Analytic Theory for High-Inclination Orbitsin the Restricted Three-Body Problem

In this study, the linear renewal equation is obtained by using the integral equation, the renewal function and the Fourier–Stieltjes transform. It is proven that the linear renewal equation can be obtained by taking the derivative of the integral equation. Analytical methods for the solution of the obtained linear renewal equation are discussed. It is shown that the linear renewal equation is a powerful tool that can model the direct relationship between stochastic processes and density functions. It is shown that the Fourier–Stieltjes transform allows the equation to be simplified in the frequency domain and analytical solutions to be obtained, and the Laplace transform provides an effective analytical solution method, especially for uniform distribution and exponential density functions. The integral equation-based linear renewal density equation derived in this study preserves the temporal and structural symmetries of the system, allowing for the analytical derivation of symmetric forms in the solution space. In the light of the findings, predictions were made about what kind of studies would be done in the future.

Purpose. Testing of numerical solution algorithm for integral equation for calculation of plane meridian magnetostatic field source distribution at interfaces of piecewise homogeneous magnetized medium by means of electrostatic analogy. Methodology. The piecewise homogeneous medium consists of three regions with different magnetic permeabilities: the shell of arbitrary meridian section, external unlimited medium outside the shell, and the medium inside the shell. For testing external homogeneous magnetic field effect on spherical shell is considered. The analytical solution of this problem on the basis of electrostatic analogy from the solution of the problem uniform electrostatic field effect on dielectric shell is obtained. We have compared results of numerical solution of integral equation with the data obtained by means of analytical solution at the variation of magnetic permeabilities of regions of medium. Results. Integral equation and the algorithm of its numerical solution for calculation of source field distribution at the boundaries of piecewise homogeneous medium is validated. Testing of integral equations correctness for calculation of fictitious magnetic charges distribution on axisymmetric boundaries of piecewise homogeneous magnetized medium and algorithms of their numerical solutions can be carried out by means of analytical solutions of problems of homogeneous electrostatic field effect analysis on piecewise homogeneous dielectric medium with central symmetry of boundaries – single-layer and multilayer spherical shells. In the case of spherical shell in wide range of values of the parameter λk, including close to ± 1, numerical solution of integral equation is stable, and relative error in calculating of fictitious magnetic charges surface density and magnetic field intensity inside the shell is from tenths of a percent up to several percent except for the cases of very small values of these quantities. Originality. The use analytical solutions for problems of calculation of external electrostatic field effect on piecewise homogeneous dielectric bodies for testing integral equations of magnetostatics and algorithms for their numerical solutions. Practical value. The described method of testing integral equations of magnetostatics and their numerical solutions can be used for calculation of magnetic fields of spacecraft control system electromagnets.

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We produce more elementary algorithms than those of Brent and Salamin for, respectively, evaluating e x {e^x} and π \pi . Although the Gauss arithmetic-geometric process still plays a central role, the elliptic function theory is now unnecessary.

In this chapter we consider how elliptic function theory and complex variable theory were finally drawn together in the 1830s and 1840s. As the recognition of the importance of the work of Abel and Jacobi grew, mathematicians came to feel that it was unsatisfactory to base the theory of elliptic functions on the inversion of many-valued integrals. One alternative would have been to adopt and develop Cauchy’s theory of complex integrals. By and large this was not done, and it is interesting to examine why. The study of elliptic integrals was felt by many to be fraught with ambiguity because of the square root in the integrand. Moreover, Cauchy’s system of definitions, based on his newly defined concepts of limit, continuity, differentiability, and integrability, was incompatible with talk of many-valued functions—Cauchy did not define continuity for a many-valued function, and indeed a many-valued function cannot be continuous according to Cauchy’s use of the term. Although a doubly periodic function is a meromorphic function defined on the whole of the complex plane, an elliptic integral makes better sense on something like a Riemann surface (a torus in this case). Thus the many-valued nature of an elliptic integral posed a challenge to mathematicians throughout the 1830s and 1840s. So the perceived problem with the foundations did not meet with a ready answer in the newly emerging theory of complex functions. Matters were to be worse with hyperelliptic integrals, because the corresponding inverse functions could not be treated as multiply-periodic functions in the plane.

