scholarly journals Approximation to Logarithmic-Cauchy Type Singular Integrals with Highly Oscillatory Kernels

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 728 ◽  
Author(s):  
SAIRA ◽  
Shuhuang Xiang
Keyword(s):  
Singular Integrals ◽  
Oscillatory Integral ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Cauchy Kernel ◽  
Cauchy Type ◽  
Highly Oscillatory ◽  
Logarithmic Singularities ◽  
Highly Oscillatory Integrals ◽  
Modified Moments

In this paper, a fast and accurate numerical Clenshaw-Curtis quadrature is proposed for the approximation of highly oscillatory integrals with Cauchy and logarithmic singularities, ⨍ − 1 1 f ( x ) log ( x − α ) e i k x x − t d x , t ∉ ( − 1 , 1 ) , α ∈ [ − 1 , 1 ] for a smooth function f ( x ) . This method consists of evaluation of the modified moments by stable recurrence relation and Cauchy kernel is solved by steepest descent method that transforms the oscillatory integral into the sum of line integrals. Later theoretical analysis and high accuracy of the method is illustrated by some examples.

scholarly journals Numerical Steepest Descent Method for Hankel Type of Hypersingular Oscillatory Integrals in Electromagnetic Scattering Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Qinghua Wu ◽  
Mengjun Sun
Keyword(s):  
Electromagnetic Scattering ◽  
High Efficiency ◽  
Integral Formula ◽  
Main Idea ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Scattering Problems ◽  
Hypersingular Integrals ◽  
Cauchy's Theorem ◽  
Highly Oscillatory

We present a fast and accurate numerical scheme for approximating hypersingular integrals with highly oscillatory Hankel kernels. The main idea is to first change the integration path by Cauchy’s theorem, transform the original integral into an integral on a , + ∞ , and then use the generalized Gauss Laguerre integral formula to calculate the corresponding integral. This method has the advantages of high-efficiency, fast convergence speed. Numerical examples show the effect of this method.

scholarly journals Differential equations with general highly oscillatory forcing terms

2014 ◽  
Vol 470 (2161) ◽  
pp. 20130490 ◽  
Author(s):  
M. Condon ◽  
A. Iserles ◽  
S. P. Nørsett
Keyword(s):  
Differential Equations ◽  
Large Family ◽  
Oscillatory Integral ◽  
Powerful Method ◽  
Initial Value ◽  
Linear Combinations ◽  
Highly Oscillatory Integral ◽  
Highly Oscillatory ◽  
Highly Oscillatory Integrals ◽  
Similar Theory

The concern of this paper is in expanding and computing initial-value problems of the form y ′= f ( y )+ h ω ( t ), where the function h ω oscillates rapidly for ω ≫1. Asymptotic expansions for such equations are well understood in the case of modulated Fourier oscillators and they can be used as an organizing principle for very accurate and affordable numerical solvers. However, there is no similar theory for more general oscillators, and there are sound reasons to believe that approximations of this kind are unsuitable in that setting. We follow in this paper an alternative route, demonstrating that, for a much more general family of oscillators, e.g. linear combinations of functions of the form e i ωg k ( t ) , it is possible to expand y ( t ) in a different manner. Each r th term in the expansion is for some ς >0 and it can be represented as an r -dimensional highly oscillatory integral. Because computation of multivariate highly oscillatory integrals is fairly well understood, this provides a powerful method for an effective discretization of a numerical solution for a large family of highly oscillatory ordinary differential equations.

Solution of inverse coefficient problem for fractional anomalous diffusion equation

2012 ◽  
Vol 9 (2) ◽  
pp. 65-70
Author(s):  
E.V. Karachurina ◽  
S.Yu. Lukashchuk
Keyword(s):  
Anomalous Diffusion ◽  
Fractional Derivatives ◽  
Descent Method ◽  
Extremum Problem ◽  
Steepest Descent Method ◽  
Coefficient Problem ◽  
Caputo Fractional Derivatives ◽  
Inverse Coefficient Problem ◽  
Residual Functional ◽  
Inverse Coefficient

An inverse coefficient problem is considered for time-fractional anomalous diffusion equations with the Riemann-Liouville and Caputo fractional derivatives. A numerical algorithm is proposed for identification of anomalous diffusivity which is considered as a function of concentration. The algorithm is based on transformation of inverse coefficient problem to extremum problem for the residual functional. The steepest descent method is used for numerical solving of this extremum problem. Necessary expressions for calculating gradient of residual functional are presented. The efficiency of the proposed algorithm is illustrated by several test examples.

scholarly journals Improved Immune Algorithm Combined with Steepest Descent Method for Optimal Design of IPMSM for FCEV Traction Motor

Energies ◽  
2021 ◽  
Vol 14 (13) ◽  
pp. 3904
Author(s):  
Ji-Chang Son ◽  
Myung-Ki Baek ◽  
Sang-Hun Park ◽  
Dong-Kuk Lim
Keyword(s):  
Optimal Design ◽  
Permanent Magnet Synchronous Motor ◽  
Torque Ripple ◽  
Descent Method ◽  
Immune Algorithm ◽  
Electric Machines ◽  
Steepest Descent Method ◽  
Superior Performance ◽  
Element Analysis ◽  
Traction Motor

In this paper, an improved immune algorithm (IIA) was proposed for the torque ripple reduction optimal design of an interior permanent magnet synchronous motor (IPMSM) for a fuel cell electric vehicle (FCEV) traction motor. When designing electric machines, both global and local solutions of optimal designs are required as design result should be compared in various aspects, including torque, torque ripple, and cogging torque. To lessen the computational burden of optimization using finite element analysis, the IIA proposes a method to efficiently adjust the generation of additional samples. The superior performance of the IIA was verified through the comparison of optimization results with conventional optimization methods in three mathematical test functions. The optimal design of an IPMSM using the IIA was conducted to verify the applicability in the design of practical electric machines.

Sign in / Sign up

Export Citation Format

Share Document