Kernel Of Integral Equation Research Articles

Kinetic theory for classical and quantum many-body chaos

For perturbative scalar field theories, the late-time-limit of the out-of-time-ordered correlation function that measures (quantum) chaos is shown to be equal to a Boltzmann-type kinetic equation that measures the total gross (instead of net) particle exchange between phase-space cells, weighted by a function of energy. This derivation gives a concrete form to numerous attempts to derive chaotic many-body dynamics from ad hoc kinetic equations. A period of exponential growth in the total gross exchange determines the Lyapunov exponent of the chaotic system. Physically, the exponential growth is a front propagating into an unstable state in phase space. As in conventional Boltzmann transport, which follows from the dynamics of the net particle number density exchange, the kernel of this kinetic integral equation for chaos is also set by the 2-to-2 scattering rate. This provides a mathematically precise statement of the known fact that in dilute weakly coupled gases, transport and scrambling (or ergodicity) are controlled by the same physics.

Bending analysis of functionally graded plates with arbitrary shapes and boundary conditions

The paper focuses on bending analysis of the functionally graded (FG) plates with arbitrary shapes and boundary conditions. The material property of FG plates is modelled by using the power law distribution. Based on the first order shear deformation plate theory (FSDT), the governing equations as well as boundary conditions are formulated and obtained by using the principle of virtual work. The coupled Boundary Element-Radial Basis Function (BE-RBF) method is established to solve the complex FG plates. The proposed methodology is developed by applying the concept of the analog equation method (AEM). According to the AEM, the original governing differential equations are replaced by three Poisson equations with fictitious sources under the same boundary conditions. Then, the fictitious sources are established by the application of a technique based on the boundary element method and approximated by using the radial basis functions. The solution of the actual problem is attained from the known integral representations of the potential problem. Therefore, the kernels of the boundary integral equations are conveniently evaluated and readily determined, so that the complex FG plates can be easily computed. The reliability of the proposed method is evaluated by comparing the present results with those from analytical solutions. The effects of the power index, the length to thickness ratio and the modulus ratio on the bending responses are investigated. Finally, many interesting features and results obtained from the analysis of the FG plates with arbitrary shapes and boundary conditions are demonstrated.

The method for calculating singular integrals in problems of axially symmetric Stokes flows

The flow of a viscous fluid at small Reynolds numbers (Stokes flow) in a three-dimensional formulation is investigated. In this case, the inertial terms in the equations of motion can be neglected. Such flows can occur in nanotubes that can be considered as inclusions in representative volume elements of nanomaterials. By using the fundamental solution of Ossen, an integral representation of the velocity is proposed. This representation is used to receive an integral equation for an unknown density. The solution of the resulting equation makes it possible to calculate the fluid pressure on the walls of the shell. The case of axially symmetric flows is investigated. For this, an integral representation of the unknown velocity in cylindrical coordinates is obtained. By integrating over the circumferential coordinate, the two-dimensional singular integral equation is reduced to one-dimensional one. It has been proved that the components of the kernels in singular operators are expressed in terms of elliptic integrals of the first and second kind. It has been proved that the singularities of the kernels of one-dimensional singular integral equations have a logarithmic character. To calculate elliptic integrals, the Gaussian algorithm based on the use of the arithmetic-geometric mean value is proposed. This procedure allows us to obtain logarithmic singular components with high accuracy, which makes it possible to use special quadrature formulas to calculate such integrals. An algorithm with usage of the boundary element method for the numerical solution of the obtained singular integral equations is proposed. The method for solving one-dimensional singular equations, where the kernels contain elliptic integrals with logarithmic singularities (i.e logarithmic singularity is not expressed explicitly) has been tested. The obtained numerical results have been compared with the well-known analytical solutions. The data obtained indicate the high efficiency of the proposed numerical method.

A collocation method for a class of Fredholm integral equations with highly oscillatory kernels

In this paper, a collocation method by using Clenshaw–Curtis points is proposed to solve the Fredholm integral equations (FIEs) with highly oscillatory kernels. The collocation method is being applied to graded and uniform meshes. Due to the highly oscillatory kernels of integral equations, the discretized collocation equation will lead to the computation of the oscillatory integrals which will be computed by using the efficient Filon-type method. Finally, the effectiveness and accuracy of the proposed method are confirmed by numerical examples.

