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Consider the Dirichlet problem for Laplace’s/Poisson’s equation in a bounded simply-connected domain S . The source function q ln | P Q * ¯ | is a fundamental solution (FS), and it can be found in many physical problems. The singularity occurs when the boundary value data affected by q ln | P Q * ¯ | as the source node Q * is located near the boundary Γ ( = ∂ S ) . So far, there is no comprehensive study on this kind of singularity. In this paper, the solution singularity is explored and the reduced convergence rates are derived for the method of particular solutions (MPS) and the method of fundamental solutions (MFS). Classic domains, such as disks, ellipses and polygons, are discussed for analysis and computation. For this new kind of solution singularity, the convergence rates of the MFS and the MPS are very low. The errors caused by numerical integration are critical to the solution accuracy. A new analytic framework for the collocation Trefftz method (CTM) involving numerical integration is established in this paper; this is an advanced development of our previous study [19]. Since the numerical solutions are poor in accuracy, removal techniques are essential in applications. New removal techniques are proposed for a node Q * located near Γ . In this paper, an additional FS as, d 0 ln | P Q 0 ¯ | , is added to the original source nodes in the traditional MFS, and the point charge d 0 ( = q ) and the source node Q 0 are unknowns to be sought by nonlinear solvers (such as the secant method). When the source node Q * is located inside S but near Γ , both simple domains (such as disks, ellipses and squares) and complicated domains (such as amoeba-like domains) are studied. The validity of the new removal techniques is supported by numerical experiments. The removal techniques in this paper may also be applied to solve source identification problems. A comprehensive study has been completed in this paper for the solitary source function q ln | P Q * ¯ | as the source node Q * is located near Γ .

It is shown that the source function on the right-hand side of the Helmholtz equation, which is a solution of a homogeneous boundary value problem for this equation, provides the excitation of an associated wave (oscillation). Since this wave (oscillation) exists only in the presence of a source of the specified type, it is proposed to classify them (wave and oscillation) as associated with the source, and to consider the problem “generating” them as self-consistent, because the excited wave (oscillation) is in synchronism with the source in all wave numbers and energy balance.

This paper is devoted to identify a space-dependent source function in a multiterm time-fractional diffusion equation with nonhomogeneous boundary condition from a part of noisy boundary data. The well-posedness of a weak solution for the corresponding direct problem is proved by the variational method. We firstly investigate the uniqueness of an inverse initial problem by the analytic continuation technique and the Laplace transformation. Then, the uniqueness of the inverse source problem is derived by employing the fractional Duhamel principle. The inverse problem is solved by the Levenberg-Marquardt regularization method, and an approximate source function is found. Numerical examples are provided to show the effectiveness of the proposed method in one- and two-dimensional cases.

An inverse source problem for an n-dimensional heat equation with a time-varying coefficient is investigated. The spatially dependent component of a source function is determined from a measurement...

The parameterization of the nonlinear energy transfer called Discrete Interaction Approximation (DIA) is optimized in the WAVEWATCH III wave model by the criterion of minimizing deviations of model predictions from the data of field measurements. Short fetches are considered for which the wind-input source function had been previously adjusted. The results of the numerical simulation and field experiment for the DIA built-in version and for the DIA with the proposed parameters are compared. The improvement of the reproduction of the main parameters of the wave spectra, significant wave height, and their mean period by the model is shown.

A modified LSM is proposed by introducing an independent micro-rotational inertia, which may help characterize the scale-dependent effect and avoid the Poisson’s ratio limitation of regular triangle lattices in two dimensions. For this method, some factors may affect data pickup and modeling accuracy, but the ‘optimal’ inputs, like stiffness ratio, numerical damping, and micro-rotational inertia, could be obtained from parameter identification by the Dakota toolkit (Adams et al., Tech Rep SAND2010–2183, 2009), when a suitable excitation source function and lattice spacing are set up. By comparing with the modified couple stress theory, we analyze the dispersion relationship of elastic waves for the estimation of the characteristic material length. It shows that this modified LSM may provide an alternative and promising way to investigate the size-dependent wave propagation in elastic media numerically.

This paper discusses local features in the source function that generate in the far zone simultaneously (a) a quadratic decay of the source spectrum and (b) the loss of radiation directivity at high frequencies. A. Gusev drew attention to this problem and suggested that a positive solution can be obtained for earthquake rupture front with a rather complex (“lace”) structure. Below we give a theoretical solution of the problem and show that the front structure can be simple enough, but not smooth.

