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Автор: Grafiati

Опубліковано: 7 серпня 2022

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1

Meher, Mehrollah, and Davood Rostamy. "Hybrid of differential quadrature and sub-gradients methods for solving the system of Eikonal equations." Nonlinear Engineering 10, no. 1 (January 1, 2021): 436–49. http://dx.doi.org/10.1515/nleng-2021-0035.

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Abstract Many important natural phenomena of wave propagations are modeled by Eikonal equations and a variety of new methods are needed to solve them. The differential quadrature method (DQM) is an effective numerical method for solving the system of differential equations that can achieve accurate numerical results using fewer grid points and therefore requires relatively little computational effort. In this paper, we focus on the implementation of the non-smooth Eikonal optimization by using a hybrid of polynomial differential quadrature (PDQ) or Fourier differential quadrature (FDQ) method and sub-gradients idea. Our goal is to develop a new Eikonal equation system design for wave propagation equations, as well as the efficiency and accuracy of new grid points to reduce errors and compare errors in various physical equation problems, especially wave propagation equations, and achieve their convergence. We explore the accuracy and stability of the Eikonal equation system by two-dimensional and three-dimensional numerical examples and the use of three types of grid points in a comprehensive manner. This article aims to create a new and innovative solution to the non-smooth Eikonal equation system. This new method is much more efficient and effective than traditional numerical solution methods same as DQ.

2

Sava, Paul, and Sergey Fomel. "3-D traveltime computation using Huygens wavefront tracing." GEOPHYSICS 66, no. 3 (May 2001): 883–89. http://dx.doi.org/10.1190/1.1444977.

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Traveltime computation is widely used in seismic modeling, imaging, and velocity analysis. The two most commonly used methods are ray tracing and numerical solutions to the eikonal equation. Eikonal solvers are fast and robust but are limited to computing only the first‐arrival traveltimes. Ray tracing can compute multiple arrivals but lacks the robustness of eikonal solvers. We propose a robust and complete method of traveltime computation. It is based on a system of partial differential equations, which is equivalent to the eikonal equation but formulated in the ray‐coordinates system. We use a first‐order discretization scheme that is interpreted very simply in terms of the Huygens’s principle. Our explicit finite‐difference solution to the eikonal equation solved in the ray‐coordinates system delivers both computational speed and stability since we use more than one point on the current wavefront at every time step. The finite‐difference method has proven to be a robust alternative to conventional ray tracing, while being faster and having a better ability to handle rough velocity media and penetrate shadow zones.

3

Penttinen, M. "Small-Angle Scattering on a System of Magnetic and Electric Charges." International Journal of Modern Physics A 12, no. 18 (July 20, 1997): 3193–204. http://dx.doi.org/10.1142/s0217751x97001687.

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The generalization of the conformal scattering method for small-angle scattering processes involving magnetic monopoles and ordinary charges is constructed. Using this generalization we show that the introduction of magnetic charges corresponds to analytical continuation of the eikonal amplitude in the complex charge plane (the imaginary part is proportional to the magnetic charge). We calculate explicitly the eikonal amplitude for scattering on a dyon and two monopoles in terms of confluent hypergeometric functions. The singularities of the corresponding amplitudes (focal points) are studied in detail.

4

KIM, YONG JOO, and MOON HOE CHA. "NON-EIKONAL PHASE SHIFT ANALYSIS FOR 12C + 12C ELASTIC SCATTERING." International Journal of Modern Physics E 09, no. 01 (February 2000): 67–76. http://dx.doi.org/10.1142/s0218301300000052.

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We present first-order non-eikonal correction to the eikonal phase shifts for heavy ion elastic scattering based on Coulomb trajectories of colliding nuclei. It has been applied satisfactorily to elastic angular distributions of the 12 C + 12 C system at E lab = 240, 360 and 1016 MeV. The refractive oscillations observed in the elastic scattering angular distributions could be explained due to interference between the near- and far-side amplitudes. The presence of a nuclear rainbow is evidenced through classical deflection function. We have found that the first-order non-eikonal effect on the imaginary potential is important when the absorptive potential is weak and the real potential is strong.

5

Filpo, Eduardo, Jessé Costa, and Jörg Schleicher. "Image-guided ray tracing and its applications." GEOPHYSICS 86, no. 3 (April 21, 2021): U39—U47. http://dx.doi.org/10.1190/geo2020-0642.1.

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Eikonal solvers have important applications in seismic data processing and inversion, the so-called image-guided methods. To this day, in image-guided applications, the solution of the eikonal equation is implemented using partial-differential-equation solvers, such as fast-marching or fast-sweeping methods. We have found that alternatively, one can numerically integrate the dynamic Hamiltonian system defined by the image-guided eikonal equation and reconstruct the solution with image-guided rays. We evaluate interesting applications of image-guided ray tracing to seismic data processing, demonstrating the use of the resulting rays in image-guided interpolation and smoothing, well-log interpolation, image flattening, and residual-moveout picking. Some of these applications make use of properties of the ray-tracing system that are not directly obtained by eikonal solvers, such as ray position, ray density, wavefront curvature, and ray curvature. These ray properties open space for a different set of applications of the image-guided eikonal equation, beyond the original motivation of accelerating the construction of minimum distance tables. We stress that image-guided ray tracing is an embarrassingly parallel problem that makes its implementation highly efficient on massively parallel platforms. Image-guided ray tracing is advantageous for most applications involving the tracking of seismic events and imaging-guided interpolation. Our numerical experiments using synthetic and real data sets indicate the efficiency and robustness of image-guided rays for the selected applications.

6

SHUTTLEWORTH, I. G. "ANALYSIS OF THE (3 × 3)-H/Cu(111) SYSTEM USING EIKONAL-LEVEL HELIUM ATOM SCATTERING SIMULATIONS." Surface Review and Letters 14, no. 06 (December 2007): 1089–93. http://dx.doi.org/10.1142/s0218625x07010810.

