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Abstract We explored nonlinear effects within the context of tsunami waveform inversion, wherein Green's functions were linearly superimposed to estimate earthquake slips. We focused on these effects while developing a source model for the 2003 Tokachi–Oki earthquake off Hokkaido, Japan. A source model for this earthquake was developed based on linear tsunami waveform inversion using Green’s functions and tsunami waveforms observed at tide gauge stations. Subsequently, tsunami waveforms from the source were simulated at the stations using nonlinear long-wave theory and compared with those estimated by inversion. The comparisons demonstrated that the waveforms had a non-negligible discrepancy that was attributed to advection effects, even for the primary wave used in the inversion at the two stations. This result strongly suggests that advection effects should be considered in the source modeling of the 2003 earthquake based on tsunami waveforms observed by tide gauges. Based on these results, a new tsunami waveform inversion technique that incorporates linearly approximated advection effects and maintain the framework of linear tsunami waveform inversion using Green’s functions is proposed and applied. The proposed method successfully mimicked the advection effects during the 2003 tsunami, reproduced better tsunami waveforms, and developed a source model for the 2003 earthquake using these effects. The peak slip amount and seismic moment were greater in the source model with advection effects than those without the effects. This finding suggests that the values in the source models developed for other earthquake events without considering these effects may have been underestimated. Graphical abstract

Abstract Carbon dioxide (CO2) emissions affect local temperature; quantifying that local response is important for learning about the earth system, the impacts of mitigation, and adaptation needs. We assume the climate system can be represented as a time‐dependent linear system, diagnosing Green's Functions for the spatial temperature response to CO2 emissions based on CMIP6 earth system models. This allows us to emulate the linear component of the temperature response to CO2. This approach is sufficient to capture the spatial temperature response of CMIP6 experiments within one standard deviation of the multimodel spread across most regions, though accuracy is lower in the Southern Ocean and the Arctic. Our approach reveals where nonlinear feedbacks are important in current CMIP6 models, and where the local system response is well represented by a time‐dependent linear differential operator. It incorporates emissions path dependency and may be useful for evaluating large ensembles of emission scenarios.

The discretization of initial and boundary value problems and their existence behaviors are of great significance in various fields. This paper explores the existence of a class of self-adjoint delta fractional difference equations. The study begins by demonstrating the uniqueness of an initial value problem of delta Riemann–Liouville fractional operator type. Based on this result, the uniqueness of the self-adjoint equation will be examined and determined. Next, we define the Cauchy function based on the delta Riemann–Liouville fractional differences. Accordingly, the solution of the self-adjoint equation will be investigated according to the delta Cauchy function. Furthermore, the research investigates the uniqueness of the self-adjoint equation including the component of Green’s functions of and examines how this equation has only a trivial solution. To validate the theoretical analysis, specific examples are conducted to support and verify our results

The present paper is dedicated to the examination of maximum and minimum results based on Green’s functions via delta fractional differences for a class of fractional boundary problems. For such a purpose, we built the corresponding Green’s functions based on the falling factorial functions. In addition, using the constructed Green’s function, the positivity of the function and its corresponding delta function are presented. We also verified the occurrence of two distinct functions with the same Green’s function. The maximality and minimality of the Green’s function show a good qualitative agreement. Finally, we considered some special examples to explain the obtained results.

In this paper, asymptotic formulae for solutions and Green's function of a boundary value problem are investigated when the equation and the boundary conditions contain a spectral parameter.

Abstract The atmospheric Green's function method is a technique for modeling the response of the atmosphere to changes in the spatial field of surface temperature. While early studies applied this method to changes in atmospheric circulation, it has also become an important tool to understand changes in radiative feedbacks due to evolving patterns of warming, a phenomenon called the “pattern effect.” To better study this method, this paper presents a protocol for creating atmospheric Green's functions to serve as the basis for a model intercomparison project, GFMIP. The protocol has been developed using a series of sensitivity tests performed with the HadAM3 atmosphere‐only general circulation model, along with existing and new simulations from other models. Our preliminary results have uncovered nonlinearities in the response of the atmosphere to surface temperature changes, including an asymmetrical response to warming versus cooling patch perturbations, and a change in the dependence of the response on the magnitude and size of the patches. These nonlinearities suggest that the pattern effect may depend on the heterogeneity of warming as well as its location. These experiments have also revealed tradeoffs in experimental design between patch size, perturbation strength, and the length of control and patch simulations. The protocol chosen on the basis of these experiments balances scientific utility with the simulation time and setup required by the Green's function approach. Running these simulations will further our understanding of many aspects of atmospheric response, from the pattern effect and radiative feedbacks to changes in circulation, cloudiness, and precipitation.