Line-driven winds of hot, luminous stars are intrinsically unstable due to the line-deshadowing instability (LDI). In non-magnetic hot stars, the LDI leads to the formation of an inhomogeneous wind consisting of small-scale, spatially separated clumps that can have great effects on observational diagnostics. However, for magnetic hot stars the LDI generated structures, wind dynamics, and effects on observational diagnostics have not been directly investigated so far. We investigated the non-linear development of LDI generated structures and dynamics in a magnetic line-driven wind of a typical O-supergiant. We employed two-dimensional axisymmetric magnetohydrodynamic (MHD) simulations of the LDI using the Smooth Source Function approximation for evaluating the assumed one-dimensional line force. To facilitate the interpretation of these magnetic models, they were compared with a corresponding non-magnetic LDI simulation as well as a magnetic simulation neglecting the LDI. A central result obtained is that the wind morphology and wind clumping properties change strongly with increasing wind-magnetic confinement. Most notably, in magnetically confined flows, the LDI leads to large-scale, shellular sheets ('pancakes') that are quite distinct from the spatially separate, small-scale clumps in non-magnetic line-driven winds. We discuss the impact of these findings for observational diagnostic studies and stellar evolution models of magnetic hot stars., 14 pages, 7 figures, accepted for publication in A&A

The wavefield is often reconstructed through solving a wave equation corresponding to an active source and using our best knowledge of the medium to reproduce the data that we seek to fit the observed data as part of a process we call full-waveform inversion (FWI). Alternatively, the wavefield can be inverted and used to invert for the velocity model, in an extended optimization problem, in which we can relax the requirement that the wavefield satisfies the wave equation. In this case, the wavefield calculation, as in FWI, requires a matrix inversion (which depends on the velocity) at practically every iteration. Thus, we formulate a bilinear optimization problem with respect to the wavefield and a modified source function, as independent variables. We specifically recast the wave equation so that velocity perturbations are included in this modified source function (which includes secondary or contrast sources); thus, it represents the velocity perturbations implicitly. The optimization includes a measure of the wavefield’s fit to the data at the sensor locations and the wavefield, as well as the modified source function, compliance with a wave equation corresponding to the background model. This problem is, however, complex with an extended model space that is ill-posed, so we use an alternating-direction method to reduce the inversion to two subproblems for inverting each of the wavefields and extended source. On the other hand, the velocity perturbations can be extracted in a separate step via direct division (deconvolution). Because we avoid using gradient methods in extracting the velocity perturbations, we are less prone to crosstalk artifacts when we use simultaneous sources. We evaluate these features on a simple two-anomalies model and the modified Marmousi model.

In this article, we have presented a static anisotropic solution of stellar compact objects for self-gravitating system by using minimal geometric deformation techniques in the framework of embedding class one space-time. For solving of this coupling system, we deform this system into two separate system through the geometric deformation of radial components for the source function $$\lambda (r)$$λ(r) by mapping: $$e^{-\lambda (r)}\rightarrow e^{-\tilde{\lambda }(r)}+\beta \,g(r)$$e-λ(r)→e-λ~(r)+βg(r), where g(r) is deformation function. The first system corresponds to Einstein’s system which is solved by taking a particular generalized form for source function $$\tilde{\lambda }(r)$$λ~(r) while another system is solved by choosing well-behaved deformation function g(r). To test the physical viability of this solution, we find complete thermodynamical observable as pressure, density, velocity, and equilibrium condition via. TOV equation etc. In addition to the above, we have also obtained the moment of inertia (I), Kepler frequency (v), compression modulus ($$K_e$$Ke) and stability for this coupling system. The M–R curve has been presented for obtaining the maximum mass and corresponding radius of the compact objects.

A non-classical one dimensional Stefan problem with thermal coefficients temperature dependent and a Robin type condition at fixed face x = 0 for a semi-infinite material is considered. The source function depends on the evolution the heat flux at the fixed face x = 0 . Existence of a similarity type solution is obtained and the asymptotic behaviour of free boundary with respect to latent heat fusion is studied. The analysis of several particular cases are given.

This article presents unsteady wall pressure measurements from a low-speed, low-solidity rotor operating in a circular duct. A circular microphone array was placed within the duct downstream of the rotor to record the unsteady surface pressure. These data were decomposed into spatial modes and then further decomposed into hydrodynamic and acoustic pressure fluctuations. The acoustic pressure was then further separated into the acoustic source function and an acoustic transfer function. The acoustic source function from the rotor compared well with a previous formulation for turbulence interaction noise. The transfer function showed a complicated relationship with frequency, with significant local maxima and minima. These features of the duct acoustics were studied using a simplified Green's function type approach, using the method of images to represent the duct end effects. The simple analytical method was able to capture many of the detailed features of the measurements with qualitative agreement.

Nowadays, as the development of Computational Fluid Dynamics (CFD) and the numerical wave tank (NWT) has advanced, numerical analysis has become increasingly useful and powerful for the ship designing and ship hydrodynamics. In this study, a momentum source wave-maker and an analytical relaxation wave absorber were embedded into 2D RANS equation model with RSM turbulence closure scheme to establish the NWT for ship designing and hydrodynamics. The VOF (volume-of-fluid) method was applied to accurately capture the water free surface. The body force-weighted scheme is chosen for pressure interpolation and the second order upwind scheme for discretization of the momentum equation. In order to calculate convection and diffusion fluxes through the control volume faces, PISO algorithm is adopted for pressure-velocity coupling. The momentum source function for wave generation and the analytical relaxation function for wave absorption were deduced for constructing the NWT (numerical wave tank). The proposed NWT was then validated by the laboratory measurements of Umeyama and the analytical solution, indicating that the constructed NWT is effective and accurate.