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The saturated (3 × 3)- H / Cu (111) system has been investigated by Eikonal-level helium atom scattering (HAS) simulations of previously reported HAS diffraction images. Basing the investigation on an unreconstructed single-domain structure with H atoms occupying three-fold hollow adsorption sites, a systematic trial of all possible structures has been performed and a series of structures has been shown to agree with the experimental images. Comparison of this fitting process to diffraction images taken at low (30°) and high (47°) angles of incidence has shown that the Eikonal approximation is a reasonable scattering algorithm for the H / Cu (111) system.

7

KIM, YONG JOO, and MOON HOE CHA. "SECOND-ORDER EIKONAL MODEL ANALYSIS OF16O+16OELASTIC SCATTERING." International Journal of Modern Physics E 10, no. 04n05 (August 2001): 373–86. http://dx.doi.org/10.1142/s0218301301000563.

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We analyze the elastic scattering angular distributions of the16O +16O system at Elab=480 MeV and 704 MeV within the framework of the second-order eikonal model based on Coulomb trajectories of colliding nuclei. The diffractive oscillatory structure observed in the elastic angular distribution could be explained due to the interference between the near- and far-side scattering amplitudes. The presence of a nuclear rainbow in this system is evidenced through a classical deflection function. The effective optical potential is developed from the second-order non-eikonal phase shifts.

8

Peletier, Mark A., and Marco Veneroni. "Stripe patterns and a projection-valued formulation of the eikonal equation." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 370, no. 1965 (April 28, 2012): 1730–39. http://dx.doi.org/10.1098/rsta.2011.0425.

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We describe recent work on striped patterns in a system of block copolymers. A by-product of the characterization of such patterns is a new formulation of the eikonal equation. In this formulation, the unknown is a field of projection matrices of the form P = e ⊗ e , where e is a unit vector field. We describe how this formulation is better adapted to the description of striped patterns than the classical eikonal equation, and illustrate this with examples.

9

Ahdid, Rachid, Said Safi, Mohamed Fakir, and Bouzid Manaut. "Geodesic Distance on Riemannian Manifold using Jacobi Iterations in 3D Face Recognition System." International Journal of Informatics and Communication Technology (IJ-ICT) 6, no. 1 (June 22, 2017): 10. http://dx.doi.org/10.11591/ijict.v6i1.pp10-19.

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In this paper, we present an automatic application of 3D face recognition system using geodesic distance in Riemannian geometry. We consider, in this approach, the three dimensional face images as residing in Riemannian manifold and we compute the geodesic distance using the Jacobi iterations as a solution of the Eikonal equation. The problem of solving the Eikonal equation, unstructured simplified meshes of 3D face surface, such as tetrahedral and triangles are important for accurately modeling material interfaces and curved domains, which are approximations to curved surfaces in R<sup>3</sup>. In the classifying steps, we use: Neural Networks (NN), K-Nearest Neighbor (KNN) and Support Vector Machines (SVM). To test this method and evaluate its performance, a simulation series of experiments were performed on 3D Shape REtrieval Contest 2008 database (SHREC2008).<strong></strong>

10

KIM, YONG JOO, and MOON HOE CHA. "COULOMB-MODIFIED EIKONAL PHASE SHIFT ANALYSIS BASED ON HYPERBOLIC TRAJECTORY FOR 12C + 12C ELASTIC SCATTERINGS." International Journal of Modern Physics E 13, no. 02 (April 2004): 439–50. http://dx.doi.org/10.1142/s0218301304002296.

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We present a Coulomb-modified eikonal model formalism based on hyperbolic trajectory for heavy-ion elastic scattering. This formalism has been applied satisfactorily to elastic scatterings of the 12 C + 12 C system at E lab =240, 360 and 1016 MeV. The presence of a nuclear rainbow in this system is evidenced through a classical deflection function. The Fraunhöfer oscillations observed in the elastic angular distributions can be explained due to interference between the near- and far-side amplitudes. We have found that the hyperbolic trajectory effect on the eikonal model is important when the absorptive potential is weak and the real potential is strong.

11

Sethian, James A., and A. Mihai Popovici. "3-D traveltime computation using the fast marching method." GEOPHYSICS 64, no. 2 (March 1999): 516–23. http://dx.doi.org/10.1190/1.1444558.

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We present a fast algorithm for solving the eikonal equation in three dimensions, based on the fast marching method. The algorithm is of the order O(N log N), where N is the total number of grid points in the computational domain. The algorithm can be used in any orthogonal coordinate system and globally constructs the solution to the eikonal equation for each point in the coordinate domain. The method is unconditionally stable and constructs solutions consistent with the exact solution for arbitrarily large gradient jumps in velocity. In addition, the method resolves any overturning propagation wavefronts. We begin with the mathematical foundation for solving the eikonal equation using the fast marching method and follow with the numerical details. We then show examples of traveltime propagation through the SEG/EAGE salt model using point‐source and plane‐wave initial conditions and analyze the error in constant velocity media. The algorithm allows for any shape of the initial wavefront. While a point source is the most commonly used initial condition, initial plane waves can be used for controlled illumination or for downward continuation of the traveltime field from one depth to another or from a topographic depth surface to another. The algorithm presented here is designed for computing first‐arrival traveltimes. Nonetheless, since it exploits the fast marching method for solving the eikonal equation, we believe it is the fastest of all possible consistent schemes to compute first arrivals.

12

KIM, YONG JOO, and MOON HOE CHA. "COULOMB-MODIFIED EIKONAL MODEL ANALYSIS OF REFRACTIVE12C+12CELASTIC SCATTERING AT Elab= 240 AND 360MeV." International Journal of Modern Physics E 11, no. 03 (June 2002): 211–19. http://dx.doi.org/10.1142/s0218301302000764.