This paper presents a novel method for analyzing the dynamic response of plane Euler-Bernoulli frames with semi-rigid connections subjected to arbitrary external loads and bending moments. The proposed solution methodology is the Green’s Functions Stiffness Method (GFSM) in the frequency domain. The GFSM is a mesh reduction method closely related with the Finite Element Method (FEM) sharing with it key components such as shape functions, fixed end forces, and stiffness matrices. By capitalizing on the strengths of both FEM and Green’s Functions, the GFSM facilitates the derivation of closed-form solutions for structural analysis. The formulation is initially established in the frequency domain and is later transformed into the time domain using the fast Fourier transform algorithm. To illustrate the applicability of the method, an example involving a one-bay, one-storey plane frame with semi-rigid connections is presented.

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.

This paper surveys a new class of electromagnetic materials. The emphasis is on the bianisotropic medium; a complex material which has never been applied in antenna design. In this context, we investigate the behavior of the magnetoelectric elements of the gyrotropic reciprocal bianisotropic medium used as a grounded substrate by studying their effects on the input impedance and mutual coupling of a printed dipole antenna. Numerical integrative equation calculations are carried out using the spectral method of moments (SMoM) based on the analytical derivation of the appropriate spectral Green’s functions for the considered dipole configuration. The obtained results reveal a change in the sign of the real part of the input impedance. This is mainly due to the complex property of the considered medium. The medium has also minimized the input impedance ∼4 times (1kΩ); this can help in device matching. This medium has shown a significant impact on the input impedance as well as the antenna array coupling. The design miniaturization possibility is an important result of this research in the design of antenna arrays.

A polar-symmetric elastic diffusion problem is considered for an orthotropic multicomponent homogeneous cylinder under uniformly distributed radial unsteady volumetric perturbations. Coupled elastic diffusion equations in a cylindrical coordinate system is used as a mathematical model. The model takes into account a relaxation of diffusion effects implying finite propagation speed of diffusion perturbations. The solution of the problem is obtained in the integral convolution form of Green's functions with functions specifying volumetric perturbations. The integral Laplace transform in time and the expansion into the Fourier series by the special Bessel functions are used to find the Green's functions. The theory of residues and tables of operational calculus are used for inverse Laplace transform. A calculus example based on a three-component material, in which two components are independent, is considered. The study of the mechanical and diffusion fields interaction in a solid orthotropic cylinder is carried out.

A detailed Volume Integral Equation (VIE) formulation for planar plasmonic nano structures with finite thickness in flat multi-layers medium is presented. The boundary condition along the localized metallic objects is expressed in terms of the unknown polarization current flowing through these objects in the form of an integral equation, which is solved using the Method of Moments (MoM). The Green's functions associated with a layered medium of practical importance are expressed in the spectral domain. The corresponding spatial domain Green's functions are obtained using the Discrete Complex Images Method (DCIM). Special treatment for the spectral function's asymptote at high spectral values is performed. The presented formulation is applied on different plasmonic structures immersed inside layered media. The structures include nano-rod and nano-patch excited by an incident plane wave. In addition, a simple band-stop filter based on quarter-wavelength stubs is considered. This filter is fed with a couple of plasmonic transmission lines. The obtained current distributions and S-parameters are compared with those obtained using a commercial full-wave electromagnetic simulator, namely CST Microwave Studio. A very good agreement is observed. The proposed integral equation formulation enjoys high degree of stability, numerical efficiency, and accuracy.

This manuscript was originally built to establish some coupled common fixed point results for rational contractive mapping in the framework of b-metric spaces. Thereafter, the existence and uniqueness of the boundary value problem for a singular coupled fractional differential equation of order ν via coupled fixed point techniques are discussed. At the last, some supportive examples to illustrate the theoretical results are presented.

Abstract This paper is devoted to prove the existence of positive solutions of a second order differential equation with a nonhomogeneous Dirichlet conditions given by a parameter dependence integral. The studied problem is a nonlocal perturbation of the Dirichlet conditions by considering a homogeneous Dirichlet-type condition at one extreme of the interval and an integral operator on the other one. We obtain the expression of the Green’s function related to the linear part of the equation and characterize its constant sign. Such a property will be fundamental to deduce the existence of solutions of the nonlinear problem. The results hold from fixed point theory applied to related operators defined on suitable cones.