A 2-D numerical wave tank (NWT) is developed using the lattice Boltzmann method (LBM) and a multi-relaxation-time (MRT) collision model coupled with an algebraic volume of fluid (VOF) scheme for free surface tracking. An external force based on the momentum source function is used to generate the waves, and a zone of porous media is used to absorb the waves. Numerical simulations of the progressive and standing waves show that the NWT can generate stable wave trains in agreement with the analytical solutions and eliminate the re-reflection waves. The NWT is used to simulate two problems encountered in practice, namely: the wave transformation over a submerged breakwater and the wave runup on a sea dike. The numerical predictions are in good agreement with the measured data.

The phase matrix in scalar radiative transfer is symmetric that grants reciprocity principle in radiative transfer. The reciprocity principle is so useful that one may compute a single transmission matrix and a single reflection matrix for a homogeneous medium regardless of in upward or downward direction. The symmetric phase matrix is also important as one only needs to solve for a real eigensolution. An eigensolution is often used in a radiative transfer solver because of its high computational efficiency. However, the phase matrix in vectorized radiative transfer is generally not symmetric which challenges the reciprocity principle and forces us to deal with a complex eigensolution that requires a major effort in computational coding tangent-linear and adjoint models. This paper introduces an approach to retain the reciprocity principle in radiative transfer and applies a Taylor expansion of analytic transmittance and reflection matrices for a base optical depth together with a doubling-adding method beyond the base in the vectorized Community Radiative Transfer Model (CRTM). The value of the base optical depth depends on the maximum absolute value of the phase matrix elements. In comparison with other forward radiative transfer models, the extended vectorized CRTM agrees well with those models. The computational efficiency among the CRTM and those models is comparable. The tangent-linear and adjoint modules of the vectorized CRTM can be used for assimilating microwave, infrared, visible and ultraviolet sensor radiances.

In this paper, the particle method for handling oscillation of floating body using variable particle spacing is reported. Heave oscillation of floating mono hull and twin hull on the free surface in a viscous fluid is performed using Improved Meshless Local Petrov-Galerkin method based on Rankine Source function (IMLPG_R). The IMLPG_R method is a particle-based method that will solve the interaction between fluid and floating body using Navier-Stokes equation. The pressure at each time step is estimated from Pressure Poisson equation, in order to incorporate the heaving of the floating bodies, some improvements like free surface identification, gradient calculations has been introduced in the present paper to handle the variable resolutions in modelling. Simulations are performed for different frequency and amplitude of oscillations for mono and twin hull forced motions to show the robustness of the developed model to handle both small and steep oscillations.

Using the source imaging technique in two-pion interferometry, we study the image of the hydrodynamic particle-emitting source with the HIJING initial conditions for relativistic heavy-ion collisions on an event-by-event basis. It is shown that the initial-state fluctuations may give rise to bumpy structures of the medium during hydrodynamical evolution, which affects the two-pion emission space and leads to a visible two-tiered shape in the source function imaged using the two-pion Bose–Einstein correlations. This two-tiered shape can be understood within a similar but more analytic granular source model and is found to be closely related to the introduced quantity $$\xi$$ , which characterizes the granular inhomogeneity of the source. By fitting the imaged source function with a granular source parametrization, we extract the granular inhomogeneity of the hydrodynamic source, which is found to be sensitive to both the Gaussian smearing width of the HIJING initial condition and the centrality of the collisions.

In order to numerically simulate the wave-current interaction problems frequently encountered by aquaculture structures, a two-dimensional numerical wave-current tank model was established here based on a mass source wave maker coupled with an analytical relaxation wave absorber. The wave-maker model and the wave-absorber model were embedded into a two-dimensional RANS solver, which was closed with RSM turbulence scheme. The volume of fluid (VOF) method was adapted to accurately capture the free surface between water and air. To generate a steady uniform current flow, the uniform current flow velocity was calculated at the left-hand-side (LHS) and right-had-side (RHS) outflow boundaries, respectively. Once the steady uniform current flow was generated over the whole computational domain, the target water wave was marked within a specified region by embedding the mass source function based on wave theory into the mass conservation equation and then propagated on the generated uniform current flow. To verify the accuracy of the numerical wave-current tank established here, some of the obtained numerical results were then compared with the experimental results and the analytical solutions, and they agreed well with each other, indicating that the model developed here has great ability in simulating water waves on uniform currents over constant water depth. The established numerical wave-current tank was then used to study the optimal layout of the mass source region and the effects of water current velocity on water surface wave parameters during regular wave coupling with uniform water currents. Meanwhile, the established model was extended to generate steep wave and apply in deep water conditions. Finally, the proposed methods were applied to investigate the wave-current-structure interaction problems and the propagation of solitary waves traveling with coplanar/counter currents. Model-data comparisons show that the developed model here is potentially useful and efficient for investigating the inevitable wave-current-structure interaction problems in aquaculture technologies.