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We present the Coulomb-modified eikonal model by using a tangential velocity at the distance of closest approach. The Fraunhöfer oscillations observed in the elastic angular distributions of the12C +12C system at Elab= 240 and 360 MeV can be explained due to the strong interference between the near- and far-side scattering amplitudes. We have found that the strongly real and weakly imaginary potentials are required to describe the refractive12C +12C elastic scatterings at Elab= 240 and 360 MeV. The refractive pattern, dominated by the far-side component of the scattering amplitude, can be shown to be sensitive to the real part of optical potential at small radius. Introducing the tangential velocity, it is shown that the Coulomb-modified eikonal model satisfactorily reproduces experimental data concerning the refractive pattern in the angular distributions of the12C +12C system at Elab= 240 and 360 MeV.

13

Kubera, P., and J. Felcman. "On a numerical flux for the pedestrian flow equations." Journal of Applied Mathematics, Statistics and Informatics 11, no. 2 (December 1, 2015): 79–96. http://dx.doi.org/10.1515/jamsi-2015-0014.

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Abstract The pedestrian flow equations are formulated as the hyperbolic problem with a source term, completed by the eikonal equation yielding the desired direction of the pedestrian velocity. The operator splitting consisting of successive discretization of the eikonal equation, ordinary differential equation with the right hand side being the source term and the homogeneous hyperbolic system is proposed. The numerical flux of the Vijayasundaram type is proposed for the finite volume solution of the hyperbolic problem. The Vijayasundaram numerical flux, originally proposed for the hyperbolic problems possessing the homogeneity property is extended for pedestrian flow, where the homogeneity property is lost. The application of the proposed numerical flux is demonstrated on the physically relevant problem.

14

Hu, Jiangtao, Junxing Cao, Huazhong Wang, Shaoyong Liu, and Xingjian Wang. "3D traveltime computation for quasi-P-wave in orthorhombic media using dynamic programming." GEOPHYSICS 83, no. 1 (January 1, 2018): C27—C35. http://dx.doi.org/10.1190/geo2016-0558.1.

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A fractured area, such as a fault area, usually induces orthorhombic anisotropy. Ignoring orthorhombic anisotropy may degrade the subsurface image by creating a well mistie and blurring the image. Traveltime computation is essential for many processing techniques, such as depth imaging and tomography. Solving the ray-tracing system and eikonal equation are two popular methods for traveltime computation in isotropic media. However, because the ray-tracing system becomes complex and the eikonal equation becomes highly nonlinear, their applications in orthorhombic media become complex. We have developed an alternative 3D traveltime computation method in orthorhombic media based on dynamic programming. To avoid solving the complex ray-tracing system and nonlinear eikonal equation, it adopts an explicitly expressed group velocity from the moveout approximation to describe the propagation of the wavepath and computes the traveltime by Fermat’s principle. Similar to depth extrapolation, it computes the traveltime from one depth to the next depth and does not suffer from a shadow zone. Besides, three strategies of traveltime computation are proposed to deal with different geologic scenarios. Because classic dynamic programming (i.e., the first strategy) computes all possible wavepaths (i.e., 24 wavepaths) across one spatial location, it may be computationally intensive. Based on the idea of wavefield decomposition (e.g., upgoing and downgoing), the second and third strategies simplify the traveltime computation and reduce the computational cost. Numerical examples on the vertical and tilted orthorhombic models indicate that the traveltime contour obtained by our method matches well with the wavefront extrapolated from the wave equation. Our method can be applied in depth imaging and tomography.

15

Zhdanov, R. Z. "Compatibility criteria and general solution of the nonlinear d'Alembert-Eikonal system." Reports on Mathematical Physics 36, no. 2-3 (October 1995): 483–88. http://dx.doi.org/10.1016/0034-4877(96)83641-x.

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16

Venetskiy, A. S., and V. A. Kaloshin. "On eikonal distribution in the aperture of a two-mirror telescopic system." Journal of Communications Technology and Electronics 57, no. 9 (September 2012): 1016–23. http://dx.doi.org/10.1134/s1064226912090124.

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17

Etikyala, R., S. Göttlich, A. Klar, and S. Tiwari. "Particle methods for pedestrian flow models: From microscopic to nonlocal continuum models." Mathematical Models and Methods in Applied Sciences 24, no. 12 (August 15, 2014): 2503–23. http://dx.doi.org/10.1142/s0218202514500274.

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A hierarchy of models for pedestrian flow is numerically investigated using particle methods. It includes microscopic models based on interacting particle system coupled to an eikonal equation, hydrodynamic models using equations for density and mean velocity, nonlocal continuum equations for the density and diffusive Hughes equations. Particle methods are used on all levels of the hierarchy. Numerical test cases are investigated by comparing the above models.

18

LIU, YU-LIANG. "EIGEN-FUNCTIONAL METHOD AND EIKONAL-TYPE EQUATIONS IN ONE-DIMENSIONAL STRONGLY CORRELATED ELECTRON SYSTEMS." International Journal of Modern Physics B 16, no. 16 (June 30, 2002): 2201–20. http://dx.doi.org/10.1142/s0217979202010385.

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Using eigen-functional method, we study one-dimensional strongly correlated electron systems with large momentum (2k F and/or 4k F ) transfer term(s), and demonstrate that this kind of problems ends in to solve the Eikonal-type equations, and these equations are universal, and independent of whether or not the system is integrable. In contrast to usual perturbation theory, this method is valid not only for weak electron interaction, but also for strong electron interaction. Comparing with exact solution of some integrable models, it can give correct results in linearization approximation. This method can also be used to study electron-phonon interaction systems, and two coupled spin chain (or quantum wire) systems.

19

Zabotin, Nikolay A., Oleg A. Godin, Paul C. Sava, and Liudmila Y. Zabotina. "Tracing Three-Dimensional Acoustic Wavefronts in Inhomogeneous, Moving Media." Journal of Computational Acoustics 22, no. 02 (April 17, 2014): 1450002. http://dx.doi.org/10.1142/s0218396x14500027.