In this paper, an analytical method is developed for the axisymmetric dynamic response of a finite thickness liquid layer overlying a transversely isotropic porous solid half-space due to body waves. Potential functions and integral transforms are used together to handle the equations of wave motion in two media. The time-harmonic excitation with axisymmetric shape is assumed to be distributed in the interface of liquid and porous media. Green’s functions of stress and displacement are derived as closed-form integral expressions. Demonstration of the effect of the liquid thickness, degree of material anisotropy, and frequency of excitation on the dynamic response is considered here. Numerical results for a uniform distributed disk load are comprised with the existing elastic and poroelastic solutions to illustrate the quality of the method. The results of the current paper can be used in analysis and modelling the rigid or flexible foundations in marine structures.

This paper deals with the dynamic characteristics for bending vibrations of Euler-Bernoulli type nanobeams taking into account the scale effects via the nonlocal motion equations. An integral method, based on the use of Green’s functions, has been used in order to obtain the corresponding eigenvalue problem. The proposed integral approach is an approximate matrix method. Effects of different boundary conditions and of an elastic foundation have been also included. The presented numerical examples show good agreement when compared to results from literature. The proposed method can be used in the case of nanodevices analysis modeled as beams (MEMS, NEMS).

This paper deals with the mathematical expressions called Sommerfeld integrals. Introduced by A. Sommerfeld in 1909, they are mathematically equivalent to inverse Hankel transforms. The original historical goal of these integrals was to provide an accurate mathematical description of the electromagnetic phenomena involved in long-distance wireless radio and telegraphy. However, their scope was quickly enlarged thanks to the so-called spectral-domain stratified theory, and now they are ubiquitous in the mathematical models associated to many electromagnetic technologies, ranging from EMC lightning modelization and ground penetrating radar to optical and plasmonic integrated devices and going through the familiar microwave and millimeter-wave planar structures using printed circuit technology. In all these areas, Sommerfeld integrals can provide direct evaluations of the involved electromagnetic fields or they can be used as Green's functions in the frame of integral equation formulations. Other disciplines involving stratified media, like seismology and geological prospection, also benefit from these integrals. After discussing the most canonical Sommerfeld integral, appearing in the so-called Sommerfeld identity, this paper reviews three classical structures, namely, the original Sommerfeld problem involving two semi-infinite media, and the microstrip and stripline geometries. It is shown that Sommerfeld integrals provide a unifying treatment of these three problems and that their mathematical features have a direct translation in terms of the physical properties exhibited by the electromagnetic fields that can exist in them.

Abstract In this paper, first we present some interesting identities associated with Green’s functions and Fink’s identity, and further we present some interesting inequalities for r-convex functions. We also present refinements of some Hardy–Littlewood–Pólya type inequalities and give an application to the Shannon entropy. Furthermore, we use the Čebyšev functional and Grüss type inequalities and present the bounds for the remainder in the obtained identities. Finally, we use the obtained identities together with Hölder’s inequality for integrals and present Ostrowski type inequalities.

In this paper we investigate the existence of positive solutions of fourth-order non autonomous differential equations with integral boundary conditions, the nonlinearity is a continuous function that depends on the spatial variable and its the second-order derivative. The approach relies an extension of Krasnoselskii's fixed point theorem in a cone. Some examples are given to illustrate our results.

Thermoacoustic imaging is a crossbred approach taking advantages of electromagnetic and ultrasound disciplines, together. A significant number of current medical imaging strategies are based on reconstruction of source distribution from information collected by sensors over a surface covering the region to be imaged. Reconstruction in thermoacoustic imaging depends on the inverse solution of thermoacoustic wave equation. Homogeneous assumption of tissue to be imaged results in degradation of image quality. In our paper, inverse solution of the thermoacoustic wave equation using layered tissue model consisting of concentric annular layers on a cylindrical cross-section is investigated for cross-sectional thermoacustic imaging of breast and brain. By using Green’s functions and surface integral methods we derive an exact analytic inverse solution of thermoacoustic wave equation in frequency domain. Our inverse solution is an extension of conventional solution to layered cylindrical structures. By carrying out simulations, using numerical test phantoms consisting of thermoacoustic sources distributed in the layered model, our layered medium assumption solution was tested and benchmarked with conventional solutions based on homogeneous medium assumption in frequency domain. In thermoacoustic image reconstruction, where the medium is assumed as homogeneous medium, the solution of nonhomogeneous thermoacoustic wave equation results in geometrical distortions, artifacts and reduced image resolution due to inconvenient medium assumptions.

The propagator of a gauge boson, like the massless photon or the massive vector bosons W± and Z of the electroweak theory, can be derived in two different ways, namely via Green’s functions (semi-classical approach) or via the vacuum expectation value of the time-ordered product of the field operators (field theoretical approach). Comparing the semi-classical with the field theoretical approach, the central tensorial object can be defined as the gauge boson projector, directly related to the completeness relation for the complete set of polarisation four-vectors. In this paper we explain the relation for this projector to different cases of the Rξ gauge and explain why the unitary gauge is the default gauge for massive gauge bosons.