Helioseismology is the study of the solar interior using observations of oscillations at the surface. It suffers from systematic errors, such as a center-to-limb error in travel-time measurements. Understanding these errors requires a good understanding of the nontrivial relationship between wave displacement and helioseismic observables. The wave displacement causes perturbations in the atmospheric thermodynamical quantities which perturb the opacity, the optical depth, the source function, and the local ray geometry, thus affecting the emergent intensity. We aim to establish the most complete relationship up to now between the displacement and the intensity perturbation by solving the radiative transfer problem in the atmosphere. We derive an expression for the intensity perturbation caused by acoustic oscillations at any point on the solar disk by applying the first-order perturbation theory. As input, we consider adiabatic modes of oscillation of different degrees. The background and the perturbed intensities are computed considering the main sources of opacity in the continuum. We find that, for all modes, the perturbations to the thermodynamical quantities are not sufficient to model the intensity. In addition, the geometrical effects due to the displacement must be taken into account as they lead to a difference in amplitude and a phase shift between the temperature at the surface and intensity perturbations. The closer to the limb, the larger the differences. This work presents improvements for the computation of the intensity perturbations, in particular for high-degree modes, and explains differences in intensity computations in earlier works. The phase shifts and amplitude differences between the temperature and intensity perturbations increase towards the limb. This should help to interpret some of the systematic center-to-limb effects observed in local helioseismology., Comment: 11 pages, 7 figures, Accepted to A&A

A new method for modeling the radiative transfer in inhomogeneous three-dimensional media illuminated by a Gaussian beam is described. This approach, called the Spectral Spherical Harmonics Discrete Ordinate Method (SSHDOM), uses the Fourier expansion method to transform the three-dimensional radiative transfer into an one-dimensional equation in the spectral domain, and the Spherical Harmonics Discrete Ordinate Method (SHDOM) for its solution. Specifically, (i) the source function is represented in the spectral domain through a spherical harmonic expansion, (ii) the spectral one-dimensional radiative transfer equation is integrated along discrete ordinates through a spatial grid, and (iii) the solution method is based on the Picard iteration. Both SSHDOM and SHDOM algorithms are implemented in a common computer code.

This study analyses the frequency content of vibrations from blasting in the near field using a vibration prediction model called full-field solution (FFS). For that purpose, the frequency content of recorded signals and synthetic waveforms obtained through the FFS using dominant and mean frequencies have been compared and correlated with the geometrical distance to the blastholes. The study shows that the model overestimates the dominant and mean frequencies in the entire frequency domain, which may require tailoring the source function to a lower frequency content. However, it attenuates the frequency content as distance increases. Additionally, the study shows that the mean frequencies may be used as a better estimator of the frequency content than the dominant frequencies.

By measuring the momentum correlations of pions created in heavy-ion collisions we can gain information about the space-time geometry of the particle emitting source. Recent experimental results from multiple different collaborations demonstrated that to properly describe the shape of the measured correlation functions, one needs to go beyond the Gaussian approximation. Some studies suggest that the Levy distribution could provide a good description of the source. While there are already many experimental results, there is very little input from the phenomenology side in explanation of the observed non-Gaussian source shapes. The EPOS model is a sophisticated hybrid model where the evolution of the newly-created system is governed by Parton-Based Gribov-Regge theory. It has already proved to be successful in describing many different experimental observations for the systems characterized by baryon chemical potential close to zero, but so far the source shape has not been explored in detail. In this paper we discuss studies of the pion emitting source based on the theoretical approach of the EPOS model., Comment: 7 pages, 5 figures, Summer XLVI-th IEEE-SPIE Joint Symposium Wilga 2020 - conference proceedings

Transient mass-transfer phenomena occurring in natural and engineered systems consist of convection, diffusion, and reaction processes. The coupled phenomena can be described by using the unsteady convection-diffusion-reaction (CDR) equation, which is classified in mathematics as a linear, parabolic partial-differential equation. The availability of analytic solutions is limited to simple cases, e.g., unsteady diffusion and steady convective diffusion. The CDR equation has been considered analytically intractable, depending on the initial and boundary conditions. If spatial adsorption and desorption of matter are super-positioned in the CDR equation as sink and source functions, respectively, then the governing equation becomes an unsteady convection-diffusion-reaction-source (CDRS) equation, of which general solutions are unknown. In this study, a general 1D analytic solution of the CDRS equation is obtained by using a one-sided Laplace transform, by assuming constant diffusivity, velocity, and reactivity. This paper also provides a general formalism to derive 1D analytic solutions for Dirichlet/Dirichlet and Dirichlet/Neumann boundary conditions. Derivations of the analytic solutions are found to be straightforward if a combination of the source function and the initial concentration provide a non-zero singularity pole of inverse Laplace transform.

