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Numerical Method for Simulation of Physical Processes Modeled by Abel’s Integral Equations
Published in Seshu Kumar Damarla, Madhusree Kundu, Fractional Order Processes, 2018
Seshu Kumar Damarla, Madhusree Kundu
is a singular integral equation (a particular case of the first-kind weakly singular Volterra integral equation) derived by Norwegian mathematician Niles Abel in 1823 to describe the motion of a particle sliding down along a smooth unknown curve, in a vertical plane, under the influence of the gravity. In Equation (3.1), f(t) is the time taken by the particle to move from the highest point of vertical height x to the lowest point 0 on the smooth curve. Among the integral equations such as Fredholm integral equation, Volterra integral equation, Cauchy integral equation, and so on, Abel’s integral equation is the most important one, the first integral equation ever treated, which directed mathematicians to the study of integral equations.
EMI and EMC Control, Case Studies, EMC Prediction Techniques, and Computational Electromagnetic Modeling
Published in David A. Weston, Electromagnetic Compatibility, 2017
The moment method is a technique for solving complex integral equations by reducing them to a system of simpler linear equations. The moment method uses a technique known as the method of weighted residuals, which has become synonymous with the moment method and with the “surface integral technique.” The MOM technique generally does an excellent job of analyzing unbounded radiation problems and analyzing perfect conductors or structures with a given conductivity. They are not well suited to structures with mixed conductivity and permittivity, although they can be used for homogeneous dielectrics alone or for very specific conductor-dielectric geometries. The MOM technique can be used to analyze a wide variety of 3D electromagnetic radiation problems.
Diffractive Elements
Published in Daniel Malacara-Hernández, Brian J. Thompson, Fundamentals and Basic Optical Instruments, 2017
Rigorous methods have led to several approaches: the integral, differential, or variational [10,11]; analytic continuation [12]; and variational methods and others [13]. The integral approach covers a range of methods based on the solution of integral equations. In some methods, the wave equations are solved by numerical integration through the grating. Recently, a method has been proposed for the calculation or the diffraction efficiency that includes the response of photosensitive materials that have a non-uniform thickness variation or erosion of the emulsion surface due to the developing process [14a, 14b].
Coupled eigenfunction expansion–boundary element method for wave scattering by thick vertical barrier over an arbitrary seabed
Published in Geophysical & Astrophysical Fluid Dynamics, 2021
A. Choudhary, S. Koley, S. C. Martha
In this section, the eigenfunction expansion method is applied to the regions and where as the BEM is applied to the region to solve the physical problem as described in section 2. The eigenfunction expansion method is used to obtain the form of the velocity potentials for regions and whilst Green's second identity along with free space Green's function is used to convert the boundary value problem into an integral equation in region . Finally, by matching the normal velocity and pressure along the auxiliary boundaries and , the contribution of the regions and are incorporated into the integral equation of region and the same is handled for solution using the BEM.
Water wave interaction with dual unequal horizontal flexible porous barriers using integral equations
Published in Ships and Offshore Structures, 2023
Integral equations are an important tool to tackle problems in many areas of mathematical physics and engineering. A wide range of boundary value problems can be converted to Fredholm integral equations. Hypersingular integral equations occur naturally in course of solving a variety of problems arising in the different fields such as potential flow past a flat plate, acoustics, fracture mechanics, etc. The application of method of hypersingular integral equations for examining wave–structure interaction problems is very convenient because arbitrary shapes and orientations of barriers can be handled by applying this method. Also, the solution technique for solving a hypersingular integral equation which involves appropriate expansion of the unknown function enables one to compute the hypersingular part analytically and takes care of the conditions at the edges of the structures. Also, it converges faster than boundary/finite element method or eigenfunction expansion-matching method. Thus the technique of hypersingular integral equations has been applied to a vertical plate by Maiti and Mandal (2010), an inclined porous plate by Gayen and Mondal (2014), a circular arc-shaped plate by Kanoria and Mandal (2002), multiple porous plates by Gayen and Mondal (2016), and Naskar et al. (2021). The utilisation of Havelock's expansion to formulate integral equation(s) was applied to vertical porous or rigid plates by Sasmal et al. (2019) and Sasmal and De (2020). All of them used the Galerkin approximation method to solve the associated integral equations. Again, an integral transform approach was used by Porter (2015) to formulate integral equation for the investigation of wave structure interaction with submerged horizontal rigid plates.
Verification of numerical solutions of thermal radiation problems in participating and nonparticipating media
Published in Numerical Heat Transfer, Part B: Fundamentals, 2023
Antonio Carlos Foltran, Carlos Henrique Marchi, Luís Mauro Moura
Radiative transfer in nonparticipating media is generally described by Fredholm integral equations of the second type. In such problems, two or more nonisothermal surfaces exchange radiation. Depending on the boundary conditions and geometrical disposal, the mathematical models varies from one algebraic equation with one or more integral terms, up to one Fredholm integral equation of the second type or systems of such integral equations.