This paper reviews some existing studies and numerical methods used for flow induced transverse vibrations analysis of flexible pipes. An integral method, based on the use of Green’s functions, already used for different straight beam dynamic analysis is adapted for the proposed subject. This approximate method leads to a matrix formulation and to an eigenvalue problem for free vibration analysis. The presented approach is able to estimate also the critical fluid velocities. Effects of boundary conditions and of elastic foundation characteristics, Coriolis terms and of other parameters on dynamic behavior of a pipe, can be included. Some numerical examples are also presented for comparisons with results obtained by FEM or with other data from literature. They show good agreement.

Abstract By using new Green’s functions and Lidstone interpolating polynomial, some new generalizations of Levinson type inequalities for (2p+1) $(2p+1)$-convex functions are obtained. In seek of applications of our results to information theory, new generalizations based on f-divergence estimates are also proven. Moreover, some inequalities for Shannon entropies are deduced as well.

This paper presents a short review of numerical methods used for lateral vibration analysis in the case of straight Euler and Timoshenko beams. A particular integral method is described with more details. This approximate integral method is based on the use of flexibility influence functions (Green’s functions). It leads to a matrix formulation and to an eigenvalue problem for vibration analysis. The presented approach is able to estimate separately the shear effects and the rotary inertia effects and also the combined effects. Simple numerical examples are also presented for comparisons with analytical and finite elements results. The results show good agreement.

Abstract In this article, we construct new Lyapunov-type inequalities for odd order boundary value problems. The aim of this article is to find the maximum of Green’s function |G2n+1(x,s)| $\vert G_{2n+1} ( x,s ) \vert $ corresponding to two-point boundary value problems. To the best of our knowledge, there is no paper dealing with Lyapunov-type inequalities for odd order boundary value problems by bounding the Green’s function of the same problem. In addition, some applications of the obtained inequalities are also given.

The paper studies the dynamic description of uniaxial and biaxial ferrimagnetics with spin s=1/2 in alternative external field. The nonlinear dynamic equations with sources are obtained, on basis on which low-frequency asymptotics of two-time Green functions in the uniaxial and biaxial cases of the ferrimagnet are obtained. Energy models are constructed that are specific functions of Casimir invariants of the algebra of Poisson brackets for magnetic degrees of freedom. On their basis, the question of the stable magnetic states has been solved for the considered systems. These equations were linearized, an explicit form of the collective excitations spectra was found, and their character was analyzed. The article studies the uniaxial case of a ferrimagnet, as well as biaxial cases of an antiferromagnet, easy-axis and easy-plane ferrimagnets. It is shown that for a uniaxial antiferromagnet the spectrum of magnetic excitations has a Goldstone character. For biaxial ferrimagnetic materials, it was found that the spectrum has either a quadratic character or a more complex dependence on the wave vector. It is shown that in the uniaxial case of an antiferromagnet the Green function of the type Gsα,sβ(k,0), Gsα,nβ(k,0) and Gsα,sβ(0,ω) have regular asymptotic behavior, and the Green function of type Gnα,nβ(k,0)≈1/k2 and Gsα,nβ(0,ω)≈1/ω, Gnα,nβ(0,ω)≈1/ω2 have a pole feature in the wave vector and frequency. Biaxial ferrimagnetic states have another type of the features of low-frequency asymptotics of the Green's functions. In the case of a ferrimagnet, the “easy-axis” of the asymptotic behavior of the Green functions Gsα,sβ(0,ω), Gsα,nβ(0,ω), Gnα,nβ(0,ω), Gsα,sβ(k,0), Gsα,nβ(k,0), Gnα,nβ(k,0) have a pole character. For the case of the “easy-plane” type ferrimagnet, the asymptotics of the Green functions Gsα,nβ(0,ω), Gnα,nβ(0,ω), Gsα,nβ(k,0), Gnα,nβ(k,0), have a pole character, and the Green function Gsα,sβ(k,ω) contains both the pole component and the regular part. A comparative analysis of the low-frequency asymptotics of Green functions shows that the nature of magnetic anisotropy significantly effects the structure of low-frequency asymptotics for uniaxial and biaxial cases of ferrimagnet. Separately, we note the non-Bogolyubov character of the Green function asymptotics for ferrimagnet with biaxial anisotropy Gnα,nβ(k,0)≈1/k4.

This work deals with structural stability analysis of straight beams, having different boundary conditions. The particular case of the beam on elastic foundation is also analyzed. The method of analysis is an approximate integral one, using structural flexibility functions (Green’s functions). The differential equations governing the Euler buckling of such beams are put in integral form. This approach is a matrix one leading to an eigenvalues problem in the case of stability analysis. Different numerical examples concerning the calculation of the critical buckling loads are presented in comparison with available data from literature. The obtained results show good agreement from engineering point of view.

