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In the paper, an inverse source problem for a 2D nonlinear elliptic partial differential equation is considered. With the over-specified boundary data given on two sides of a rectangle, we do not need the data on bottom and top sides to estimate an unknown 2D source function. As a consequence, we encounter a quite difficult inverse problem for solving the inverse source problem in an environment of the inverse Cauchy problem. Generating from the given Cauchy data, we introduce a homogenization function and then a sequence of boundary functions are derived, which combined with the zero element constitute a linear space. An energetic functional in terms of boundary functions is derived and each energetic boundary function preserves the energy identity. The novel algorithm with energetic boundary functions as the bases and the linear system to decide the expansion coefficients are developed from the energetic boundary functional method (EBFM). At each spatial step we solve a nonlinear scalar equation by the Newton method to determine the multiplier in the energetic boundary function. The accuracy and robustness of the EBFM are confirmed, when we compare the estimated results to the exact solutions. Being an extension of the nonlinear inverse source problem to an arbitrary domain, we propose an inner homogenization functions method to solve the problem, with two examples demonstrating the accuracy and efficiency.

The study of van der Waals interactions plays a central role in the understanding of bonding across a range of biological, chemical and physical phenomena. The presence of van der Waals interactions can be identified through analysis of the reduced density gradient, a fundamental parameter at the core of Density Functional Theory. An extension of Bader’s Quantum Theory of Atoms in Molecules is developed here through combination with the analysis of the reduced density gradient. Through this development, a new quantum chemical topological tool is presented: the volumetric source function. This technique allows insight into the atomic composition of van der Waals interactions, offering the first route towards applying the highly successful source function to these disperse interactions. A new algorithm has been implemented in the open-source code, CRITIC2, and tested on acetone, adipic and maleic acids molecular crystals, each stabilized by van der Waals interactions. This novel technique for studying van der Waals interactions at an atomic level offers unprecedented opportunities in the fundamental study of intermolecular interactions and molecular design for crystal engineering, drug design and bio-macromolecular processes.

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.

In this paper, we consider diffusion equations with a power-type coefficient and nonlinear source. We pay the main attention to constructing exact solutions in terms of the Lambert function. We subject the source function to certain conditions that guarantee the existence of exact solutions of a given type. For illustrating the obtained results, we give several examples of exact solutions to nonlinear diffusion equations, including those with polynomial and fractional-rational source functions.

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.

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.

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 problem of heat distribution in semiconductor materials irradiated with sharply focused lowenergy electron beams in the absence of heat exchange between the target and external medium is considered by methods of mathematical modeling. The model is based on solving a multidimensional steady-state heatconduction equation using Green’s function. The source function is a model applicable to a wide class of solids and a wide range of energies of primary electrons, based on separate description of the contributions of absorbed and backscattered electrons to the energy dissipated in the target. Using the features of such an approach, a nonmonotonic dependence of the temperature of maximum heating of the target on the energy of primary electrons is explained. The results of calculations using the considered model of heating the target with an electron probe are presented for various semiconductor materials of electronic engineering.

This work is concerned with an inverse problem of identifying the current source distribution of the time-harmonic Maxwell's equations from multi-frequency measurements. Motivated by the Fourier method for the scalar Helmholtz equation and the polarization vector decomposition, we propose a novel method for determining the source function in the full vector Maxwell's system. Rigorous mathematical justifications of the method are given and numerical examples are provided to demonstrate the feasibility and effectiveness of the method.

inverse source problem of identifying the source function f in the abstract Cauchy problem $$u_t+Au=f(t),\, 0

The acoustic approximation, even for anisotropic media, is widely used in current industry imaging and inversion algorithms mainly because P-waves constitute most of the energy recorded in seismic exploration. The resulting acoustic formulas tend to be simpler, resulting in more efficient implementations, and they depend on fewer medium parameters. However, conventional solutions of the acoustic-wave equation with higher-order derivatives suffer from S-wave artifacts. Thus, we separate the quasi-P-wave propagation in anisotropic media into the elliptic anisotropic operator (free of the artifacts) and the nonelliptic anisotropic components, which form a pseudodifferential operator. We then develop a separable approximation of the dispersion relation of nonelliptic-anisotropic components, specifically for transversely isotropic media. Finally, we iteratively solve the simpler lower-order elliptical wave equation for a modified source function that includes the nonelliptical terms represented in the Fourier domain. A frequency-domain Helmholtz formulation of the approach renders the iterative implementation efficient because the cost is dominated by the lower-upper decomposition of the impedance matrix for the simpler elliptical anisotropic model. In addition, the resulting wavefield is free of S-wave artifacts and has a balanced amplitude. Numerical examples indicate that the method is reasonably accurate and efficient.