A propulsively managed stationkeeping strategy for a spacecraft orbiting a collinear equilibrium point in a three-body system is introduced. A nominal circular solution with nonuniform speed, which is derived from the Jacobi integral equation, employing elliptic integral theory, is used in a plane perpendicular to the line joining the two primaries. Thrust control inputs, which are found to be nonlinear functions of time, are used to negate the instability ofthe nominal orbit. Analytical relationships for control requirements are developed. Orbit parameters are analyzed and chosen so that the cost of maintaining the nominal orbit is minimized. Resulting thrust requirements are assessed for feasibility with respect to existing propulsion capabilities. Significant freedoms are found to exist for tailoring thrust demands. If proven feasible, the thrust managed orbit could find applications in communications, in situ space measurements, observation platforms, forecasting-warning systems, and loitering. An in situ magnetosphere sampling mission based on a lunar synchronized geopolar orbit is proposed and analyzed using the new results.

We present a new method for the numerical solution of singular integral equations on the real axis. The method’s value stems from a new formula for the Cauchy integral of a rational function with an oscillatory exponential factor. The inner product of such functions is also computed explicitly. With these tools in hand, the GMRES algorithm is applied to both non-oscillatory and oscillatory singular integral equations. In specific cases, ideas from Fredholm theory and Riemann–Hilbert problems are used to motivate preconditioners for these singular integral equations. A significant acceleration in convergence is realized for these examples. This presents a useful link between the theory of singular integral equations and the numerical analysis of such equations. Furthermore, this method presents a first step towards a solver for the inverse scattering transform that does not require the deformation of a Riemann–Hilbert problem.

This paper presents an analytical investigation and numerical solutions to the problems of wave radiation and wave diffraction by a submerged sphere. We assume that the fluid is homogeneous, inviscid, and incompressible and the fluid motion is irrotational. The surge and the heave motions are studied and the hydrodynamic coefficients of these motions are calculated. The analytical solutions are obtained by expanding the velocity potential into a series of associated legendre functions in a spherical coordinate system. The problem is also solved numerically based on the boundary integral equation method (BIEM) using non-singular form of the Green's formula. The major advantages of this modified fomulation are that the exact geometry of the body can be applied to solve of the problem and the integral equations are amenable to solution by the quadrature formulas. The results of the computations are presented graphically in the form of non-dimensional parameters and good agreement between analytical and numerical solutions is observed. It is proposed that this modified formulation can be extended to the problem of motion of submerged or floating bodies in waves at forward speed.

The modified Laplace plane (modified by including solar radiation pressure as a perturbation) gives a surface of frozen orbits that could be advantageous for spacecraft. The modified Laplace plane is defined by an angle and circular orbit radius from a small body where the perturbations from the Sun, J2 and solar radiation pressure are balanced. The modified Laplace plane will be applied to an object around an asteroid. The modified Laplace plane and various other initial conditions will be investigated at three different asteroids, Bennu, 66391 (1999 KW4) and Vesta. These asteroids have different masses, shape, spin axes and orbits around the Sun that will lead to various results. A spacecraft of varying mass-to-area ratio will also be investigated. These different parameters will help determine what causes instability with regard to escape or collision of the spacecraft. The analysis will focus on determining how stable the modified Laplace plane and the Sun terminator plane are to escapes and collisions.