This paper presents a new method of describing the electronic spectrum and electrical conductivity of disordered crystals based on the Hamiltonian of electrons and phonons. Electronic states of a system are described by the tight-binding model. Expressions for Green’s functions and electrical conductivity are derived using the diagram method. Equations are obtained for the vertex parts of the mass operators of the electron–electron and electron–phonon interactions. A system of exact equations is obtained for the spectrum of elementary excitations in a crystal. This makes it possible to perform numerical calculations of the energy spectrum and to predict the properties of the system with a predetermined accuracy. In contrast to other approaches, in which electron correlations are taken into account only in the limiting cases of an infinitely large and infinitesimal electron density, in this method, electron correlations are described in the general case of an arbitrary density. The cluster expansion is obtained for the density of states and electrical conductivity of disordered systems. We show that the contribution of the electron scattering processes to clusters is decreasing, along with increasing the number of sites in the cluster, which depends on a small parameter.

Abstract We study in this paper a wave–Schrödinger transmission system for its stability. By analyzing carefully Green’s functions for the infinitesimal generator of the semigroup associated with the system under consideration, we obtain a useful resolvent estimate on this generator which can be applied to derive the decaying property. Our study is inspired by L. Lu & J.-M. Wang [Appl. Math. Lett., 54:7–14, 2016] whose energy decay result is improved upon in our paper. Our method, different from the one used in the previous reference, can be adapted to study stability problems for other 1-D transmission systems.

The paper deals with the problem of determining the stress-strain state of a thermoelastic multicomponent medium with plane boundaries (layer and half-space) taking into account the presence of diffusion fluxes in each medium component. The effect of changes in the concentration and temperature on the stress-strain state of the medium has been studied with the help of a locally equilibrium model of thermoelastic diffusion, which includes the coupled system of equations of motion, heat transfer, and mass transfer. The solution has been found using the Laplace transform, as well as using the Fourier expansion for the layer and the sine-cosine transform for the half-space. The surface Green's functions have been expressed and analyzed. Test calculation has been performed.

We investigate the existence of positive solutions for a class of fractional differential equations of arbitrary order δ>2, subject to boundary conditions that include an integral operator of the fractional type. The consideration of this type of boundary conditions allows us to consider heterogeneity on the dependence specified by the restriction added to the equation as a relevant issue for applications. An existence result is obtained for the sublinear and superlinear case by using the Guo–Krasnosel’skii fixed point theorem through the definition of adequate conical shells that allow us to localize the solution. As additional tools in our procedure, we obtain the explicit expression of Green’s function associated to an auxiliary linear fractional boundary value problem, and we study some of its properties, such as the sign and some useful upper and lower estimates. Finally, an example is given to illustrate the results.

Coal-fired power units, now balancing power shortages in the power system, must be characterised by increasingly higher flexibility of operation. This means faster start-ups and the capacity for frequent decreases and increases in the power output. These processes cause large temperature gradients in elements of the power unit and the turbine and lead to an increase in the stress level. At such an operating regime it is impossible to ensure safety based on start-up characteristics only—it becomes necessary to constantly monitor stress levels in critical areas of machinery and equipment elements. The stress level in turbine elements can be monitored on-line using algorithms based on Green’s functions and Duhamel’s integral. This paper presents examples of modifications of stress calculations in turbine valves and casings during start-ups. By modifying basic algorithms, it is possible to take into account the impact of the variability of heat transfer coefficients on the thermal stress level. Additionally, individual Green’s functions and correction factors were determined for specific stages of start-ups. Due to modifications, it is possible to obtain satisfactory agreement with the results obtained from FEM-based calculations for the entire heating process. Equations are also given that enable estimation of values of the heat transfer coefficient in turbine valves. The proposed modification of the algorithm will substantially improve the accuracy of stress modelling in transient states of the turbine operation. On-line stress monitoring will enable an increase in the flexibility of the power unit operation and facilitate operational control, ensuring safety of individual elements at the same time. The stress values calculated in the on-line mode can also be used to estimate fatigue life consumption and forecast the residual lifetime of individual components.