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.

This paper is devoted to the study of an inverse semilinear parabolic problem. The problem contains an unknown solely time-dependent source function p and a homogeneous Dirichlet boundary condition. Moreover, an integral measurement of the total energy/mass in the domain is given. A full-discrete finite element scheme to approximate the unique weak solution is designed. For the time discretization backward Euler’s method is used. For the space discretization the finite element method is applied. Various error estimates are derived, depending on the regularity of the data and on the choice of the finite elements.

A new approach to a rigorous theory of aperture antennas is proposed, which is based on solution of the self-consistent problem of radiation. It is ascertained that this formulation of the problem leads to a system of homogeneous Fredholm integral equations of the second kind—i.e., a problem for eigenfunctions and eigenvalues that provides a basis for solving the problem of field generation in the open space. The passage to Fredholm integral equations of the first kind allows one to solve the task of synthesis of the source function for a given radiation field.

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.

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.

A novel method is proposed in this paper to extract acoustic emission (AE) source using Helmholtz potentials approach. In order to characterize the source in terms of potentials the underlying physics is to detect the AE signals and then develop an inverse algorithm to characterize the source. According to the new concept, the source can be characterized it provides the excitation potential information from the crack and that can be used to diagnose the crack (crack type and crack growth etc.). An AE experiment was designed to measure the acoustic emissions from the fatigue crack growth. A test specimen was made of 1 mm thick 304-steel material. A small hole (1 mm diameter) was drilled at the center of the specimen to initiate the fatigue crack. The specimen was subjected to the cyclic loadings by using the hydraulic MTS machine. AE waveforms are generated by convolutions of AE source functions, plate transfer functions. An inverse algorithm was developed to characterize the source of AE signal. AE signal analysis are done to determine AE source function using deconvolution process.

In the previous chapter, multi-scale microscopic flow models in shale gas reservoirs considering various complex flow mechanisms have been derived. These models are the basis for the study of the transient flow theory for fractured wells in the shale reservoirs. According to the research on the transient flow theory for fractured wells ( Aguilera, 1995 , Zhang et al., 2004 ), two popular methods for this theory are frequently used: a source function and an orthogonal transformation. The latter one is often used for linear flow, and the former is the main method to consider for interference among fractures or interference between wellbores and fractures. In this chapter, the microscopic flow models established in the previous chapter are first solved based on the basic oil and gas diffusivity equations. Then the continuous point source solutions are derived based on the source function theory, together with applications of mathematic methods including the Delta function, Laplace transformation, orthogonal transformation, and superposition principle ( Carslaw and Jaeger, 1959 ).

In this paper, we study two stochastic problems for time-fractional RayleighStokes equation including the initial value problem and the terminal value problem. Here, two problems are perturbed by Wiener process, the fractional derivative are taken in the sense of Riemann-Liouville, the source function and the time-spatial noise are nonlinear and satisfy the globally Lipschitz conditions. We attempt to give some existence results and regularity properties for the mild solution of each problem.

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.

We develop Second Order Asymptotical Regularization (SOAR) methods for solving inverse source problems in elliptic partial differential equations with both Dirichlet and Neumann boundary data. We show the convergence results of SOAR with the fixed damping parameter, as well as with a dynamic damping parameter, which is a continuous analog of Nesterov’s acceleration method. Moreover, by using Morozov’s discrepancy principle together with a newly developed total energy discrepancy principle, we prove that the approximate solution of SOAR weakly converges to an exact source function as the measurement noise goes to zero. A damped symplectic scheme, combined with the finite element method, is developed for the numerical implementation of SOAR, which yields a novel iterative regularization scheme for solving inverse source problems. Several numerical examples are given to show the accuracy and the acceleration effect of SOAR. A comparison with the state-of-the-art methods is also provided.

