LLD energy becomes larger and more long-ranged with the atomic-size misfit between Al and X elements and also with the sp-d interaction of Al-X (X = d element). [ABSTRACT FROM AUTHOR]

We investigate the nonexistence of positive blow up boundary solutions to Δα/2 u = uγ in the unit ball of Rd. [ABSTRACT FROM AUTHOR]