A reliable automatic simulation of singular electroanalytical transients, by the adaptive Huber method for Volterra integral equations

9R1. Natural Boundary Integral Method and Its Applications. - De-hao Yu (Inst of Comput Math and Sci/Eng Comput, Chinese Acad of Sci, Beijing, ROC). Kluwer Acad Publ, Dordrecht, Netherlands. Distributed in USA by Kluwer Acad Publ, Norwell MA. 2002. 539 pp. ISBN 1-4020-0457-5. $155.00.Reviewed by D Givoli (Dept of Aerospace Eng, Technion-Israel, Haifa, 32000, Israel).Kang Feng (1923-1993) was a well-known Chinese mathematician, and until his death was the President of the Chinese Society of Computational Mathematics. De-hao Yu, the author of this book, was his PhD student. This is an English version of a previous monograph by Yu that appeared in Chinese ten years ago. The book summarizes Yu’s research work based on the mathematical foundation laid by Feng. Most of Feng’s and Yu’s publications have appeared in Chinese and have not been generally accessible to western readers. This monograph, published by Kluwer, is especially welcome since it allows general access to the interesting work of both Feng and Yu. The book focuses on the concept of Natural Boundary Reduction (NBR) in the context of two-dimensional (2D) elliptic problems, and in particular problems governed by Laplace’s equation, the biharmonic equation, plane elasticity equations, and Stokes’ equations. The idea is to replace the differential equation in the given 2D domain Ω by an integral equation on its boundary Γ, and to solve this integral equation numerically on Γ via a variational formulation. This sounds like the basic idea of the Boundary Element Method (BEM); however, in NBR the reduction of the differential equation in Ω to an integral equation on Γ is performed differently, using the so-called natural integral operator, which is also known as the Dirichlet-to-Neumann (DtN) map. The book is written beautifully. It is very clear, interesting, and not dry despite being mathematically rigorous. Although the author is a mathematician, he wrote this book in a way that makes it accessible to mathematically-oriented graduate students and researchers in computational and applied mechanics and engineering. The basic ideas are described clearly and in a well-organized way. In addition, one can learn quite a lot from this book about areas as diverse as singular integrals, harmonic functions, and domain decomposition. The summary on singular integrals in Section 1.4, for example, is superb. The book, with 540 pages, is divided into seven chapters. Chapter 1 introduces the main ideas of NBR, and is followed by Chapters 2–5, which apply the method to the four differential equations mentioned above. Chapter 6 discusses the coupling of NBR and Finite Elements. This is similar to the DtN method, which has been devised independently in the west primarily for wave problems (which are not discussed at all in this book). The last chapter shows how to solve unbounded domain problems using NBR combined with iterative Domain Decomposition. The book also includes a preface (telling about Prof Feng), a list of more than 200 references (about half of which are Chinese publications), and an index. The NBR method is described on the back cover as a competitor to the standard BEM. Such a comparison is not totally appropriate. The BEM is a general method that can handle problems in irregular geometries. It makes use of the full-space Green’s function associated with the operator involved, which does not depend on the given geometry. On the other hand, NBR makes use of the natural integral operator, which depends on the specific geometry under consideration. Finding this operator on Γ (analytically) is equivalent to solving the problem in Ω analytically. Thus, in its simplest form, the NBR method can be applied in practice only in cases where the analytic solution is already known! Of course, there is no point in using NBR in such a manner. However, this does not mean that the idea underlying NBR is not useful. It is definitely useful when the domain of the problem under consideration involves a region where the problem is analytically-solvable and a region where it is not. Such problems are discussed in the last two chapters, where the practical usefulness of the method is demonstrated. In addition, understanding the various ways in which a problem can be represented (via partial differential equations, as a variational problem, by an integral equation, etc) is always beneficial and may lead to interesting insights regarding the properties of solutions and to new computational methods. Natural Boundary Integral Method and Its Applications is neither a course textbook nor a state-of-the-art research book on a wide scientific area, but it is a satisfying self-contained summary of a very interesting piece of work that has been hidden from the western reader so far. This monograph is highly recommended as an enjoyable and eye-opening reading for the mathematically-oriented researcher and practitioner of applied mechanics.