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We present a numerical implementation of an alternative formulation of the geometrical, or ray, acoustics, where wavefronts rather than rays are the primary objects. Rays are recovered as a by-product of wavefront tracing. The alternative formulation of the geometrical acoustics is motivated, first, by the observation that wavefronts are often more stable than rays at long-range sound propagation, and, second, by a need for computationally efficient modeling of high-frequency acoustic fields in three-dimensionally inhomogeneous, moving or motionless fluids. Wavefronts are found as a finite-difference solution to a system of partial differential equations, which is equivalent to the eikonal equation and is a direct implementation of the intuitive Huygens' wavefront construction. The finite-difference algorithm is an extension of the approach originally developed in the framework of an open source Madagascar project. Benchmark problems, which admit exact, analytic solutions of the eikonal equation, are formulated and utilized to verify the finite-difference wavefront tracing algorithm. Huygens' wavefront tracing (HWT) is applied to modeling sound propagation in three-dimensionally inhomogeneous ocean and atmosphere.

20

Rong, Bosheng, Hui Zhao, Shaohua Cui, and Cuiping Zhang. "Continuum Dynamic Traffic Assignment Model for Autonomous Vehicles in a Polycentric Urban City with Environmental Consideration." Mathematical Problems in Engineering 2018 (November 13, 2018): 1–15. http://dx.doi.org/10.1155/2018/8345979.

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This paper proposed a continuum dynamic model for autonomous vehicles in a polycentric urban city by considering the environment impact of traffic emission. The model assumes that homogeneous autonomous vehicles are continuously distributed over the urban areas which tend to choose a path to minimize their total travel cost from origin to destination. To describe the path choice behavior of travelers, we presented the continuum dynamic traffic assignment model which consists of a two-dimensional hyperbolic system of nonlinear conservation laws with source terms and an Eikonal-type equation. The elastic demand is considered using a function which associating each copy of flow with its total instantaneous travel cost. For the environmental impacts, here we consider the influence of CO emission and include the cost of emission into the actual transportation cost. A solution algorithm for the model is designed as a cell-centered finite volume method for conservation law equations and a fast sweeping method for Eikonal-type equations on unstructured grids. Numerical examples are given to demonstrate the model and the proposed solution algorithm. Further, the results of the travel cost considering CO emissions and not considering CO emissions are compared.

21

Guo, Gaoshan, Haiqiang Lan, Xiaole Zhou, Youshan Liu, Umair Bin Waheed, and Jingyi Chen. "Topography-dependent eikonal tomography based on the fast-sweeping scheme and the adjoint-state technique." GEOPHYSICS 87, no. 2 (December 27, 2021): U29—U41. http://dx.doi.org/10.1190/geo2021-0116.1.

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First-arrival traveltime tomography has been widely used for upper crustal velocity modeling, but it usually suffers from the problem of complex surface topography. To overcome this problem, we have developed a new topography-dependent eikonal tomography scheme that combines a developed accurate and efficient traveltime modeling method and introduces a flexible and robust adjoint inversion scheme in the presence of irregular topography. A surface-flattening scheme is used to handle the irregular surface, where the real model is discretized by curvilinear grids and the irregular free surface is mathematically flattened through the transformation from Cartesian to curvilinear coordinates. Based on this parameterization, the forward traveltime modeling is conducted by a monotone fast-sweeping method that discretizes the factored topography-dependent eikonal equation with a point-source condition. This algorithm can circumvent the source-singularity problem and decrease the numerical error in the vicinity of a point source in the curvilinear system. Then, the gradient-based inversion is used to minimize the misfit function, which is achieved by a matrix-free adjoint-state method without cumbersome ray tracing and explicit estimation of the Fréchet derivative matrix in the curvilinear coordinate system. The new tomographic scheme is evaluated through numerical examples with different seismic structures with complex topography, and then applied to a wide-angle profile acquired in the northeastern Tibetan Plateau. The results validate the effectiveness and efficiency of our tomography scheme in constructing shallow crustal velocity models with irregular topography.

22

Dacorogna, Bernard, Roland Glowinski, and Tsorng-Whay Pan. "Numerical methods for the solution of a system of Eikonal equations with Dirichlet boundary conditions." Comptes Rendus Mathematique 336, no. 6 (March 2003): 511–18. http://dx.doi.org/10.1016/s1631-073x(03)00024-4.

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23

Zhdanov, Renat Z. "On Integration of the Nonlinear d’Alembert-Eikonal System and Conditional Symmetry of Nonlinear Wave Equations." Journal of Nonlinear Mathematical Physics 4, no. 1-2 (January 1997): 49–61. http://dx.doi.org/10.2991/jnmp.1997.4.1-2.6.

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24

Yehorchenko, Irina. "Solutions of the system of d'Alembert and eikonal equations, and classification of reductions of PDEs." Journal of Physics: Conference Series 621 (June 11, 2015): 012018. http://dx.doi.org/10.1088/1742-6596/621/1/012018.

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25

Tew, R. H. "Diffraction of sound by a surface inhomogeneity at a fluid-solid interface." European Journal of Applied Mathematics 3, no. 2 (June 1992): 115–45. http://dx.doi.org/10.1017/s0956792500000747.

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The diffraction of a plane sound wave in a fluid by an adjacent elastic solid containing a surface flaw is analysed using ray techniques. By solving the eikonal equation with suitable boundary data, the pattern of the rays leaving the boundary and propagating into the fluid and solid respectively is established, with the corresponding amplitudes being furnished by the appropriate system of transport equations. For the acoustic and elastic cylindrical bulk waves that emanate from the flaw itself, the amplitude directivities cannot be found from this ray analysis alone.

26

Müller, L., A. Meurer, F. Schneider, and A. Klar. "A numerical investigation of flux-limited approximations for pedestrian dynamics." Mathematical Models and Methods in Applied Sciences 27, no. 06 (April 11, 2017): 1177–97. http://dx.doi.org/10.1142/s0218202517400127.

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A hierarchy of models for pedestrian flow with fixed speed is numerically investigated. The starting point is a microscopic model based on a stochastic interacting particle system coupled to an eikonal equation. Starting from this model a nonlocal and nonlinear flux-limited maximum-entropy equation for density and mean velocity is derived via a mean field kinetic equation. Finally, associated classical scalar equations for the density are considered for comparison. These models are compared to each other for different test cases showing the superiority of the flux-limited approach, in particular for situations with smaller values of the stochastic noise.