The interval coupled lattice Boltzmann equations for electrons and phonons are used to analyse the heating process of thin metal films. The interval lattice Boltzmann method (ILBM) with the uncertainly defined external source function associated with the laser irradiation is used to simulate the heat transfer. The solution of the interval Boltzmann transport equations has been obtained taking into account the rules of directed interval arithmetic. A similar analysis has been done using the sensitivity model where the Boltzmann transport equations and boundary-initial conditions have been differentiated with respect to the no-interval laser parameter. The knowledge of the sensitivity function distribution and the application of the Taylor formula allow one to find the border solutions of the problem analysed which correspond to the solution obtained assuming the uncertainly defined source function. In the final part of the paper the results of numerical computations obtained using both methods are presented.

The Jacobi spectral Galerkin method for Volterra integral equations of the second kind with a weakly singular kernel is proposed in this paper. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors (in the L ω α , β 2 -norm and the L ∞ -norm) will decay exponentially provided that the source function is sufficiently smooth. Numerical examples are given to illustrate theoretical results.

The use of multiple fractured horizontal wells (MFHWs) is considered an effective means for developing shale gas reservoirs, and a variety of models have recently been introduced to study the pressure behaviors of MFHWs in shale gas reservoirs. However, most of the existing models are based on some ideal assumptions that may not always be true in practice. This paper presents a novel semi-analytical model for finite-conductivity MFHWs in shale gas reservoirs with consideration of more actual conditions, such as finite-conductivity fractures, different fracture lengths, different fracture intervals, various angles between the fractures and the horizontal well, fracture asymmetry about the horizontal well, and partially penetrating fractures. Laplace transform, source function, numerical discrete method, and Gaussian elimination method are applied to solve the present model. The solution is presented in Laplace space so that the effects of the wellbore storage and skin can be easily incorporated by Duhamel's principle. Type curves of the pressure responses are plotted, possible flow regimes are identified, and a detailed analysis of the pressure characteristics is presented. The present model is much closer to the conditions of an actual reservoir, and thus it can be applied to an accurate interpretation of the pressure data of an MFHW in a shale gas reservoir.

Two explanations for the pattern of Moreton waves observed in the solar atmosphere in Hα are considered: a model with a wave-front cloud located in the upper chromosphere and moving radially downward, and a model in which the entire Hα absorption line is shifted. It was not possible to obtain the observed brightness curve for the front in the former model for any optical parameters of the cloud: the source function S, optical depth τ, Doppler width Δλd, and Doppler shift of the cloud absorption within the Hα line Δλsh (due to the radial motion of the wave-front cloud). The observed pattern of the wave front can be obtained only if the entire Hα absorption line is shifted. On this basis, it is concluded that this wave propagates in the region of formation of the Hα absorption, i.e., in the photosphere and lower chromosphere. This conclusion is supported by the fact that Moreton waves are not observed in the upper chromosphere. It is also shown that these waves cannot propagate in the corona, since the time for for the coronal gas to cool to temperatures near 10 000 K exceeds the wave period by an order of magnitude. Moreton waves are also not shocks, since the observed wave-front profiles do not display the discontinuities characteristic of shocks.

In this paper, we discuss the application of spectral Jacobi-collocation methods to a certain class of weakly singular Volterra integral equations. First, we use some function transformations and variable changes to transform the equation into a Volterra integral equation defined on the standard interval [ - 1 , 1 ] . Then the Jacobi–Gauss quadrature formula is used to approximate the integral operator. For the spectral Jacobi-collocation method, a rigorous error analysis in both the L ∞ and weighted L 2 norms is given under the assumption that both the kernel function and the source function are sufficiently smooth. Finally, some numerical examples are provided to illustrate the theoretical results.

