Generation of harmonics treated by a new integral equation method

1993 ◽  
Vol 71 (7-8) ◽  
pp. 340-346 ◽  
Author(s):  
S Varró ◽  
F. Ehlotzky
Keyword(s):  
Integral Equation ◽  
Laser Field ◽  
Integral Equation Method ◽  
Bound State ◽  
Multiphoton Processes ◽  
Matrix Elements ◽  
Short Range Potential ◽  
High Laser ◽  
Transition Matrix Elements ◽  
Generation Of Harmonics

We derive a new integral equation for the nonperturbative evaluation of transition matrix elements of multiphoton processes at high laser field intensities. Our approach to the problem is partly based on the use of the Kramers–Henneberger transformation. As a specific example, we apply our method to the generation of harmonics by an electron in a short range potential, supporting a single bound state. We evaluate the rates R2m+1 for the odd harmonics created in this process for a set of specifically chosen parameters and we compare our results with presently available experimental data.

scholarly journals The volume integral equation method in magnetostatic problem

2019 ◽  
Vol 27 (1) ◽  
pp. 60-69
Author(s):  
Pavel G Akishin ◽  
Andrey A Sapozhnikov
Keyword(s):  
Integral Equation ◽  
Large Scale ◽  
Singular Integrals ◽  
Integral Equation Method ◽  
Equation Method ◽  
Matrix Elements ◽  
System Matrix ◽  
Volume Integral ◽  
Volume Integral Equation ◽  
Volume Integral Equation Method

This article addresses the issues of volume integral equation method application to magnetic system calculations. The main advantage of this approach is that in this case finding the solution of equations is reduced to the area filled with ferromagnetic. The difficulty of applying the method is connected with kernel singularity of integral equations. For this reason in collocation method only piecewise constant approximation of unknown variables is used within the limits of fragmentation elements inside the famous package GFUN3D. As an alternative approach the points of observation can be replaced by integration over fragmentation element, which allows to use approximation of unknown variables of a higher order.In the presented work the main aspects of applying this approach to magnetic systems modelling are discussed on the example of linear approximation of unknown variables: discretisation of initial equations, decomposition of the calculation area to elements, calculation of discretised system matrix elements, solving the resulting nonlinear equation system. In the framework of finite element method the calculation area is divided into a set of tetrahedrons. At the beginning the initial area is approximated by a combination of macro-blocks with a previously constructed two-dimensional mesh at their borders. After that for each macro-block separately the procedure of tetrahedron mesh construction is performed. While calculating matrix elements sixfold integrals over two tetrahedra are reduced to a combination of fourfold integrals over triangles, which are calculated using cubature formulas. Reduction of singular integrals to the combination of the regular integrals is proposed with the methods based on the concept of homogeneous functions. Simple iteration methods are used to solve non-linear discretized systems, allowing to avoid reversing large-scale matrixes. The results of the modelling are compared with the calculations obtained using other methods.

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