The two-particle momentum correlation function of a K^-p pair from high-energy nuclear collisions is evaluated in the KbarN-piSigma-piLambda coupled-channels framework. The effects of all coupled channels together with the Coulomb potential and the threshold energy difference between K^-p and Kbar0 n are treated completely for the first time. Realistic potentials based on the chiral SU(3) dynamics are used which fit the available scattering data. The recently measured correlation function is found to be well reproduced by allowing variations of the source size and the relative weight of the source function of piSigma with respect to that of KbarN. The predicted K^-p correlation function from larger systems, which is less affected by the piSigma source function, indicates that the investigation of its source size dependence is useful in providing further constraints in the study of the KbarN interaction., Comment: 5 pages, 3 figures, published version

In this study, further development of an in-house code, named a constraint interpolation profile (CIP)-based model is presented. A momentum source function is introduced as an internal wave maker to generate regular and focused waves accurately without being affected by reflected waves. A self-adaptive source region method is proposed to optimize the width of the momentum source region to different wave conditions. The proposed model is applied to different cases. Test of regular wave reflection from a vertical wall is first carried out. Regular wave decomposition over a submerged breakwater is then simulated. Focused wave generation and propagation over a submerged breakwater are finally studied and compared with a newly performed experiment. Good agreements are obtained. All the results indicate that the CIP-based model with a non-reflection internal wave-maker can work as a powerful tool to study wave-structure interaction in coastal engineering.

This paper is devoted to solve an inverse time-dependent source problem for a time-fractional diffusion-wave equation. The source function in the time-fractional diffusion-wave equation is ...

Temperature distribution in biological tissue is described by dual-phase lag model supplemented by appropriate boundary and initial conditions. Laser–tissue interactions are taken into account in the source function occurring in this model. The problem is solved using the finite difference method. Next, the Arrhenius integral which is a measure of tissue destruction is calculated. The inverse problem formulated here is to estimate the laser intensity assuring the destruction of this tissue region for which the Arrhenius integral exceeds the critical value.

The spectrum of symmetric modes forming a specified distribution of the radiation field is calculated using solution to the self-consistent problem of radiation from the end of a circular dielectric waveguide. The self-consistent problem is reduced to solution of a system of integral equations. In the case of the synthesis of the source function in the space of eigenfunctions of the boundary value problem for a circular dielectric waveguide, these equations are the Fredholm integral equations of the first kind.

A momentum source function derived from the linear wave theory is introduced into the weakly compressible smoothed particle hydrodynamic (SPH) model for the long-time simulation of regular and irregular wave generation problems. Wave absorption is realized by adding a velocity attenuation term into the governing momentum equation. The performances of the wave maker are tested under different wave conditions. The wave maker is then applied to the study of the challenging processes, such as random wave breaking on the reef-face or the reef-crest, wave setup and spectral transfer of wave energy from the peak frequency to lower frequencies over the reef-flat. The predicted results are compared with the experimental data and a good agreement is obtained. The SPH model with a non-reflective spectral wave maker can be used as a practical tool for studying wave interaction with coastal structures.

Exact solutions of a nonlinear Boltzmann kinetic equation with a source are constructed in the case of an isotropic distribution function and Maxwell model of isotropic scattering. These solutions are constructed with the use of an equivalence group such that one of its transformations uniquely identifies the class of the source functions that are linear in terms of the distribution function; moreover, the transformed equation has a zero right side. As a result, invariant solutions of the type of the Bobylev–Krook–Wu solutions can be explicitly found, in particular, those that admit physical interpretation.

The quasi-mobility function of charge carriers with a specified energy for describing their dynamics using the kinetic equation is studied in the important case of two-term isotropic approximation. In the stationary case, the quasi-mobility function is independent of the source function of charge carriers and makes it possible to calculate the integral mobility. The correlation between the quasi-mobility and parameters of the system is analyzed. It is proved that this characteristic does not generally describe the contribution of charge carriers with a specified energy to the integral mobility. In the case of almost elastic scattering, the quasi-mobility, as is known, can have a clear physical meaning; however, in the case of the scattering of charge carriers at acoustic phonons in a solid, this quasi-mobility interpretation is found to be incorrect due to the specific features of the collision integral and the form of the quasi-mobility function.

The kinetic energy (KE) fluxes into subsurface currents (EFc) is an important boundary condition for ocean circulation models. Traditionally, numerical models assume the KE flux from wind (EFair) is identical to EFc, that is, no net KE is gained (or lost) by surface waves. This assumption, however, is invalid when the surface wave field is not fully developed, and acquires KE when it grows in space or time. In this study, numerical experiments are performed to investigate the KE flux budget across the air-sea interface under both uniform and idealized tropical cyclone (TC) winds. The wave fields are simulated using the WAVEWATCH III model under different wind forcing. The difference between EFair and EFc is estimated using an air-sea KE budget model. To address the uncertainty of these estimates resides in the variation of source functions, two source function packages are used for this study: the ST4 source package (Ardhuin et al, 2010), and the ST6 source package (Babanin, 2011). The modeled EFc is significantly reduced relative to EFair under growing seas for both the uniform and TC experiments. The reduction can be as large as 20%, and the variation of this ratio is highly dependent on the choice of source function for the wave model. Normalized EFc are found to be consistent with analytical expressions by Hwang and Sletten (2008) and Hwang and Walsh (2016) and field observations by Terray et al. (1996) and Drennan et al. (1996), while the scatters are more widely in the TC cases due to the complexity of the associated wave field. The waves may even give up KE to subsurface currents in the left rear quadrant of fast moving storms. Our results also suggest that the normalized KE fluxes may depend on both wave age and friction velocity (u*).