Kinetic models are used extensively in science, engineering, and medicine. Mathematically, they are a set of coupled differential equations including a source function, otherwise known as an input function. We investigate whether parametric modeling of a noisy input function offers any benefit over the non-parametric input function in estimating kinetic parameters. Our analysis includes four formulations of Bayesian posteriors of model parameters where noise is taken into account in the likelihood functions. Posteriors are determined numerically with a Markov chain Monte Carlo simulation. We compare point estimates derived from the posteriors to a weighted non–linear least squares estimate. Results imply that parametric modeling of the input function does not improve the accuracy of model parameters, even with perfect knowledge of the functional form. Posteriors are validated using an unconventional utilization of the chi square test. We demonstrate that if the noise in the input function is not taken into account, the resulting posteriors are incorrect.

Summary This paper extended the conventional multiple hydraulic fractured horizontal (MFH) well into a composite model to describe the stimulated reservoir volume (SRV) caused by hydraulic fracturing. Employing the Laplace transform, Source function, and Dirac delta function methods, the continuous linear source function for general composite dual-porosity is derived, and the solution of the MFH well in a composite gas reservoir is obtained with the numerical discrete method. Through the Stehfest numerical algorithm and Gauss elimination method, the transient pressure responses for well producing at a constant production rate and the production rate vs. time for constant bottomhole pressure are analyzed. The effects of related parameters such as natural permeability and radial of the SRV region, formation permeability and interporosity coefficient on transient pressure and production performance are analyzed as well. The presented model and obtained results in this paper not only enrich the well testing models of such unconventional reservoir, but also can use to interpret on-site data which have significance on efficient reservoir development.

International audience; This paper is concerned with the uniqueness on two inverse moving source problems in electrody-namics with partial boundary data. We show that (1) if the temporal source function is compactly supported, then the spatial source profile function or the orbit function can be uniquely determined by the tangential trace of the electric field measured on part of a sphere; (2) if the temporal function is given by a Dirac distribution, then the impulsive time point and the source location can be uniquely determined at four receivers on a sphere.

The January 2011 calibration explosions at the Sayarim Military Range in the Negev Desert, Israel, provide a unique dataset to study seismic and acoustic partitioning for surface sources at near‐source and local distances. We present an analysis of the seismic and overpressure/acoustic signals generated from two explosions, which included 10,240 and 102,080 kg of mainly ANFO with some Composition B, detonated on the surface, on 24 and 26 January, respectively, at different times of the day. A temporary seismo‐acoustic‐overpressure network was deployed at distances between 0.1 and 39 km to supplement data from permanent stations in Israel. The near‐source data confirm the explosions produced overpressure signals consistent with complete and simultaneous surface detonation of 7.4 and 76.8 metric tons of TNT‐equivalent explosives. We observe acoustic amplification at local distances (>5 km) south of the larger explosion that can be explained by a low‐altitude southward flowing wind at the detonation time. For the smaller shot, the southward flow was not present and amplification was not observed in the pressure data. We present results of modeling these overpressure data with blast pressure scaling models with and without meteorological data. Seismic phases generated by the surface shots include P waves, fundamental and higher mode Rayleigh waves, and Love waves that appear to have been generated at or very near the explosion source. The seismic ground motion is less than would be expected for fully coupled explosions of 7.6 and 76.8 tons, and at near‐source distances (

In this paper, we are going to consider the following problem (1) Δ u = k 2 u + f ( x , y ) , ( x , y ) ∈ [ 0 , π ] × [ 0 , 1 ] , u ( 0 , y ) = u ( π , y ) = 0 , 0 ⩽ y ⩽ 1 , u y ( x , 0 ) = g ( x ) , 0 ⩽ x ⩽ π , u ( x , 0 ) = φ ( x ) , 0 ⩽ x ⩽ π with corresponding measured data functions ( φ e , g e ) , a given function f is known as the “source function”. We want to determine the solution u ( · , y ) for 0 y ⩽ 1 from the Cauchy data ( u , u y ) at y = 0 . The problem is ill-posed, as the solution exhibits unstable dependence on the given data functions. The aim of this paper is to introduce new efficient regularization methods, such as truncation of high frequency and quasi-boundary-type methods, with explicit error estimates for an extended case (i.e. the inhomogeneous problem with f ≠ 0 in Eq. (1) ). In addition, we also carry out numerical experiments and compare numerical results of our methods with Qin and Wei’s methods [1] in homogeneous case, as well as to compare our numerical solutions with exact solutions in inhomogeneous case. It shows that our truncation and Qin and Wei’s truncation methods giving almost same results. However, our quasi-boundary-type methods show a better results than quasi-reversibility method of Qin and Wei in term of error estimation and convergence speed.