27

ZABOTIN, NIKOLAY A., OLEG A. GODIN, PAUL C. SAVA, and LIUDMILA Y. ZABOTINA. "ACOUSTIC WAVEFRONT TRACING IN INHOMOGENEOUS, MOVING MEDIA." Journal of Computational Acoustics 20, no. 03 (September 2012): 1250009. http://dx.doi.org/10.1142/s0218396x12500099.

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We extend the Huygens wavefront tracing algorithm, which is a part of an open source Madagascar project, to sound propagation in inhomogeneous, moving media and apply it to a series of benchmark tasks. One set of tasks admits exact analytic solutions and serves the purpose of validation of the new algorithm. Another set of calculations demonstrates applicability of the algorithm to the studies of wavefront dynamics and stability in ocean and atmospheric acoustics. The method is based on a system of differential equations equivalent to the eikonal equation, but formulated in the ray coordinate system. In this paper, we present a first-order, two-dimensional discretization scheme that is interpreted very simply in terms of the Huygens' principle. The method has proved to be a convenient alternative to conventional ray tracing.

28

KIM, YONG JOO, and MOON HOE CHA. "EIKONAL MODEL ANALYSIS FOR 9Li+p AND 11Li+p ELASTIC SCATTERINGS AND HALO EFFECT." International Journal of Modern Physics E 10, no. 02 (April 2001): 91–98. http://dx.doi.org/10.1142/s0218301301000435.

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The elastic angular distributions for 9 Li+p and 11 Li+p at E lab /A = 60 and 62 MeV/nucleon, respectively, have been analyzed within the framework of the eikonal model. It is assumed that 11 Li has the halo structure of a 9 Li core plus two weakly bound neutrons and is treated such that the nuclear potential has a sum of the core and halo neutron contributions. We have found that the break-up effect of halo neutrons is essential to understand the elastic cross sections of 11 Li+p system at E lab /A = 62 MeV/nucleon. The near- and the far-side decompositions of elastic cross sections have also been performed by following the Fuller's formalism.

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KIM, YONG JOO, and MOON HOE CHA. "TANGENTIAL VELOCITY AND HALO EFFECTS IN THE 9Li+p AND 11Li+p ELASTIC SCATTERINGS." International Journal of Modern Physics E 14, no. 07 (October 2005): 1051–61. http://dx.doi.org/10.1142/s0218301305003727.

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We present the Coulomb-modified eikonal model formalism based on a hyperbolic trajectory by using a tangential velocity at the distance of closest approach. By assuming that 11 Li has a halo structure of two weakly bound neutrons around the 9 Li core, we take the total nuclear optical potential as a sum of the core and the two halo neutrons. The theoretical calculations for the elastic scattering of 9 Li + p and 11 Li + p at E lab /A=60 and 62 MeV, respectively are in good agreements with the observed data. It is found that the break-up effect of halo neutrons and the tangential velocity at the distance of closest approach are both important to understand the elastic cross sections of 11 Li + p system at E lab /A=62 MeV .

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Kim, Yong Joo, and Moon Hoe Cha. "Tangential Velocity Corrections to a Second-Order Coulomb-Modified Eikonal Model for Heavy-Ion Elastic Scattering." International Journal of Modern Physics E 12, no. 04 (August 2003): 479–92. http://dx.doi.org/10.1142/s0218301303001399.

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We present the tangential velocity corrections to the second-order Coulomb-modified eikonal model at the distance of closest approach. It has been applied to elastic angular distributions of the 16 O +16 O system at E lab =350 and 480 MeV. The calculated results with tangential velocity show better agreements with the experimental data compared to those with asymptotic velocity. The Fraunhöfer oscillations observed in the elastic angular distributions can be explained by the strong interference between the near- and the far-side amplitudes. Airy structures can be shown by reducing the effective imaginary potential strength. It is found that the Airy minimum is more visible as the effective imaginary potential strength is reduced. Deep real potentials associated with rather weak imaginary ones are found to be essential to describe the refractive 16 O +16 O elastic scatterings at E lab =350 and 480 MeV.

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Abdul Salam, Parveena Shamim, Wolfgang Bock, Axel Klar, and Sudarshan Tiwari. "Disease contagion models coupled to crowd motion and mesh-free simulation." Mathematical Models and Methods in Applied Sciences 31, no. 06 (April 9, 2021): 1277–95. http://dx.doi.org/10.1142/s0218202521400066.

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Modeling and simulation of disease spreading in pedestrian crowds have recently become a topic of increasing relevance. In this paper, we consider the influence of the crowd motion in a complex dynamical environment on the course of infection of the pedestrians. To model the pedestrian dynamics, we consider a kinetic equation for multi-group pedestrian flow based on a social force model coupled with an Eikonal equation. This model is coupled with a non-local SEIS contagion model for disease spread, where besides the description of local contacts, the influence of contact times has also been modeled. Hydrodynamic approximations of the coupled system are derived. Finally, simulations of the hydrodynamic model are carried out using a mesh-free particle method. Different numerical test cases are investigated, including uni- and bi-directional flow in a passage with and without obstacles.

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Sevastianov, Anton L. "Asymptotic method for constructing a model of adiabatic guided modes of smoothly irregular integrated optical waveguides." Discrete and Continuous Models and Applied Computational Science 28, no. 3 (December 15, 2020): 252–73. http://dx.doi.org/10.22363/2658-4670-2020-28-3-252-273.