The propagation of unsteady kinematic or electromagnetic perturbations in an isotropic ball given on its surface has been studied. Along with the Maxwell equations and the linearized Ohm's law, we have studied the linear equations of motion for an elastic ball, the right side of which includes the Lorentz force as the body force. The radial and tangential components of the unknown quantities expand into series of the Legendre and Gegenbauer polynomials. The initial-boundary value problem is solved by means of the Laplace transform by time and the expansion of the series coefficients into power series in a small parameter connecting the mechanical and electromagnetic characteristics of the continuum. The expansion in the power series allows to construct a recurrent sequence of boundary value problems with respect to the unknown components of the mechanical and electromagnetic fields. Each individual problem is solved by means of generalized convolutions of Green's functions with the unknown functions corresponding to the preceding terms of the recurrent sequence. The quasi-static analogs have been used as Green's functions for the electromagnetic field. For the mechanical field, the explicit form of Green's bulk functions found using computer algebra and complex analysis methods has been used.

Projectile trajectories calculated under non-standard conditions are considered to be perturbed. The tools utilized for the analysis of perturbed trajectories are weighting factor functions (WFFs) which are a special kind of sensitivity functions. WFFs are used for calculation of meteo ballistic elements μB (ballistic wind wB, virtual temperature τB, pressure pB, density ρB, speed of sound a) as well. An effect of weapon system parameters can be incorporated into calculations through the reference height of trajectory - RHT. RHT are also calculated from WFFs. Methods based on RHT are far more effective than traditional methods that use weighting factors q. We have found that the existing theory of RHT has several shortcomings due to we created an improved theory of generalized RHT which represent a special sensitivity parameters of dynamical systems. Using this theory will improve methods for designing firing tables, fire control systems algorithms, and meteo message generation algorithms.

This article is devoted to the study of the parameter's set where the Green's function related to a general linear nth-order operator, depending on a real parameter, T_n[M], coupled with many different two point boundary value conditions, is of constant sign. This constant sign is equivalent to the strongly inverse positive (negative) character of the related operator on suitable spaces related to the boundary conditions.

In this article, we establish new Lyapunov-type inequalities for third-order linear differential equations $$ y'''+q( t) y=0, $$ under the three-point boundary conditions $$ y( a) =y( b) =y( c) =0 $$ and $$ y( a) =y''( d) =y( b) =0 $$ by bounding Green's functions G(t,s) corresponding to appropriate boundary conditions. Thus, we obtain the best constants of Lyapunov-type inequalities for three-point boundary value problems for third-order linear differential equations in the literature.

Instead of usual approach, applying displacement-displacement Green’s functions, the momentum-momentum Green’s functions will be used to calculate the diffusion tensor. These functions enter into Kubo’s formula defining diffusion properties of the system. Calculation of the diffusion tensor requires solving of the system of difference equations. It is shown that the elements of the diffusion tensor express discrete behaviour of the dispersion law of elementary excitations and, more important - that they are temperature independent.

We consider the following second order differential equation with delay $$ \begin{cases} \begin{split} (Lx)(t)&\equiv{x''(t)+\sum_{j=1}^p {a_{j}(t)x'(t-\tau_{j}(t))}+\sum_{j=1}^p {b_{j}(t)x(t-\theta_{j}(t))}}=f(t), & t\in[0,\omega] \end{split} \\ x(t_k)=\gamma_{k}x(t_k-0), \ x'(t_k)=\delta_{k}x'(t_k-0), \quad k=1,2,\dots,r. \end{cases} $$ In this paper we find sufficient conditions of positivity of Green's functions for this impulsive equation coupled with two-point boundary conditions in the form of theorems about differential inequalities. Choosing the test function in these theorems, we obtain simple sufficient conditions.

Presented paper is dedicated to problems of deformation of the Earth's crust as a response to the surface loading caused by continental waters. The aim of this study was to specify areas particularly vulnerable to studied deformation and to compare calculated and observed displacements. Information of the continental water volume was taken from the WaterGAP Global Hydrological Model. Calculated values of the deformations were verified with the results obtained with programs SPOTL and grat. Vertical deformations were almost 10 times higher than the deformation in the horizontal plane, for which reason later part of the paper focuses on the former. In order to check agreement of the calculated and observed deformation 23 stations of International GNSS Service (IGS) were selected and divided into three groups (inland, near the shoreline and islands). Before comparison outliers and discontinuities were removed from GNSS observations. Modelled and observed signals were centred. The analysed time series of the vertical displacements showed that only for the inland stations it is possible to effectively remove displacements caused by mass transfer in the hydrosphere. For stations located in the coastal regions or islands, it is necessary to consider additional movement effects resulting from indirect ocean tidal loading or atmosphere loading.

Propagators approximated by meromorphic functions with complex conjugated poles are widely used to model the infrared behavior of QCD Green’s functions. In this paper, analytical solutions for two point correlators made out of functions with complex conjugated poles or branch points have been obtained in the Minkowski space for the first time. As a special case the Gribov propagator has been considered as well. The result is different from the naive analytical continuation of the correlator primarily defined in the Euclidean space. It is free of ultraviolet divergences and instead of Lehmann it rather satisfies Gribov integral representation.