We present in this paper the convergence properties of Jacobi spectral collocation method when used to approximate the solution of multidimensional nonlinear Volterra integral equation. The solution is sufficiently smooth while the source function and the kernel function are smooth. We choose the Jacobi–Gauss points associated with the multidimensional Jacobi weight function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega ({\mathbf{x}})=\Pi _{i=1}^d(1-x_i)^\alpha (1+x_i)^\beta ,\; -1

Consider the scattering of the two- or three-dimensional Helmholtz equation where the source of the electric current density is assumed to be compactly supported in a ball. This paper concerns the stability analysis of the inverse source scattering problem which is to reconstruct the source function. Our results show that increasing stability can be obtained for the inverse problem by using only the Dirichlet boundary data with multi-frequencies., arXiv admin note: text overlap with arXiv:1607.06677

In this paper we propose a novel approach to the problem of model-based source identification in steady-state transport phenomena given a set of noisy measurements. We formulate the problem as an optimization problem in function space and utilize the adjoint method to calculate the gradient. To obtain a finite dimensional representation of this problem we employ proper orthogonal decomposition, which provides a small number of basis functions that best approximate the function space in which the concentration function lives. Similarly, we parametrize the source function by nonlinear tower functions, which allow us to reduce the size of the problem from thousands of unknowns to a handful of variables. The above approximations give rise to a low dimensional nonlinear optimization problem, for which we provide explicit expressions for the gradient and Hessian that can be used with available optimization techniques to solve for the desired source function. We provide simulation results that demonstrate a drastic reduction in computation time. At the same time we are able to solve complex advection-diffusion problems in non-convex environments.

Synchrotron emission and absorption determine the observational appearance of many astronomical systems. In this paper, we describe a numerical scheme for calculating synchrotron emissivities and absorptivities in all four Stokes parameters for arbitrary gyrotropic electron distribution functions, building on earlier work by Leung, Gammie, and Noble. We use this technique to evaluate the emissivities and the absorptivities for a thermal (Maxwell-J\"uttner), isotropic power-law, and isotropic kappa distribution function. The latter contains a power-law tail at high particle energies that smoothly merges with a thermal core at low energies, as is characteristic of observed particle spectra in collisionless plasmas. We provide fitting formulae and error bounds on the fitting formulae for use in codes that solve the radiative transfer equation. The numerical method and the fitting formulae are implemented in a compact C library called ${\tt symphony}$. We find that: the kappa distribution has a source function that is indistinguishable from a thermal spectrum at low frequencies and transitions to the characteristic self-absorbed synchrotron spectrum, $\propto \nu^{5/2}$, at high frequency; the linear polarization fraction for a thermal spectrum is near unity at high frequency; and all distributions produce $O(10\%)$ circular polarization at low frequency for lines of sight sufficiently close to the magnetic field vector., Comment: Accepted for publication in ApJ

The Legendre spectral Galerkin method for Volterra integral equations of the second kind with a weakly singular kernel is proposed in this paper. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors (in the L2 -norm and the L∞ -norm ) will decay exponentially provided that the source function is sufficiently smooth. Numerical examples are given to illustrate the theoretical results.

Linear response laws and causality (the effect cannot precede the cause) are of fundamental importance in physics. In the context of classical electrodynamics, students often have a difficult time grasping these concepts because the physics is obscured by the intermingling of the time and frequency domains. In this paper, we analyse the linear response laws and causality in the time and frequency domains with the aim of pedagogical clarity. We will show that it is easy to violate causality in the frequency domain by making a vanishing absorption approximation. Furthermore, we will show that there can be subtle differences between Fourier transforming Maxwell equations and using a monochromatic source function. We discuss how these concepts can be obscured and offer some suggestions to improve the situation.

