scholarly journals The Vlasov?Maxwell Operators in Self-adjoint Form

1973 ◽  
Vol 26 (3) ◽  
pp. 301
Author(s):  
DP Mason
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Fredholm Integral Equation ◽  
Scalar Product ◽  
Separation Of Variables ◽  
Systematic Method ◽  
Maxwell Operator ◽  
One Dimensional ◽  
Bounded Plasma

A systematic method for obtaining the scalar product with respect to which a given Vlasov-Maxwell operator is self-adjoint is illustrated by considering the operator introduced by Shure (1964) to describe one-dimensional longitudinal oscillations in a bounded plasma. By separation of variables the problem is reduced to the solution of a singular integral equation for the weight function of the scalar product. The solution of this equation is not explicit but is in the form of a simple linear Fredholm integral equation of the second kind which is easily solved. The method should be applicable to similar operators in this field.

Approximate Solution of a Singular Integral Equation with a Cauchy Kernel on the Euclidean Plane

2018 ◽  
Vol 18 (4) ◽  
pp. 741-752
Author(s):  
Dorota Pylak ◽  
Paweł Karczmarek ◽  
Paweł Wójcik
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Euclidean Plane ◽  
Singular Integral Equations ◽  
Cauchy Kernel ◽  
Challenging Problem ◽  
One Dimensional ◽  
Interpolating Polynomials

AbstractMultidimensional singular integral equations (SIEs) play a key role in many areas of applied science such as aerodynamics, fluid mechanics, etc. Solving an equation with a singular kernel can be a challenging problem. Therefore, a plethora of methods have been proposed in the theory so far. However, many of them are discussed in the simplest cases of one–dimensional equations defined on the finite intervals. In this study, a very efficient method based on trigonometric interpolating polynomials is proposed to derive an approximate solution of a SIE with a multiplicative Cauchy kernel defined on the Euclidean plane. Moreover, an estimation of the error of the approximated solution is presented and proved. This assessment and an illustrating example show the effectiveness of our proposal.

scholarly journals Investigation of the semi-strip’s stress state in the case of steady-state oscillations

2019 ◽  
pp. 54-57
Author(s):  
N. D. Vaysfeld ◽  
Z. Yu. Zhuravlova ◽  
O. P. Moyseenok ◽  
V. V. Reut
Keyword(s):  
Integral Equation ◽  
Steady State ◽  
General Solution ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Partial Solution ◽  
Initial Problem ◽  
Dimensional Problem ◽  
One Dimensional ◽  
The One

The elastic semi-strip under the dynamic load concentrated at the centre of the semi-strip’s short edge is considered. The lateral sides of the semi-strip are fixed. The case of steady-state oscillations is considered. The initial problem is reduced to the one-dimensional problem with the help of the semi-infinite sin-, cos-Fourier’s transform. The one-dimensional problem is formulated in the vector form. Its solution is constructed as a superposition of the general solution for the homogeneous equation and the partial solution for the inhomogeneous equation. The general solution for the homogeneous vector equation is found with the help of the matrix differential calculations. The partial solution is expressed through Green’s matrixfunction, which is constructed as the bilinear expansion. The inverse Fourier’s transform is applied to the derived expressions for the displacements. The solving of the initial problem is reduced to the solving of the singular integral equation. Its solution is searched as the series of the orthogonal Chebyshev polynomials of the second kind. The orthogonalization method is used for the solving of the singular integral equation. The stress-deformable state of the semi-strip is investigated regarding both the frequency of the applied load, and the load segment’s length.