Ionization potentials (IPs) and electron affinities (EAs) are important quantities input into most models for calculating the open-circuit voltage (Voc) of organic solar cells. We assess the semi-empirical density-functional tight-binding (DFTB) method with the third-order self-consistent charge (SCC) correction and the 3ob parameter set (the third-order DFTB (DFTB3) organic and biochemistry parameter set) against experiments (for smaller molecules) and against first-principles GW (Green’s function, G, times the screened potential, W) calculations (for larger molecules of interest in organic electronics) for the calculation of IPs and EAs. Since GW calculations are relatively new for molecules of this size, we have also taken care to validate these calculations against experiments. As expected, DFTB is found to behave very much like density-functional theory (DFT), but with some loss of accuracy in predicting IPs and EAs. For small molecules, the best results were found with ΔSCF (Δ self-consistent field) SCC-DFTB calculations for first IPs (good to ± 0.649 eV). When considering several IPs of the same molecule, it is convenient to use the negative of the orbital energies (which we refer to as Koopmans’ theorem (KT) IPs) as an indication of trends. Linear regression analysis shows that KT SCC-DFTB IPs are nearly as accurate as ΔSCF SCC-DFTB eigenvalues (± 0.852 eV for first IPs, but ± 0.706 eV for all of the IPs considered here) for small molecules. For larger molecules, SCC-DFTB was also the ideal choice with IP/EA errors of ± 0.489/0.740 eV from ΔSCF calculations and of ± 0.326/0.458 eV from (KT) orbital energies. Interestingly, the linear least squares fit for the KT IPs of the larger molecules also proves to have good predictive value for the lower energy KT IPs of smaller molecules, with significant deviations appearing only for IPs of 15–20 eV or larger. We believe that this quantitative analysis of errors in SCC-DFTB IPs and EAs may be of interest to other researchers interested in DFTB investigation of large and complex problems, such as those encountered in organic electronics.

Theoretical and analytical thermal and electrical properties are studied through the 2,7-Di([1,1′-biphenyl]-4-yl)-9H-fluorene aromatic system as a prototype of a molecular switch. Variations of the dihedral angles between the two Benzene rings at each end of the molecule have been considered, thus determining the dependence on the structural variation of the molecule when the aromatic system is connected between metal contacts. The molecule is modeled through a Tight-Binding Hamiltonian where—from the analytical process of decimation and using Green’s functions—the probability of transmission (T) is calculated by using the Fisher–Lee relationship. Consequently, the thermal and electrical transport properties such as I − V curves, quantum noise (S), Fano factor (F), electrical conductance (G), thermal conductance ( κ ), Seebeck coefficient (Q), and merit number ( Z T ) are calculated. The available results offer the possibility of designing molecular devices, where the change in conductance or current induced by a stereoelectronic effect on the molecular junctions (within the aromatic system) can produce changes on the insulating–conductive states.

In this review paper, we stress the importance of the higher transcendental Wright functions of the second kind in the framework of Mathematical Physics. We first start with the analytical properties of the classical Wright functions of which we distinguish two kinds. We then justify the relevance of the Wright functions of the second kind as fundamental solutions of the time-fractional diffusion-wave equations. Indeed, we think that this approach is the most accessible point of view for describing non-Gaussian stochastic processes and the transition from sub-diffusion processes to wave propagation. Through the sections of the text and suitable appendices, we plan to address the reader in this pathway towards the applications of the Wright functions of the second kind.

We present a multiscale model based on many-body Green’s functions theory in the GW approximation and the Bethe–Salpeter equation ( GW -BSE) for the simulation of singlet and triplet exciton transport in molecular materials. Dynamics of coupled electron–hole pairs are modeled as a sequence of incoherent tunneling and decay events in a disordered morphology obtained at room temperature from molecular dynamics. The ingredients of the rates associated to the events, i.e. reorganization energies, site energies, lifetimes, and coupling elements, are determined from a combination of GW -BSE and classical polarizable force field techniques. Kinetic Monte Carlo simulations are then employed to evaluate dynamical properties such as the excitonic diffusion tensor and diffusion lengths. Using DCV5T-Me(3,3), a crystalline organic semiconductor, we demonstrate how this multiscale approach provides insight into the fundamental factors driving the transport processes. Comparing the results obtained via different calculation models, we investigate in particular the effects of charge-transfer mediated high exciton coupling and the influence of internal site energy disorder due to conformational variations. We show that a small number of high coupling elements indicative of delocalized exciton states does not impact the overall dynamics perceptively. Molecules with energies in the tail of the excitonic density of states dominate singlet decay, independent of the level of disorder taken into account in the simulation. Overall, our approach yields singlet diffusion lengths on the order of 10 nm as expected for energetically disordered molecular materials.