A circular coil equipped with magnetic core of finite length and constant permeability is difficult to analyse by traditional analytical methods in consequence of the complicated boundary conditions. The difficulties can be overcome by using the truncated region eigenfunction expansion method, and with the aid of a special source function K ( z ) introduced in the work, this method can be simplified remarkably and concise expressions of the inductance of coils with magnetic cores are obtained. In addition, the calculations can be extended readily to a pair of coaxial coils with magnetic cores of finite length and different permeability. The eigenvalues and inversions of matrices are implemented numerically in the solving process, while most of calculations are accomplished by analytic integrations and matrix algebra, by which the deduced expressions are quite suitable for numerical evaluations. The proposed method is verified by the numerical results compared with those of finite-element method simulations, from which it is confirmed that the proposed method is much faster and more memory efficient.

A Wave Boundary Layer Model (WBLM) is implemented in the third-generation ocean wave model SWAN to improve the wind-input source function under idealized, fetch-limited condition. Accordingly, the white capping dissipation parameters are re-calibrated to fit the new wind-input source function to parametric growth curves. The performance of the new pair of wind-input and dissipation source functions is validated by numerical simulations of fetch-limited evolution of wind-driven waves. As a result, fetch-limited growth curves of significant wave height and peak frequency show close agreement with benchmark studies at all wind speeds (5 ∼ 60 ms−1) and fetches (1 ∼ 3000 km). The WBLM wind-input source function explicitly calculates the drag coefficient based on the momentum and kinetic energy conservation. The modeled drag coefficient using WBLM wind-input source function is in rather good agreement with field measurements. Thus, the new pair of wind-input and dissipation source functions not only improve the wave simulation but also have the potential of improving air-sea coupling systems by providing reliable momentum flux estimation at the air-sea interface. This article is protected by copyright. All rights reserved.

An attraction-repulsion chemotaxis model with nonlinear chemotactic sensitivity functions and growth source is considered. The global-in-time existence and boundedness of solutions are proved under some conditions on the nonlinear sensitivity functions and growth source function. Our results improve the earlier ones for the linear sensitivity functions.

This paper considers nonstationary monochromatic radiative transfer in an infinite onedimensional homogeneous medium. The medium is considered to be illuminated by a momentary isotropic point energy source. The optical properties of the medium are characterized by the absorption coefficient α, the single-scattering albedo λ, the mean time t 1 of photon stay in the absorbed state, and the mean time t 2 of its stay on the path between two consecutive scatterings. The exact solution of the nonstationary radiative transfer equation has been obtained for the case t 1 = t 2. Asymptotic expressions have been derived for the source function, for the average intensity, and for radiation flux when points of the medium are located at large optical distances from the power source |τ| ≫ 1 and for small absorption of light in the medium (1 − λ ≪ 1), assuming that t 1 ≫ t 2, t 1 ≪ t 2, or t 1 = t 2. These expressions are more precise than the ones previously known.

In this study, we have simulated the effect of light sources with different spectral output functions on the generation of A-scans in optical coherence tomography using a fundamental physics-based interferometric model. Many different source function were examined, and compared to a standard Gaussian source. These sources included truncated Gaussians, multiple Gaussians, other non-Gaussian, Lorentzian, square and triangular sources. Only the pure Gaussian source produced A-scans without false artefacts such as satellite peaks, that could produce misinterpretation of real OCT images that may be used for patient diagnosis. A triangular source produces the next best response with small extraneous peaks, whereas all other sources have significant false artefacts present in their A-scans.

Context. Knowledge about hot, massive stars is usually inferred from quantitative spectroscopy. To analyse non-spherical phenomena, the existing 1D codes must be extended to higher dimensions, and corresponding tools need to be developed. Aims. We present a 3D radiative transfer code that is capable of calculating continuum and line scattering problems in the winds of hot stars. By considering spherically symmetric test models, we discuss potential error sources, and indicate advantages and disadvantages by comparing different solution methods. Further, we analyse the ultra-violet (UV) resonance line formation in the winds of rapidly rotating O stars. Methods. We consider both a (simplified) continuum model including scattering and thermal sources, and a UV resonance line transition approximated by a two-level-atom. We applied the short-characteristics (SC) method, using linear or monotonic Bézier interpolations, for which monotonicity is of prime importance, to solve the equation of radiative transfer on a non-uniform Cartesian grid. To calculate scattering dominated problems, our solution method is supplemented by an accelerated Λ-iteration scheme using newly developed non-local operators. Results. For the spherical test models, the mean relative error of the source function is on the 5 − 20% level, depending on the applied interpolation technique and the complexity of the considered model. All calculated line profiles are in excellent agreement with corresponding 1D solutions. Close to the stellar surface, the SC methods generally perform better than a 3D finite-volume-method; however, they display specific problems in searchlight-beam tests at larger distances from the star. The predicted line profiles from fast rotating stars show a distinct behaviour as a function of rotational speed and inclination. This behaviour is tightly coupled to the wind structure and the description of gravity darkening and stellar surface distortion. Conclusions. Our SC methods are ready to be used for quantitative analyses of UV resonance line profiles. When calculating optically thick continua, both SC methods give reliable results, in contrast to the alternative finite-volume method.