Forward problem of nonlinear Fredholm integral equation in reference medium via velocity‐weighted wavefield function

Geophysics ◽  
1997 ◽  
Vol 62 (2) ◽  
pp. 650-656 ◽  
Author(s):  
Li‐Yun Fu ◽  
Yong‐Guang Mu ◽  
Huey‐Ju Yang
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Fredholm Integral Equation ◽  
Scattered Field ◽  
Singularity Analysis ◽  
Forward Problem ◽  
Velocity Function ◽  
Host Medium ◽  
Nonlinear Fredholm Integral Equation

A nonlinear Fredholm integral equation of the first kind for the perturbation potential can be derived by interpreting the acoustic velocity as a perturbation of a reference velocity. We present an accurate and efficient method to formulate and numerically solve this equation with no restriction on the size of the perturbation by introducing a velocity‐weighted wavefield function (i.e., the scalar product of the potential function with the perturbed velocity function inside a scatterer), which is an intermediate function associating the observed wavefield with the perturbed velocity function. We start with a singularity analysis of the Fredholm integral equation when the observation point coincides with the scattering point to establish a linear singular integral equation with respect to the velocity‐weighted wavefield function. Then we formulate the relation between the velocity‐weighted wavefield and scattered field in the f‐k domain. A numerical scheme is developed to solve the forward problem for the velocity‐weighted wavefield inside a scatterer. Then the observed wavefield can be calculated through back substitutions of the Fredholm integral equation. The total wavefield and scattered field from a triangular object in a host medium observed from four sides are computed and compared with those given by the boundary‐element (BE) method and the approach using Born approximation. A second model computes both the common shot‐point (CSP) and common depth‐point (CMP) records of three velocity layers in a host medium. In addition, theoretical analyses and numerical experiments show that we can also use the singular integral equation to decompose accurately the velocity‐weighted wavefield given by inverse algorithms to recover the perturbed velocity function.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

Approximate Solution Of A Singular Integral Equation With A Multiplicative Cauchy Kernel In The Half-Plane

2008 ◽  
Vol 8 (2) ◽  
pp. 143-154 ◽  
Author(s):  
P. KARCZMAREK
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Half Plane ◽  
Singular Integral ◽  
Trigonometric Polynomials ◽  
Cauchy Kernel

AbstractIn this paper, Jacobi and trigonometric polynomials are used to con-struct the approximate solution of a singular integral equation with multiplicative Cauchy kernel in the half-plane.

A semianalytical and finite-element solution to the unbonded contact between a frictionless layer and an FGM-coated half-plane

2017 ◽  
Vol 24 (2) ◽  
pp. 448-464 ◽  
Author(s):  
Jie Yan ◽  
Changwen Mi ◽  
Zhixin Liu
Keyword(s):  
Integral Equation ◽  
Finite Element ◽  
Singular Integral Equation ◽  
Contact Problem ◽  
Singular Integral ◽  
Material Properties ◽  
Elastic Layer ◽  
Coated Substrate ◽  
Receding Contact ◽  
Contact Size

In this work, we examine the receding contact between a homogeneous elastic layer and a half-plane substrate reinforced by a functionally graded coating. The material properties of the coating are allowed to vary exponentially along its thickness. A distributed traction load applied over a finite segment of the layer surface presses the layer and the coated substrate against each other. It is further assumed that the receding contact between the layer and the coated substrate is frictionless. In the absence of body forces, Fourier integral transforms are used to convert the governing equations and boundary conditions of the plane receding contact problem into a singular integral equation with the contact pressure and contact size as unknowns. Gauss–Chebyshev quadrature is subsequently employed to discretize both the singular integral equation and the force equilibrium condition at the contact interface. An iterative algorithm based on the method of steepest descent has been proposed to numerically solve the system of algebraic equations, which is linear for the contact pressure but nonlinear for the contact size. Extensive case studies are performed with respect to the coating inhomogeneity parameter, geometric parameters, material properties, and the extent of the indentation load. As a result of the indentation, the elastic layer remains in contact with the coated substrate over only a finite interval. Exterior to this region, the layer and the coated substrate lose contact. Nonetheless, the receding contact size is always larger than that of the indentation traction. To validate the theoretical solution, we have also developed a finite-element model to solve the same receding contact problem. Numerical results of finite-element modeling and theoretical development are compared in detail for a number of parametric studies and are found to agree very well with each other.

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