Some well-known numerical methods for approximating conformal mapping of doubly regions onto an annulus or the unit disk are presented. There is a definite need for a simple yet accurate method for mapping a general doubly connected region onto a circular annulus. According to Kantorovich and Krylov (1958, p. 362) the problem of finding the conformal modulus is ‘one of the difficult problems of the theory of conformal transformation’. As such, analytic solutions have been determined for a very restricted class of doubly connected regions, like those mentioned in Muskhelishvili (1963, §48). Numerical solutions are also confined to a limited class of regions where either one boundary is circular or axisymmetric. Most common methods use integral equations, iterations, polynomial approximations, and kernels. We shall develop Symm’s integral equations and the related orthonormal polynomial method. A dipole formulation that leads to the method of reduction of connectivity shall be presented. Another useful method for multiply connected regions, based on Mikhlin’s integral equation, that also works for simply and doubly connected regions as well will be discussed in Chapter 13.KeywordsUnit DiskConformal MappingParallel StripPeriodic StripConformal ModulusThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A three-dimensional curvilinear hybrid finite-element–integral equation approach is developed. The hybrid finite-element–integral equation method is formulated in general curvilinear coordinates so that arbitrarily curved geometries can be modeled without approximations. The advantages of the finite-element and the integral equation methods are used to eliminate the disadvantages of both methods. For modeling regions of inhomogeneous or anisotropic materials, in which the integral equation method is difficult to implement, the finite-element method is used. The problem with the finite-element method of accurately terminating the computational mesh is eliminated by application of the exact boundary integral equation. The hybrid finite-element–integral equation code is validated on a range of composite curvilinear geometries against measured data or analytical solutions.

The boundary integral equations (BIEs), in their original forms, which govern the electromagnetic (EM) wave scattering in three-dimensional space contain at least a hypersingularity (1/R/sup 3/) or a Cauchy-singularity (1/R/sup 2/), usually both. Thus, obtaining reliable numerical solutions using such equations requires considerable care, especially when developing systematic numerical integration procedures for realistic problems. Regularized BIEs for the numerical computation of time-harmonic EM scattering fields due to arbitrarily-shaped scatterers are introduced. Two regularization approaches utilizing an isolation method plus a mapping are presented to remove all singularities prior to numerical integration. Both approaches differ from all existing approaches to EM scattering problems. Both work for integral equations initially containing either hypersingularities or Cauchy-singularities, without the need to introduce surface divergences or other derivatives of the EM fields on the boundary. Also, neither approach is limited to flat surfaces nor flat-element models of curved surfaces. The Muller linear combination of the electric- and magnetic-field integral equations (EFIE) and (MFIE) is used to avoid the resonance difficulty that is usually associated with integral equation-based formulations. Some preliminary numerical results for EM scattering due to single and multiple dielectric spheres are presented and compared with analytical solutions.

The two-bodies problem can be fully solved, and was solved by Kepler (1609) and Newton (1687). The general three-body problem is often given as an example of a mathematical problem that ‘can’t be solved’. So, there is no general analytical solution. This problem can be significant and a special case of this problem is the Circular Restricted Three-Body Problem (CRTBP), which can be applied to the Earth-Moon system with a spacecraft, the Sun-Earth system with an asteroid, etc. In this paper, let’s focus on the motion of a test particle near the triangular Lagrange points L4 and L5 in the Earth-Moon and the Sun-Earth systems. Studying the movement of an object around these points is especially important for space mission design. To generate a trajectory around these points, the non-linear equations of motion for the circular restricted three-body problem were numerically integrated into MATLAB® 2023 software and the results are presented in the plane (x, y) and the phase plane (x, vx) and (y, vy). By numerical orbit integration, it is possible to investigate what happens when the displacement is relatively large or short from the Lagrange points. Then the small astronomical body may vibrate around these points. The results in this paper are shown in the rotating and inertia axes. Various initial positions near the Lagrange points and velocities are used to produce various paths the test particle can take. The same examples of numerical studies of trajectories associated with Lagrange points are shown in the inertial and the rotating coordinates system and are discussed. From the results of the numerical tests performed in MATLAB® 2023, it is possible to saw that there are different types of periodic, quasi-periodic, and chaotic orbits