The quark gluon plasma is formed in heavy-ion collisions, and it can be described by solutions of relativistic hydrodynamics. In this paper we utilize perturbative hydrodynamics, where we study first order perturbations on top of a known solution. We investigate the perturbations on top of the Hubble flow. From this perturbative solution we can give the form of the particle emitting source and calculate observables of heavy-ion collisions. We describe the source function and the single-particle momentum spectra for a spherically symmetric solution., Comment: 7 pages, 3 figures

In this work, the motion of a two‐dimensional rectangular freely floating body under waves is simulated using Improved Meshless Local Petrov‐Galerkin method with Rankine Source function (IMLPG_R) with variable spacing resolutions. The IMLPG_R method is a particle method that solves Navier–Stokes equations using the fractional step method to capture the wave properties. However, many existing particle methods are computationally intensive to model the wave‐floating body due to the requirement of fine particles, needing uniform distribution throughout the domain. To improve the computational efficiency and capture the body response properly, variable spaced particle distribution with fine resolution near the floating body and coarse resolution far from the body is implemented. Numerical schemes to handle variable resolutions are reported. An iterative scheme to handle the wave‐floating body is implemented in the particle method. Two test cases, one with small wave and another with steep waves, are simulated for uniform particle distribution and the result shows good agreement with literature. Based on this, the performance of the variable spaced particle distribution is tested in coupling with floating body solver. The application of the method for wave impact load from the green water loading of the floating structure is also simulated.

Ships often have to pass each other in proximity in harbor areas and waterways in dense shipping-traffic environment. Hydrodynamic interaction occurs when a ship is overtaking (or being overtaken) or encountering other ships. Such an interactive effect could be magnified in confined waterways, e.g., shallow and narrow rivers. Since Yeung published his initial work on ship interaction in shallow water, progress on unsteady interaction among multiple ships has been slow, though steady, over the following decades. With some exceptions, nearly all the published studies on ship-to-ship problem neglected free-surface effects, and a rigid-wall condition has often been applied on the water surface as the boundary condition. When the speed of the ships is low, this assumption is reasonably accurate as the hydrodynamic interaction is mainly induced by near-field disturbances. However, in many maneuvering operations, the encountering or overtaking speeds are actually moderately high (Froude number Fn > 0.2, where (equation) U is ship speed, g is the gravitational acceleration, and L is the ship length), especially when the lateral separation between ships is the order of ship length. Here, the far-field effects arising from ship waves can be important. The hydrodynamic interaction model must take into account the surface-wave effects. Classical potential-flow formulation is only able to deal with the boundary value problem when there is only one speed involved in the free-surface boundary condition. For multiple ships traveling with different speeds, it is not possible to express the free surface boundary condition by a single velocity potential. Instead, a superposition method can be applied to account for the velocity field induced by each vessel with its own and unique speed. The main objective of the present article is to propose a rational superposition method to handle the unsteady free-surface boundary condition containing two or more speed terms, and validate its feasibility in predicting the hydrodynamic behavior in ship encountering. The methodology used in the present article is a three-dimensional boundary-element method based on a Rankine-type (infinite-space) source function, initially introduced by Bai and Yeung. The numerical simulations are conducted by using an in-house-developed multibody hydrodynamic interaction program "MHydro." Waves generated and forces (or moments) are calculated when ships are encountering or passing each other. Published model-test results are used to validate our calculations, and very good agreement has been observed. The numerical results show that free-surface effects need to be taken into account for Fn > 0.2.

This article applies perturbation theory to obtain an asymptotic expansion that approximates the Radiative Transfer Equation (RTE) Green’s function. The integral (source function) formulation of the RTE yields a recursive operator; applying the operator once to a point source collimated beam yields a closed-form extension to Beer’s Law. In the weakly scattering range (i.e., albedo below 0.36 or domain optical thickness below 0.85), the approximation correctly describes the energy transported in phase space for any phase function (mean squared errors below 10%). Besides the theoretical methodology’s value, the closed-form approximation allows developing a deterministic numerical method, free of directional meshes, that computes the global irradiance field in three-dimensional enclosures under different internal and boundary conditions. For the cases tested, its accuracy and versatility are comparable to a Monte Carlo method in the weakly scattering range, while its computational efficiency resembles that of the P1 approximation.

This short communication interprets the delta-fit technique in a context of similarity transformation and the correction to the source function, and derives the analogous form of the method to be applied for the scattering phase matrix. To adapt the delta-fit method to vector radiative transfer, the mathematically exact form of the similarity principle is used in the theoretical development. Some examples of relevant radiative transfer simulations are also presented for atmospheric ice particles. The performance of the adopted delta-fit method is comparable to the delta-M method with single scattering correction except for worse delta-fit performance for polarized radiance calculations in forward directions.