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The paper considers a class of smoothly irregular integrated optical multilayer waveguides, whose properties determine the characteristic features of guided propagation of monochromatic polarized light. An asymptotic approach to the description of such electromagnetic radiation is proposed, in which the solutions of Maxwells equations are expressed in terms of the solutions of a system of four ordinary differential equations and two algebraic equations for six components of the electromagnetic field in the zero approximation. The gradient of the phase front of the adiabatic guided mode satisfies the eikonal equation with respect to the effective refractive index of the waveguide for the given mode.The multilayer structure of waveguides allows one more stage of reducing the model to a homogeneous system of linear algebraic equations, the nontrivial solvability condition of which specifies the relationship between the gradient of the radiation phase front and the gradients of interfaces between thin homogeneous layers.In the final part of the work, eigenvalue and eigenvector problems (differential and algebraic), describing adiabatic guided modes are formulated. The formulation of the problem of describing the single-mode propagation of adiabatic guided modes is also given, emphasizing the adiabatic nature of the described approximate solution of Maxwells equations.

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Kuziv, Yaroslav Yu. "Software for the numerical solution of first-order partial differential equations." Discrete and Continuous Models and Applied Computational Science 27, no. 1 (December 15, 2019): 42–48. http://dx.doi.org/10.22363/2658-4670-2019-27-1-42-48.

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Partial differential equations of the first order, arising in applied problems of optics and optoelectronics, often contain coefficients that are not defined by a single analytical expression in the entire considered domain. For example, the eikonal equation contains the refractive index, which is described by various expressions depending on the optical properties of the media that fill the domain under consideration. This type of equations cannot be analysed by standard tools built into modern computer algebra systems, including Maple.The paper deals with the adaptation of the classical Cauchy method of integrating partial differential equations of the first order to the case when the coefficients of the equation are given by various analytical expressions in the subdomains G1, . . . , Gk , into which the considered domain is divided. In this case, it is assumed that these subdomains are specified by inequalities. This integration method is implemented as a Python program using the SymPy library. The characteristics are calculatednumerically using the Runge-Kutta method, but taking into account the change in the expressions for the coefficients of the equation when passing from one subdomain to another. The main functions of the program are described, including those that can be used to illustrate the Cauchy method. The verification was carried out by comparison with the results obtained in the Maple computer algebra system.

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Миловский, Н. Д., та А. О. Климин. "Геометрическая оптика во вращающемся диэлектрике". Журнал технической физики 128, № 12 (2020): 1958. http://dx.doi.org/10.21883/os.2020.12.50335.150-20.

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Rotating solid-state homogeneous isotropic dielectric without dispersion in the accompanying rotation of the reference frame turns out to be an inhomogeneous anisotropic medium due to the influence of two competing physical mechanisms: the inhomogeneity of free space caused by rotation and entrainment of light by the moving medium. In a rotating dielectric in the geometric optics approximation the eikonal equation and the corresponding system of ordinary differential equations in characteristic form are obtained. The solutions of the equations with the calculated parameters determined using of the first integrals of the system are pairs "opposite “R-ray trajectories and f - phase trajectories fronts. A formula for the intensity of a light pulse propagating along an arbitrary R-trajectories was obtained. "Oncoming “trajectories of both types do not have common points and are shifted to opposite sides of a straight line between end points. Their structural parameters (minimum distance to the axis of rotation, the length of the arc, the region of determination by the azimuthal coordinate, the optical length, etc.) change under the influence of both physical mechanisms, depending on the speed of rotation. Closed optical paths in the RRF for the relay network and for the two-mirror Fabry-Perot resonator in generating laser can be created using two types of mirrors that are adaptable to the frequency of rotation which normals to the reflecting surfaces must have certain different angles with the ray vectors and wavefronts of radiation arriving at the reflector. The Sagnac effect is a consequence inhomogeneity (deformation) of free space along the azimuthal coordinate, and its value is the result of the competing influence of both physical mechanisms.

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Esposito, Giampiero, Emmanuele Battista, and Elisabetta Di Grezia. "Bicharacteristics and Fourier integral operators in Kasner spacetime." International Journal of Geometric Methods in Modern Physics 12, no. 05 (May 2015): 1550060. http://dx.doi.org/10.1142/s0219887815500607.

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The scalar wave equation in Kasner spacetime is solved, first for a particular choice of Kasner parameters, by relating the integrand in the wave packet to the Bessel functions. An alternative integral representation is also displayed, which relies upon the method of integration in the complex domain for the solution of hyperbolic equations with variable coefficients. In order to study the propagation of wave fronts, we integrate the equations of bicharacteristics which are null geodesics, and we are able to express them, for the first time in the literature, with the help of elliptic integrals for another choice of Kasner parameters. For generic values of the three Kasner parameters, the solution of the Cauchy problem is built through a pair of integral operators, where the amplitude and phase functions in the integrand solve a coupled system of partial differential equations. The first is the so-called transport equation, whereas the second is a nonlinear equation that reduces to the Eikonal equation if the amplitude is a slowly varying function. Remarkably, the analysis of such a coupled system is proved to be equivalent to building first an auxiliary covariant vector having vanishing divergence, while all nonlinearities are mapped into solving a covariant generalization of the Ermakov–Pinney equation for the amplitude function. Last, from a linear set of equations for the gradient of the phase one recovers the phase itself. This is the parametrix construction that relies upon Fourier–Maslov integral operators, but with a novel perspective on the nonlinearities in the dispersion relation. Furthermore, the Adomian method for nonlinear partial differential equations is applied to generate a recursive scheme for the evaluation of the amplitude function in the parametrix. The resulting formulas can be used to build self-dual solutions to the field equations of noncommutative gravity, as has been shown in the recent literature.

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Bijker, R., and J. N. Ginocchio. "Eikonal scattering from complex systems." Physical Review C 45, no. 6 (June 1, 1992): 3030–33. http://dx.doi.org/10.1103/physrevc.45.3030.

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KIM, YONG JOO, JONG-KWAN WOO, and MOON HOE CHA. "ANALYTIC FIRST-ORDER EIKONAL MODEL FOR HEAVY-ION ELASTIC SCATTERINGS." International Journal of Modern Physics E 19, no. 10 (October 2010): 1947–60. http://dx.doi.org/10.1142/s0218301310016430.