CONTACT PROBLEM FOR A TWO-LAYERED CYLINDER

Introduction. The investigation of the contact problems for cylindrical bodies is urgent due to the engineering contact strength analysis on shafts, cores and pipe-lines. In the present paper, a new contact problem of elastostatics on the interaction between a rigid band and an infinite two-layered cylinder, which consists of an internal continuous cylinder and an outer hollow one, with a frictionless contact between the cylinders, is studied. The outer cylindrical band of finite length is press fitted. By using a Fourier integral transformation, the problem is reduced to an integral equation with respect to the unknown contact pressure.Materials and Methods. Different combinations of linearly elastic materials of the composite cylinder are considered. Asymptotics of the symbol function of the integral equation kernel at zero and infinity is analyzed. This plays an important role for the application of the analytical solution methods. A key dimensionless geometric parameter is introduced, and a singular asymptotic technique is employed to solve the integral equation.Research Results. On the basis of the symbol function properties, a special easily factorable approximation being applicable in a wide variation range of the problem parameters is suggested. The Monte-Carlo method is used to determine the approximation parameters. The asymptotic formulas are derived both for the contact pressure, and for its integral characteristic. Calculations are made for different materials and for various relative thickness of the cylindrical layer including thin-walled layers.Discussion and Conclusions. The asymptotic solutions are effective for relatively wide bands when the contact zone length is bigger than the diameter of the composite cylinder. It is significant that the method is applicable also for those cases when the outer cylindrical layer is treated as a cylindrical shell. The asymptotic solutions can be recommended to engineers for the contact strength analysis of the elastic barrels with a flexible coating of another material.

A cut-off criterion for calculating more reliable adsorption energy distributions from gas adsorption isotherms

Adsorption energy distributions from experimental gas adsorption isotherms are capable to characterize the energetic heterogeneity of a solid surface. Unfortunately, they can only be computed by the adsorption integral equation, which represents an ill-posed problem, i.e., the solution is highly sensitive to errors in the input data. Ill-posed problems are usually solved by means of regularization, but general regularization schemata do not provide useful criteria to estimate the approximation quality. In this paper, a former presented solution strategy tailor-made for the Langmuir kernel of the adsorption integral equation is extended to Fourier transform. This yields a simple and effective cut-off criterion for the Fourier cosine transform of the adsorption energy distribution. The cut-off criterion is applied to calculate adsorption energy distributions from synthetic and experimental adsorption isotherms.

Using Ultra-High Expansion Orders of Max-Ortho Basis Functions for Analysis of Axially Symmetric Metallic Antennas

Implementation of max-ortho basis functions is proposed in a method for analysis of axially symmetric metallic antennas based on exact kernel of electric field integral equation in combination with Galerkin testing. High-precision evaluation of matrix elements is enabled by: a) representing them as a linear combination of impedance integrals due to the Legendre polynomials and their first derivatives; b) using the singularity cancelation techniques; and c) evaluating the Legendre polynomials and their first derivatives by well-known recurrent formulas. Applicability of max-ortho bases up to expansion order of $n =128$ is illustrated on a full-wave thick dipole antenna.

Contact and crack problems in generalized materials and their relationships

A new term of generalized material is introduced here. This definition covers all anisotropic magneto-electro-elastic materials and one-, two- and tree-dimensional generally anisotropic quasi-crystals and hopefully, some new not yet discovered materials. We consider a half-space $$x_{3}\ge 0$$ , made of generalized material and subjected to arbitrary point sources or point dislocations, which can be interpreted also as electric and/or magnetic influence. General solution was obtained by using two-dimensional Fourier transform. The final results are presented as single integrals over a unit circle. Some components of the surface Green’s functions were computed in a finite form, no computation of any integral is needed. The theory of generalized functions was used. This result allows us to derive the governing integral equations for the normal and tangential contact and crack problems. We also establish certain relationships between the Fourier transforms of the kernels of the relevant integral equations. As a bonus, some interesting properties of the determinants, which might be new, were established.