The paper investigates the dynamics of the thermal field of the ACCC (aluminum conductor composite core) line. The system was heated by solar radiation and current flow. Conductor cooling was modeled using the total heat transfer coefficient as the sum of convective and radiative components. The temperature increase generated by the current is described by a system of parabolic differential equations with an appropriate set of boundary, initial and continuity condition. The mentioned boundary-initial problem was solved by a modified Green’s method, adapted to the layered structure of the system. For this purpose, Green’s functions, as the kernels of integral operators inverse to differential ones, were determined. Aluminum resistivity and heat transfer coefficient change significantly with temperature. For this reason, the solution to the problem is presented in the form of a lower and upper estimation of the heating curve and local time constant. A steady-state current rating was also determined. The results are presented graphically and verified by other methods (power balance and finite element). The physical interpretation of the presented solution is also given.

This work considers a second order impulsive coupled system with full nonlinearities, generalized impulse functions and mixed boundary conditions. This is the first time where such coupled systems are considered with nonlinearities with dependence on both unknown functions and their derivatives, together impulsive functions given by more general framework allowing jumps on the both functions and both derivatives. The arguments apply the fixed point theory, Green's functions echnique, L1-Carathéodory functions theory and Schauder's fixed point theorem. An application to the transverse vibration system of elastically coupled double-string is presented in the last section.

The subject of the paper are Green’s functions for the stress intensity factors of modes I, II and III. Green’s functions are defined as a solution to the problem of an elastic, transversely isotropic solid with a penny-shaped or an external crack under general axisymmetric loadings acting along a circumference on the plane parallel to the crack plane. Exact solutions are presented in a closed form for the stress intensity factors under each type of axisymmetric ring forces as fundamental solutions. Numerical examples are employed and conclusions which can be utilized in engineering practice are formulated.

We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic heat equations of the following form: \(D_t^{(\alpha)} u(t, x)=\textit{B}u+u\cdot \dot W^H\), where \(D_t^{(\alpha)}\) is the Caputo fractional derivative of order \(\alpha\in (0,1)\) with respect to the time variable \(t\), \(\textit{B}\) is a second order elliptic operator with respect to the space variable \(x\in\mathbb{R}^d\) and \(\dot W^H\) a time homogeneous fractional Gaussian noise of Hurst parameter \(H=(H_1, \cdots, H_d)\). We obtain conditions satisfied by \(\alpha\) and \(H\), so that the square integrable solution \(u\) exists uniquely.

We propose a graphene nanoribbon-based heterojunction, where a defect-free interface separates two zigzag graphene nanoribbons prepared in opposite antiferromagnetic spin configurations. This heterospin junction is found to allow the redirecting of low-energy electrons from one edge to the other. The basic scattering mechanisms and their relation to the system’s geometry are investigated through a combination of Landauer–Green’s function and the S-matrix and eigen-channel methods within a tight-binding + Hubbard model validated with density functional theory. The findings demonstrate the possibility of using zigzag-edged graphene nanoribbons (zGNRs) in complex networks where current can be transmitted across the entire system, instead of following the shortest paths along connected edges belonging to the same sub-lattice.

We consider the following second order differential equation with delay \[\begin{cases} (Lx)(t)\equiv{x''(t)+\sum_{j=1}^p {b_{j}(t)x(t-\theta_{j}(t))}}=f(t), \quad t\in[0,\omega],\\ x(t_j)=\gamma_{j}x(t_j-0), x'(t_j)=\delta_{j}x'(t_j-0), \quad j=1,2,\ldots,r. \end{cases}\] In this paper we find necessary and sufficient conditions of positivity of Green's functions for this impulsive equation coupled with one or two-point boundary conditions in the form of theorems about differential inequalities. By choosing the test function in these theorems, we obtain simple sufficient conditions. For example, the inequality \(\sum_{i=1}^p{b_i(t)\left(\frac{1}{4}+r\right)}\lt \frac{2}{\omega^2}\) is a basic one, implying negativity of Green's function of two-point problem for this impulsive equation in the case \(0\lt \gamma_i\leq{1}\), \(0\lt \delta_i\leq{1}\) for \(i=1,\ldots ,p\).

In this paper, we prove the existence of solutions of nonlinear boundary value problems of arbitrary even order using the lower and upper solutions method. In particular, we point out the fact that the existence of a pair of lower and upper solutions of a considered problem could imply the existence of solution of another one with different boundary conditions. We consider Neumann, Dirichlet, mixed and periodic boundary conditions.