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We present analytic expressions for the zero-order eikonal phase shift and its first-order correction by approximating a distance between two colliding nuclei. This formalism has been applied to elastic scatterings of the 12 C + 40 Ca and the 12 C + 90 Zr systems at E lab = 420 MeV , and the 16 O + 40 Ca and the 16 O + 90 Zr ones at E lab = 1503 MeV . The calculated angular distributions, taking into account up to the analytic first-order eikonal phase shift, are found to be in fairly good agreement with the observed data. The reaction cross-sections obtained from the present model produce very excellent agreements with ones of exact first-order eikonal model calculations. We have found that analytic eikonal phase shift including the first-order correction is one theoretical method to the analysis of heavy-ion elastic scattering.

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Gharti, Hom Nath, Volker Oye, Michael Roth, and Daniela Kühn. "Automated microearthquake location using envelope stacking and robust global optimization." GEOPHYSICS 75, no. 4 (July 2010): MA27—MA46. http://dx.doi.org/10.1190/1.3432784.

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Most earthquake location methods require phase identification and arrival-time measurements. These methods are generally fast and efficient but not always applicable to microearthquake data with low signal-to-noise ratios because the phase identification might be very difficult. The migration-based source location methods, which do not require an explicit phase identification, are often more suitable for such noisy data. Whereas some existing migration-based methods are computationally intensive, others are limited to a certain type of data or make use of only a particular phase of the signal. We have developed a migration-based source location method especially applicable to data with relatively low signal-to-noise ratios. We projected seismograms onto the ray coordinate system for each potential source-receiver configuration and subsequently computed their envelopes. The envelopes were time shifted according to synthetic P- and S-wavearrival times (computed using an eikonal solver) and stacked for a predefined time window centered on the arrival time of the corresponding phase. This was done for each component and phase individually, and the squared sum of the stacks was defined as the objective function. We applied a robust global optimization routine called differential evolution to maximize the objective function and thereby locate the seismic event. Our source location method provides a complete algorithm with only a few control parameters, making it suitable for automatic processing. We applied this method to single and multicomponent data using P and/or S phases. We conducted controlled tests using synthetic seismograms contaminated with a minimum of 30% white noise. The synthetic data were computed for a complex and heterogeneous model of the Pyhäsalmi ore mine in Finland. We also successfully applied the method to real seismic data recorded with the in-mine seismic network of the Pyhäsalmi mine.

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PELETIER, MARK A., and MARCO VENERONI. "STRIPE PATTERNS IN A MODEL FOR BLOCK COPOLYMERS." Mathematical Models and Methods in Applied Sciences 20, no. 06 (June 2010): 843–907. http://dx.doi.org/10.1142/s0218202510004465.

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We consider a pattern-forming system in two space dimensions defined by an energy [Formula: see text]. The functional [Formula: see text] models strong phase separation in AB diblock copolymer melts, and patterns are represented by {0, 1}-valued functions; the values 0 and 1 correspond to the A and B phases. The parameter ε is the ratio between the intrinsic, material length-scale and the scale of the domain Ω. We show that in the limit ε → 0 any sequence uε of patterns with uniformly bounded energy [Formula: see text] becomes stripe-like: the pattern becomes locally one-dimensional and resembles a periodic stripe pattern of periodicity O(ε). In the limit the stripes become uniform in width and increasingly straight. Our results are formulated as a convergence theorem, which states that the functional [Formula: see text] Gamma-converges to a limit functional [Formula: see text]. This limit functional is defined on fields of rank-one projections, which represent the local direction of the stripe pattern. The functional [Formula: see text] is only finite if the projection field solves a version of the Eikonal equation, and in that case it is the L2-norm of the divergence of the projection field, or equivalently the L2-norm of the curvature of the field. At the level of patterns the converging objects are the jump measures |∇uε| combined with the projection fields corresponding to the tangents to the jump set. The central inequality from Peletier and Röger, Arch. Rational Mech. Anal.193 (2009) 475–537, provides the initial estimate and leads to weak measure-function pair convergence. We obtain strong convergence by exploiting the non-intersection property of the jump set.

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Schneider, William A. "Robust and efficient upwind finite‐difference traveltime calculations in three dimensions." GEOPHYSICS 60, no. 4 (July 1995): 1108–17. http://dx.doi.org/10.1190/1.1443839.

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First‐arrival traveltimes in complicated 3-D geologic media may be computed robustly and efficiently using an upwind finite‐difference solution of the 3-D eikonal equation. An important application of this technique is computing traveltimes for imaging seismic data with 3-D prestack Kirchhoff depth migration. The method performs radial extrapolation of the three components of the slowness vector in spherical coordinates. Traveltimes are computed by numerically integrating the radial component of the slowness vector. The original finite‐difference equations are recast into unitless forms that are more stable to numerical errors. A stability condition adaptively determines the radial steps that are used to extrapolate. Computations are done in a rotated spherical coordinate system that places the small arc‐length regions of the spherical grid at the earth’s surface (z = 0 plane). This improves efficiency by placing large grid cells in the central regions of the grid where wavefields are complicated, thereby maximizing the radial steps. Adaptive gridding allows the angular grid spacings to vary with radius. The computation grid is also adaptively truncated so that it does not extend beyond the predefined Cartesian traveltime grid. This grid handling improves efficiency. The method cannot compute traveltimes corresponding to wavefronts that have “turned” so that they propagate in the negative radial direction. Such wavefronts usually represent headwaves and are not needed to image seismic data. An adaptive angular normalization prevents this turning, while allowing lower‐angle wavefront components to accurately propagate. This upwind finite‐difference method is optimal for vector‐parallel supercomputers, such as the CRAY Y-MP. A complicated velocity model that generates turned wavefronts is used to demonstrate the method’s accuracy by comparing with results that were generated by 3-D ray tracing and by an alternate traveltime calculation method. This upwind method has also proven successful in the 3-D prestack Kirchhoff depth migration of field data.