A Reverse Operation Self-Consistent Evaluation Approach for Singular Integrals in the SIE Analysis of PEC Targets

Discussed herein is a novel approach entitled reverse operation self-consistent evaluation (ROSE). Specifically, we applied ROSE for the treatment of weak-singular and hypersingular kernels in surface integral equations (SIEs) for the computation of electromagnetic (EM) scattering from perfect electrical conductors (PECs). Different from the existing forward methods, ROSE is based on reverse operations. The singular entries are not evaluated directly by analytical or numerical evaluation; instead, they are introduced as unknowns and are solved algebraically. For EM scattering from PEC, SIEs can be set up by using the Stratton–Chu representation formula. An auxiliary problem with the same meshed boundary is then formulated, of which any plane waves are the self-consistent solutions. It is shown that by applying a consistent discretization, the matrix of the auxiliary problem can be made identical with the original one. As a consequence, these self-consistent solutions can be utilized to solve the singular entries of the original system. Explicitly, a set of plane waves are interpolated using the appropriate basis functions. Subsequently, the corresponding coefficient vectors are utilized to construct small linear systems to solve the troublesome singular entries in the underlying matrix. The proposed ROSE approach is applicable to different singular kernels as well as different discretization schemes. In this paper, we restrict our discussion to the weak-singular and hypersingular kernels using, respectively, div-conformal Rao–Wilton–Glisson basis functions and nonconformal piecewise constant vector basis functions. Accelerated by the multilevel fast multipole algorithm, the proposed method is applicable for real-life applications.

Correction to: Green’s functions for the magneto-electro-elastic anisotropic half-space and their applications to contact and crack problems

The following text should be changed in the abstract: Using the theory of generalized functions, some basic parameters at the half-space boundary are defined in a finite form, and no computation of any integral is needed. Knowledge of some of the Green’s functions in finite form allows us to derive the governing integral equations for the normal contact and crack problems, as well as to establish certain relationship between the integrands in Fourier transforms of the kernels of the relevant integral equations.

Asymptotic behavior of the solutions for the Laplace equation with a large spectral parameter and the inhomogeneous Robin type conditions

Reduced problems are elliptic problems with a large parameter (as the spectral parameter) given by the Laplace transform of time dependent problems. In this paper, asymptotic behavior of the solutions of the reduced problem for the classical heat equation in bounded domains with the inhomogeneous Robin type conditions is discussed. The boundary of the domain consists of two disjoint surfaces, outside one and inside one. When there are inhomogeneous Robin type data at both boundaries, it is shown that asymptotics of the value of the solution with respect to the large parameter at a given point inside the domain is closely connected to the distance from the point to the both boundaries. It is also shown that if the inside boundary is strictly convex and the data therein vanish, then the asymptotics is different from the previous one. The method for the proof employs a representation of the solution via single layer potentials. It is based on some non trivial estimates on the integral kernels of related integral equations which are previously established and used in studying an inverse problem for the heat equation via the enclosure method.

Nonlinear bending analysis of nonlocal nanoplates with general shapes and boundary conditions by the boundary-only method

In this paper, the geometrically nonlinear bending analysis of nanoplates with general shapes and boundary conditions is highlighted. The governing equations are derived based on the classical plate theory using nonlocal differential constitutive relation of Eringen and von Kármán's nonlinear strains. The boundary-only method is developed by using the principle of the analog equation (PAE). According to the PAE, the original governing differential equations are replaced by three uncoupled equations with fictitious sources under the same boundary conditions, namely two Poisson equations and one biharmonic equation. Subsequently, the fictitious sources are established using a technique based on the boundary element method and approximated by using the radial basis functions. The solution of the actual problem is attained from the known integral representations of the potential and plate problems. Therefore, the kernels of the boundary integral equations are conveniently established and readily calculated that the complex nanoplates can be easily analyzed. The accuracy of the proposed methodology is evaluated by comparing the obtained results with available solutions. Moreover, the influences of nonlocal parameter on the various characteristics of effective distributed loads are elucidated. Finally, the effects of nonlocal parameter, von Kármán's nonlinearity and aspect ratio on nonlinear bending responses are studied.

