On Love's integral equation using wavelets
Swaraj Paul (SRM University, Chennai)
Speaker |
Swaraj Paul (SRM University, Chennai)
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When |
Sep 20, 2024
from 04:00 PM to 05:00 PM |
Where | LH-006, Ground Floor |
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SEMINAR TALK
Title: On Love's integral equation using wavelets.
Abstract: Most real-life physical phenomena can be modeled using Partial Differential Equations (PDEs) or Integral Equations (IEs). Depending on the problem, handling the equivalent IEs is sometimes easier than dealing with PDEs. One integral equation with significant applications in various fields, such as water wave scattering, elasticity, potential theory, diffraction, and quantum mechanics, is the Love Integral Equation (LIE), first studied by E. R. Love in 1949.
This is a regular integral equation with a Poisson kernel on the upper-half plane, involving a small parameter. By employing the approximate identity, it can be shown that the solution tends to u(x) = 1/2 for -1<x<1, and u(1) = 3/4 as the small parameter approaches zero. Due to the singularity at the endpoint, producing a numerical solution becomes quite challenging. Therefore, reproducing such a solution is difficult. Since 1949, numerous numerical methods have been developed to solve the LIE, including the Nyström method, quadrature rules, collocation methods, Gauss–Legendre quadrature, and others. Wavelet methods have also proven effective for detecting singularities and handling singular integrals.
In this talk, we will implement a Legendre multiwavelet-based method to solve a LIE with a very small parameter. We will demonstrate the efficiency of the wavelet in detecting singularities. Finally, we will validate the method by solving one LIE.
This is a regular integral equation with a Poisson kernel on the upper-half plane, involving a small parameter. By employing the approximate identity, it can be shown that the solution tends to u(x) = 1/2 for -1<x<1, and u(1) = 3/4 as the small parameter approaches zero. Due to the singularity at the endpoint, producing a numerical solution becomes quite challenging. Therefore, reproducing such a solution is difficult. Since 1949, numerous numerical methods have been developed to solve the LIE, including the Nyström method, quadrature rules, collocation methods, Gauss–Legendre quadrature, and others. Wavelet methods have also proven effective for detecting singularities and handling singular integrals.
In this talk, we will implement a Legendre multiwavelet-based method to solve a LIE with a very small parameter. We will demonstrate the efficiency of the wavelet in detecting singularities. Finally, we will validate the method by solving one LIE.
Speaker's Bio: Research Assistant Professor Department of Mathematics College of Engineering and Technology SRM Institute of Science and Technology Kattankulathur - 603203 Chengalpet District Tamilnadu, India.