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BAYE, DANIEL. "THREE AND FOUR-BODY BREAKUP REACTIONS." International Journal of Modern Physics E 17, no. 10 (November 2008): 2301–9. http://dx.doi.org/10.1142/s0218301308011513.

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Breakup reactions provide spectroscopic information on exotic nuclei. Coulomb breakup indirectly measures the astrophysical S factor for radiative-capture reactions. The validity of first-order perturbation theory is limited for extended systems such as halo nuclei. More elaborate reaction models are necessary: semi-classical time-dependent Schrödinger equation, eikonal and dynamical eikonal approximations, continuum-discretized coupled-channel method. Breakup experiments do not provide much information on the structure of a two-cluster halo nucleus but accurate exclusive experiments should be more interesting for three-cluster nuclei.

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KOPIETZ, PETER. "BOSONIZATION AND THE EIKONAL EXPANSION: SIMILARITIES AND DIFFERENCES." International Journal of Modern Physics B 10, no. 17 (July 30, 1996): 2111–24. http://dx.doi.org/10.1142/s0217979296000969.

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We compare two non-perturbative techniques for calculating the single-particle Green’s function of interacting Fermi systems with dominant forward scattering: our recently developed functional integral approach to bosonization in arbitrary dimensions, and the eikonal expansion. In both methods the Green’s function is first calculated for a fixed configuration of a background field, and then averaged with respect to a suitably defined effective action. We show that, after linearization of the energy dispersion at the Fermi surface, both methods yield for Fermi liquids exactly the same non-perturbative expression for the quasi-particle residue. However, in the case of non-Fermi liquid behavior the low-energy behavior of the Green’s function predicted by the eikonal method can be erroneous. In particular, for the Tomonaga-Luttinger model the eikonal method neither reproduces the correct scaling behavior of the spectral function, nor predicts the correct location of its singularities.

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Wereszczyński, A. "Generalized eikonal knots and new integrable dynamical systems." Physics Letters B 621, no. 1-2 (August 2005): 201–7. http://dx.doi.org/10.1016/j.physletb.2005.06.050.

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Al-Khalili, J. S., J. A. Tostevin, and J. M. Brooke. "Beyond the eikonal model for few-body systems." Physical Review C 55, no. 3 (March 1, 1997): R1018—R1022. http://dx.doi.org/10.1103/physrevc.55.r1018.

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45

Waheed, Umair bin, Ivan Pšenčík, Vlastislav Červený, Einar Iversen, and Tariq Alkhalifah. "Two-point paraxial traveltime formula for inhomogeneous isotropic and anisotropic media: Tests of accuracy." GEOPHYSICS 78, no. 5 (September 1, 2013): WC65—WC80. http://dx.doi.org/10.1190/geo2012-0406.1.

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On several simple models of isotropic and anisotropic media, we have studied the accuracy of the two-point paraxial traveltime formula designed for the approximate calculation of the traveltime between points [Formula: see text] and [Formula: see text] located in the vicinity of points [Formula: see text] and [Formula: see text] on a reference ray. The reference ray may be situated in a 3D inhomogeneous isotropic or anisotropic medium with or without smooth curved interfaces. The two-point paraxial traveltime formula has the form of the Taylor expansion of the two-point traveltime with respect to spatial Cartesian coordinates up to quadratic terms at points [Formula: see text] and [Formula: see text] on the reference ray. The constant term and the coefficients of the linear and quadratic terms are determined from quantities obtained from ray tracing and linear dynamic ray tracing along the reference ray. The use of linear dynamic ray tracing allows the evaluation of the quadratic terms in arbitrarily inhomogeneous media and, as shown by examples, it extends the region of accurate results around the reference ray between [Formula: see text] and [Formula: see text] (and even outside this interval) obtained with the linear terms only. Although the formula may be used for very general 3D models, we concentrated on simple 2D models of smoothly inhomogeneous isotropic and anisotropic ([Formula: see text] and [Formula: see text] anisotropy) media only. On tests, in which we estimated two-point traveltimes between a shifted source and a system of shifted receivers, we found that the formula may yield more accurate results than the numerical solution of an eikonal-based differential equation. The tests also indicated that the accuracy of the formula depends primarily on the length and the curvature of the reference ray and only weakly depends on anisotropy. The greater is the curvature of the reference ray, the narrower its vicinity, in which the formula yields accurate results.

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Walther, Adriaan. "Angle eikonals for a perfect zoom system." Journal of the Optical Society of America A 18, no. 8 (August 1, 2001): 1968. http://dx.doi.org/10.1364/josaa.18.001968.

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47

Venetskiy, A. S., and V. A. Kaloshin. "On eikonal aberrations in axisymmetric double-reflector telescopic systems." Journal of Communications Technology and Electronics 61, no. 4 (April 2016): 385–94. http://dx.doi.org/10.1134/s1064226916040136.

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MULHOLLAND, A. J., and J. GOMATAM. "PATTERN FORMATION IN EXCITABLE REACTION–DIFFUSION SYSTEMS: THE EIKONAL ANALYSIS ON THE TORUS." Journal of Biological Systems 03, no. 04 (December 1995): 1013–19. http://dx.doi.org/10.1142/s0218339095000903.

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The excitable reaction–diffusion (R–D) systems of biological and chemical origin harbour a wealth of patterns and structures, not all of which have been modelled by the full R-D equations. The analytical and numerical facility offered by the eikonal approach to the R-D equation is exploited here in the demonstration of existence and stability of a class of solutions on a torus.

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Soravia, Pierpaolo. "Degenerate Eikonal equations with discontinuous refraction index." ESAIM: Control, Optimisation and Calculus of Variations 12, no. 2 (March 22, 2006): 216–30. http://dx.doi.org/10.1051/cocv:2005033.

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Waheed, Umair bin, Ehsan Haghighat, Tariq Alkhalifah, Chao Song, and Qi Hao. "PINNeik: Eikonal solution using physics-informed neural networks." Computers & Geosciences 155 (October 2021): 104833. http://dx.doi.org/10.1016/j.cageo.2021.104833.

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