Green’s functions for the magneto-electro-elastic anisotropic half-space and their applications to contact and crack problems

A general solution is obtained for a magneto-electro-elastic half-space \(x_{3}\ge 0\) subjected to arbitrary point forces or arbitrary point dislocations, as well as electric and magnetic influence by using two-dimensional Fourier transform. The final results are presented as single integrals over a unit circle. Using the theory of generalized functions, all basic parameters at the half-space boundary are defined in a finite form, and no computation of any integral is needed. Knowledge of Green’s functions in finite form allows us to derive the governing integral equations for the normal and tangential contact and crack problems, as well as to establish certain relationship between the kernels of the relevant integral equations. We also established some interesting general properties of the determinants, which might be new.

On the applicability of asymptotic formulas of retrieving “optical” turbulence parameters from pulse lidar sounding data: II–results of numerical simulation

The applicability of formulas derived in the first part of this work has been studied in a numerical experiment. The geometrical optics approximation is shown to be applicable only on short paths whose length is less than the corresponding diffraction lengths by a factor of hundreds. The restrictions are caused by oscillations of the kernel of the initial integral equation. They lead to strong oscillations of the third derivative of the measured data. The formulas based on the asymptotic formula of the kernel for a point receiver are slightly sensitive to oscillations of the measured data. Applying the formulas for a point receiver in the case of receivers with a finite radius smoothes the retrieved distributions and shifts them with respect to the given ones. A technique of taking these factors into account in the process of retrieving has been proposed. Together with smoothing of the retrieved distributions, applying the point receiver approximation leads to partial loss of the information about the turbulence spectrum in the retrieved data. This allows one to simplify the retrieval procedure by reducing it to calculating usual derivatives of the second order.

Triton Binding Energy of Kharkov Potential

The Kharkov potential is a recent field theoretical model of nucleon–nucleon (NN) interaction that has been built up in the framework of the instant form of relativistic dynamics starting with the total Hamiltonian of interacting meson and nucleon fields and using the method of unitary clothing transformations. The latter connect the representation of “bare” particles and the representation of “clothed” particles, i.e., the particles with physical properties. Unlike many available NN potentials each of which is the kernel of the corresponding nonrelativistic Lippmann–Schwinger (LS) equation this potential being dependent in momentum space on the Feynman-like propagators and covariant cutoff factors at the meson–nucleon vertices is the kernel of relativistic integral equations for the NN bound and scattering states. Therefore we do not need to invent any transform of a given nonrelativistic potential to its relativistic counterpart. As a feasible study, we have started with the so-called 5ch Faddeev calculation for three-nucleon bound state (triton) and obtained a reasonable value of its binding energy ( $$-7.42$$ MeV).

Nonsingular Field-Only Surface Integral Equations for Electromagnetic Scattering

A boundary integral formulation of electromagnetics that involves only the components of $\boldsymbol{E}$ and $\boldsymbol{H}$ is derived without the use of surface currents that appear in the classical PMCHWT formulation. The kernels of the boundary integral equations for $\boldsymbol{E}$ and $\boldsymbol{H}$ are non-singular so that all field quantities at the surface can be determined to high precision and also geometries with closely spaced surfaces present no numerical difficulties. Quadratic elements can readily be used to represent the surfaces so that the surface integrals can be calculated to higher numerical precision than using planar elements for the same numbers of degrees of freedom.

A New and Efficient Boundary Element-Free Method for 2-D Crack Problems

An efficient boundary element-free method is established for 2-D crack problems by combining a pair of boundary integral equations and the moving-least square approximation. The displacement boundary integral equation is collated on the on-crack boundary, and a new traction boundary integral equation is applied on the crack surface without the separate consideration of the upper and lower sides. In virtue of integration by parts, only singularity in order 1/r is involved in the integral kernels of new traction boundary integral equation, which brings convenience to the numerical implementation. Meanwhile, the integration by parts produces the new variables, the displacement density, and displacement dislocation density, and they are the coexisting unknowns along with the displacement and displacement dislocation. With the high-order continuity of the moving-least square approximation, these new variables are directly approximated with the nodal displacement or displacement dislocation, and the final system of equations contains the unknowns of nodal displacements and displacement dislocations only. The boundary element-free computational scheme is established, and several examples show the efficiency and flexibility of the proposed method.