This paper is concerned with an iterative algorithm for inverse evaluation of the source function for two elliptic systems. The algorithm starts with an initial guess for the unknown source function, obtains a background field and, obtains the working equations for the error field. The correction to the assumed value appears as a source term for the error field. It formulates two well-posed problems for the error field which makes it possible to obtain the correction term. The algorithm can also recover the source function with partial data at the boundary. We consider 2-D as well as 3-D domains. The method can be applied to both Poisson and Helmholtz operators. Numerical results indicate that the algorithm can recover close estimates of the unknown source functions based on measurements collected at the boundary.

A source-function wavemaker for wave-resolving models is evaluated for its capability to reproduce random directionally spread wave fields in the sea-swell band (0.04–0.3 Hz) relevant for realistic nearshore applications. The wavemaker is tested with a range of input wave characteristics defined by the non-dimensional amplitude (a/h), wavenumber (kh), wavemaker width, mean wave angle and directional spread. The (a/h) and kh dependency of modeled results are collapsed with the Ursell number (Ur = (a/h)/(kh)2). For monochromatic waves, the wavemaker accurately reproduced the input wave height for Ur

In this study, Donelan, M.A., Babanin, A.V., Young, I.R., Banner, M.L., 2006. J. Phys. Oceanogr. 36, 1672–1688 source function is used to calculate drag coefficients from both the scanning radar altimeter (SRA) measured two dimensional wave spectra obtained during hurricane Ivan in 2004 and the WAVEWATCH III simulated wave spectra. The drag coefficients disagree between the SRA and model spectra mainly in the right/left rear quadrant of the hurricane where the observed spectra appear to be bimodal while the model spectra are single peaked with more energy in the swell frequencies and less energy in the wind sea frequencies. These results suggest that WAVEWATCH III is currently not capable of providing sensible stress calculations in the rear quadrants of the hurricane.

The two flares of 19 and 30 July 1999 were observed in the Hα line using the multichannel flare spectrograph (MFS) at the Astronomical Institute in Ondřejov, Czech Republic. We use a modified cloud method to fit the Hα line profiles which avoids using the background profile. We obtain the four parameters of the two flares: the source function, the optical thickness at line center, the line-of-sight velocity and the Doppler width. The observed asymmetry profiles have been reproduced by the theoretical ones based on our model. A discussion is made about the results of strong and weak flares using the present method.

In this study, a mass source wavemaker and a numerical sponge layer were embedded into a three-dimensional (3D) Navier–Stokes equations model to study wave–structure interaction problems. The capability of the numerical sponge layer was first examined and an optimal layout of the sponge layer was determined based on a series of two-dimensional (2D) numerical experiments. Then, the scheme was extended to 3D geometry. In addition, an approach for designing the mass source function for an internal wavemaker was developed. The proposed method was applied to directional wave cases. Regular, irregular, and solitary waves were examined. The numerical results were compared with the analytical solutions and some numerical results obtained using the momentum source method, with good agreements observed for a wide range of relative water depths. Finally, the proposed model was applied to simulate 2D and 3D wave–structure interaction problems. Model-data comparisons show that the proposed model is potentially useful and efficient for examining wave–structure interactions.

The solar chromosphere can be observed well through strong absorption lines. We infer the physical parameters of chromospheric plasmas from these lines using a multilayer spectral inversion. This is a new technique of spectral inversion. We assume that the atmosphere consists of a finite number of layers. In each layer the absorption profile is constant and the source function varies with optical depth with a constant gradient. Specifically, we consider a three-layer model of radiative transfer where the lowest layer is identified with the photosphere and the two upper layers are identified with the chromosphere. The absorption profile in the photosphere is described by a Voigt function, and the profile in the chromosphere by a Gaussian function. This three-layer model is fully specified by 13 parameters. Four parameters can be fixed to prescribed values, and one parameter can be determined from the analysis of a satellite photospheric line. The remaining 8 parameters are determined from a constrained least-squares fitting. We applied the multilayer spectral inversion to the spectral data of the Hα and the Ca II 854.21 nm lines taken in a quiet region by the Fast Imaging Solar Spectrograph (FISS) of the Goode Solar Telescope (GST). We find that our model successfully fits most of the observed profiles and produces regular maps of the model parameters. The combination of the inferred Doppler widths of the two lines yields reasonable estimates of temperature and nonthermal speed in the chromosphere. We conclude that our multilayer inversion is useful to infer chromospheric plasma parameters on the Sun.

This study provides basic instantaneous source functions for different source-reservoir geometries in infinite fractal reservoirs (FRs) and infinite slab FRs. The concept of fractal geometry (FG) has already been adopted to analyse and model well-reservoir systems with complex structures. All these complexities have been modelled through a partial differential equation called fractal diffusivity equation (FDE). In this paper, by use of the source/Green’s function technique, we analytically solve the FDE for different boundary conditions, i.e., different source-reservoir geometries. The wellbore pressure response for horizontal wells in an infinite slab FR (synthetic example) is derived and analysed to illustrate the applications of the provided analytical source solutions and how to use these solutions.