Imaging analysis of two-pion interferometry is performed for an evolving particle-emitting source in heavy ion collisions at HIRFL-CSR energy. The source evolution is described by the relativistic hydrodynamics in (2+1) dimensions. The model-independent characteristic quantities of the source are investigated and compared with the interferometry results obtained by the usual Gaussian formula fit. It is found that the first-order source function moments can describe the source sizes. The ratio of the normalized standard deviation to the first-order moment , /, is sensitive to the shape of the source function.

This work is to provide spectral and pseudo-spectral Jacobi-Galerkin approaches for the second kind Volterra integral equation. The Gauss-Legendre quadrature formula is used to approximate the integral operator and the inner product based on the Jacobi weight is implemented in the weak formulation in the numerical implementation. For some spectral and pseudo-spectral Jacobi-Galerkin methods, a rigorous error analysis in both the infinity and weighted norms is given provided that both the kernel function and the source function are sufficiently smooth. Numerical experiments validate the theoretical prediction.

This paper is devoted to high-order numerical time integration of first-order wave equation systems originating from spatial discretization of Maxwell's equations. The focus lies on the accuracy of high-order composition in the presence of source functions. Source functions are known to generate order reduction, and this is most severe for high-order methods. For two methods based on two well-known fourth-order symmetric compositions, convergence results are given assuming simultaneous space-time grid refinement. Herewith physical sources and source functions emanating from Dirichlet boundary conditions are distinguished. Among other things it is shown that the reduction can cost two orders. On the other hand, when a certain perturbation of a source function is used, the reduction is generally diminished by one order. In that case, reduction is absent for physical sources and for Dirichlet sources the order is equal to at least three under stable simultaneous space-time grid refinement.

Three-dimensional source images for mid-rapidity, low transverse momentum kaon and pion pairs have been extracted from central Au + Au collisions data at √sNN = 200 GeV by the STAR experiment at RHIC. The pion source function displays significant non-Gaussian features implying a finite pion emission duration. On the other hand, the kaon source function is essentially Gaussian, consistent with instantaneous emission from the fireball.

In this paper, we analyze the physical properties of dark mottles in the chromospheric network using two-dimensional spectroscopic observations in Hα obtained with the Gottingen Fabry-Perot Spectrometer in the Vacuum Tower Telescope at the Observatory del Teide, Tenerife. Cloud modeling was applied to measure the mottles' optical thickness, source function, Doppler width, and line-of-sight velocity. Using these measurements, the number density of hydrogen atoms in levels 1 and 2, total particle density, electron density, temperature, gas pressure, and mass density parameters were determined with the method of Tsiropoula & Schmieder (1997). We also analyzed the temporal behaviour of a mottle using cloud parameters. Our result shows that it is dominated by 3 minute signals in source function, and 5 minutes or more in velocity (© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this paper we consider a class of nonlinear singular boundary value problems − ( x α y ′ ( x ) ) ′ + x α f ( x , y ( x ) , x α y ′ ( x ) ) = 0 , 0 x 1 , y ′ ( 0 ) = y ′ ( 1 ) = 0 , for α ≥ 1 . We assume that the source function f ( x , y , x α y ′ ) is Lipschitz in x α y ′ and one-sided Lipschitz in y . The initial approximations are an upper solution u 0 ( x ) and a lower solution v 0 ( x ) which can be ordered in one way, v 0 ( x ) ≤ u 0 ( x ) , or the other, u 0 ( x ) ≤ v 0 ( x ) . We propose an iterative scheme and establish the existence of solutions bounded by v 0 and u 0 , and allow ∂ f / ∂ y to take both positive and negative values. The method is constructive in nature and can be used to generate solutions of the nonlinear singular boundary value problems.

This work is to provide general spectral and pseudo-spectral Jacobi-Petrov- Galerkin approaches for the second kind Volterra integro-differential equations. The Gauss-Legendre quadrature formula is used to approximate the integral operator and the inner product based on the Jacobi weight is implemented in the weak formulation in the numerical implementation. For some spectral and pseudo-spectral Jacobi-Petrov- Galerkin methods, a rigorous error analysis in both L 2 !�,� and L ∞ norms is given pro- vided that both the kernel function and the source function are sufficiently smooth. Numerical experiments validate the theoretical prediction.