Direct numerical solution of the Lippmann-Schwinger equation in coordinate space without partial-wave decomposition.

Direct numerical solution of the coordinate-space integral-equation version of the two-particle Lippmann-Schwinger (LS) equation is considered without invoking the traditional partial-wave decomposition. The singular kernel of the three-dimensional LS equation in coordinate space is regularized by a subtraction technique. The resulting nonsingular integral equation is then solved via the Nystrom method employing a direct-product quadrature rule for three variables. To reduce the computational burden of discretizing three variables, advantage is taken of the fact that, for central potentials, the azimuthal angle can be integrated out, leaving a two-variable reduced integral equation. A regularization method for the kernel of the two-variable integral equation is derived from the treatment of the singularity in the three-dimensional equation. A quadrature rule constructed as the direct product of single-variable quadrature rules for radial distance and polar angle is used to discretizethetwo-variable integral equation. These two- and three-variable methods are tested on the Hartree potential. The results show that the Nystrom method for the coordinate-space LS equation compares favorably in terms of its ease of implementation and effectiveness with the Nystrom method for the momentum-space version of the LS equation.

Analytical estimations of pulse parameters in the modified integral neutron kinetics model for pulsed reactor and subcritical block

Application of modified integral neutron kinetic model to calculate principal characteristics of pulsed coupled reactor system consisting of pulsed reactor and subcritical block is discussed. The model is based on the use of respective time-dependent kernels of integral equation for reactor power and space-time Green's function for the subcritical block. It is possible to reduce the set of integral equations to the set of elementary algebraic and first-order differential equations by using exponential approximation of the kernels and Green's function.Approximations of «inertialess» reactivity dumping and jump reactivity boost on prompt neutrons are used as the «reactivity-power» feedback in order to close the mathematical model. This allows integrating corresponding kinetic equations in analytical form notwithstanding the fact that the kinetic equation for reactor is nonlinear.Analytical relations allowing estimating basic characteristics of the system such as energy and maximum pulse power in the reactor and in the subcritical block with accuracy sufficient in engineering practices were obtained.The performed calculations showed applicability of the analytical dependences of energy characteristics of the system on the impact coefficient of subcritical block on the reactor, on the lifetime of neutrons in the reactor and on the «time» constant of the block for fixed value of pulse energy in the reactor. The obtained ratio is valid for the reactor within the whole range of variation of system parameters while for the subcritical block it is correct only for the system operated with fast neutron spectrum in the reactor and with thermal neutron spectrum in the subcritical block when the so-called “delta” approximation of the reactor pulse is realized. In the case when such approximation is not valid the “Gaussian” approximation to the shape of the reactor pulse is applied for which more accurate analytical formulas were also obtained for estimation of maximum pulse energy in the block. These formulas depend on the ratio of duration of start-up period of the pulsed coupled reactor system to the value of “time” constant of the subcritical block and are correct for the systems with similar neutron spectra.The obtained analytical relations can be applied for optimization of parameters of coupled reactor-laser systems.

Bloch–Siegert Effect in Magnetic-Resonance Sounding

The magnetic-resonance sounding (MRS) method is intended for groundwater prospecting. The Earth’s magnetic field used in MRS is of the order of 5 × 10−5 T. The radiofrequency (RF) field produced in MRS by the surface antenna is linearly polarized and may be comparable with the geomagnetic field. The use of RF pulses at Larmor frequency leads to the appearance of the mean resonance offset in rotating frame (Bloch–Siegert shift) which is caused by the influence of the off-resonance counter rotating component of the RF field. The solution of the inverse problem for ground water distribution must take account of the proton spin resonance frequency shift, because an inaccurate kernel of the respective integral equation may lead to significant errors in the estimation of aquifer characteristics. The value for Bloch–Siegert effect with one order of approximation higher than usual has been calculated. The MRS technique with 100-m-diameter antenna has been used at the frozen Ob reservoir near Novosibirsk, Russia, to detect subice water to prove the improved physical model of MRS. There is a good agreement between measured and calculated data taking into account the Bloch–Siegert and the next-order effects. New physical model accounting the Bloch–Siegert effect allowed a significant improvement of the reliability of the MRS data interpretation.