Many processes in seismic data analysis and seismic imaging can be identified with solution operators of evolution equations. These include data downward continuation and velocity continuation. We have addressed the question of whether isochrons defined by imaging operators can be identified with wavefronts of solutions to an evolution equation. Rays associated with this equation then would provide a natural way of implementing prestack map migration. Assuming absence of caustics, we have developed constructive proof of the existence of a Hamiltonian describing propagation of isochrons in the context of common-offset depth migration. In the presence of caustics, one should recast to a sinking-survey migration framework. By manipulating the double-square-root operator, we obtain an evolution equation that describes sinking-survey migration as a propagation in two-way time with surface data being a source function. This formulation can be viewed as an extension of the exploding reflector concept from zero-offset to sinking-survey migration. The corresponding Hamiltonian describes propagation of extended isochrons (fronts with constant two-way time) connected by extended isochron rays. The term extended reflects the fact that two-way time propagation now takes place in high-dimensional space with the following coordinates: subsurface midpoint, subsurface offset, and depth. Extended isochron rays can be used in a natural manner for implementing sinking-survey migration in a map-migration fashion.

Let H be a Hilbert space with the inner product and the norm , a positive self-adjoint unbounded time-dependent operator on H and . We establish stability estimates of Holder type and propose a regularization method with error estimates of Holder type for the ill-posed backward semi-linear parabolic equation with the source function f satisfying a local Lipschitz condition.

Many important physical phenomena are described by wave or diffusion-wave type equations. Recent work has shown that a transform domain signal description from linear system theory can give meaningful insight to multi-dimensional wave fields. In N. Baddour [AIP Adv. 1, 022120 (2011)], certain results were derived that are mathematically useful for the inversion of multi-dimensional Fourier transforms, but more importantly provide useful insight into how source functions are related to the resulting wave field. In this short addendum to that work, it is shown that these results can be applied with a Gaussian source function, which is often useful for modelling various physical phenomena.

Context. In the past, chromospheric features were mostly studied by analyzing observations performed along the H α profile, but several aspects related to their formation and dynamics remained uncertain and poorly understood. Recently, new investigations have been carried out using data obtained along the Ca II line at 854.21 nm, providing new inputs for clarification of some of these aspects. Aims. In order to give a further contribution to the comprehension of the physical processes occurring in small-scale chromospheric features, we analyzed high spatial and temporal resolution images, acquired along the Ca II (λ = 854.21 nm) line with the Interferometric Bidimensional Spectrometer (IBIS). We studied four chromospheric structures such as mottles and arches belonging to an arch filament system (AFS) inside a bipolar region, observed on October 3, 2006. Methods. We evaluated the plasma velocity along the line of sight (LOS) using two methods: the Doppler shift of the centroid of the line profile and the cloud model. Also, we deduced the mean temperature, the Doppler width, the optical thickness and the source function in the structures to which we could apply the cloud model. Results. The pattern of the LOS velocity in the four mottles showed different behaviors. A mottle, initially, showed positive and negative velocities in eastern and western endpoints, respectively, then the plasma motion seems to reverse over a period of about 4 mn. In another mottle a motion characterized by alternate upward and downward plasma flow along the main axis was recorded. Irregular upward and downward motions along the other two mottles confirm previous results. The LOS velocities measured in the AFS, observed during the decay phase of the region, are of the same order of magnitude as those measured in short-lived active regions during their emergence phase. Conclusions. The observations carried out in the Ca II line allowed us to obtain information on small-scale magnetic features, like mottles and AFS, observed in the chromosphere. These results, on one hand, confirm previous results obtained using data acquired in the H α line, and on the other hand, provide new clues to the dynamic similarities between mottles and dynamic fibrils. Moreover, this study allowed us to single out the presence of an AFS during a phase characterized by decreasing magnetic flux and the approach of the opposite polarities.

Making use of the maximum entropy method, we study the most probable source function in heavy ion collisions. An anisotropic Gaussian source is deduced by simply assuming that the particles are emitted within a finite proper-time. The general relations between the most probable source function and the minimal assumptions are discussed, which are instructive in constructing a self-consistent source function from observed Hanbury–Brown/Twiss(HBT) correlations.