Integral Equation Of First Kind Research Articles

Superconvergence results for hypersingular integral equation of first kind by Chebyshev spectral projection methods

In this article, we propose Chebyshev spectral projection methods to solve the hypersingular integral equation of first kind. The presence of strong singularity in Hadamard sense in the first part of the integral equation makes it challenging to get superconvergence results. To overcome this, we transform the first kind hypersingular integral equation into a second kind integral equation. This is achieved by defining a bounded inverse of the hypersingular integral operator in some suitable Hilbert space. Using iterated Chebyshev spectral Galerkin method on the equivalent second kind integral equation, we obtain improved convergence of O(N−2r), where N is the highest degree of Chebyshev polynomials employed in the approximation space and r is the smoothness of the solution. Further, using commutativity of projection operator and inverse of the hypersingular integral operator, we are able to obtain superconvergence of O(N−3r) and O(N−4r), by Chebyshev spectral multi-Galerkin method (CSMGM) and iterated CSMGM, respectively. Finally, numerical examples are presented to verify our theoretical results.

Low rank approximation in the computation of first kind integral equations with TauToolbox

Tau Toolbox is a mathematical library for the solution of integro-differential problems, based on the spectral Lanczos' Tau method. Over the past few years, a class within the library, called polynomial, has been developed for approximating functions by classical orthogonal polynomials and it is intended to be an easy-to-use yet efficient object-oriented framework.In this work we discuss how this class has been designed to solve linear ill-posed problems and we provide a description of the available methods, Tikhonov regularization and truncated singular value expansion. For the solution of the Fredholm integral equation of the first kind, which is built from a low-rank approximation of the kernel followed by a numerical truncated singular value expansion, an error estimate is given.Numerical experiments illustrate that this approach is capable of efficiently compute good approximations of linear discrete ill-posed problems, even facing perturbed available data function, with no programming effort. Several test problems are used to evaluate the performance and reliability of the solvers.The final product of this paper is the numerical solution of a first-kind integral equation, which is constructed using only two inputs from the user: the kernel and the right-hand side.

Immersed Boundary Double Layer method: An introduction of methodology on the Helmholtz equation

The Immersed Boundary (IB) method of Peskin (1977) [1] is useful for problems that involve fluid-structure interactions or complex geometries. By making use of a regular Cartesian grid that is independent of the geometry, the IB framework yields a robust numerical scheme that can efficiently handle immersed deformable structures. Additionally, the IB method has been adapted to problems with prescribed motion and other PDEs with given boundary data. IB methods for these problems traditionally involve penalty forces which only approximately satisfy boundary conditions, or they are formulated as constraint problems. In the latter approach, one must find the unknown forces by solving an equation that corresponds to a poorly conditioned first-kind integral equation. This operation can therefore require a large number of iterations of a Krylov method, and since a time-dependent problem requires this solve at each step in time, this method can be prohibitively inefficient without preconditioning. This work introduces a new, well-conditioned IB formulation for boundary value problems, called the Immersed Boundary Double Layer (IBDL) method. In order to lay the groundwork for similar formulations of Stokes and Navier-Stokes equations, this paper focuses on the Poisson and Helmholtz equations to introduce the methodology and to demonstrate its efficiency over the original constraint method. In this double layer formulation, the equation for the unknown boundary distribution corresponds to a well-conditioned second-kind integral equation that can be solved efficiently with a small number of iterations of a Krylov method without preconditioning. Furthermore, the iteration count is independent of both the mesh size and spacing of the immersed boundary points. The method converges away from the boundary, and when combined with a local interpolation, it converges in the entire PDE domain. Additionally, while the original constraint method applies only to Dirichlet problems, the IBDL formulation can also be used for Neumann boundary conditions.

Quasi-local and frequency-robust preconditioners for the Helmholtz first-kind integral equations on the disk

We propose preconditioners for the Helmholtz scattering problems by a planar, disk-shaped screen in ℝ3. Those preconditioners are approximations of the square-roots of some partial differential operators acting on the screen. Their matrix-vector products involve only a few sparse system resolu- tions and can thus be evaluated cheaply in the context of iterative methods. For the Laplace equation (i.e. for the wavenumber k = 0) with Dirichlet condition on the disk and on regular meshes, we prove that the preconditioned linear system has a bounded condition number uniformly in the mesh size. We further provide numerical evidence indicating that the preconditioners also perform well for large values of k and on locally refined meshes.

RADIOTHERAPY TUMOR CONTROL CUMULATIVE PROBABILITY 3D INTEGRAL EQUATION WITH TCP IMPROVED MODEL

A triple-variable Integral Equation for Tumor Control Cumulative Probability (TCCP) is developed with the addition of the important radiotherapy Treatment-Time parameter. Based on a previous double TCCP Integral Equation depending on Biological Model parameters (α , β ), the NEffective model is implemented in Tumor Control Probability (TCP) equation. Results comprise a TCCP Integral Equation model with three essential parameters for Radiation Therapy precise determination of TCCP. Solution shows the method to obtain a 3D Integral Equation of first kind. Results applications could involve better optimization for TCCP in radiotherapy biological models. Radiotherapy Treatment Planning Optimization (TPO) and Radioprotection consequences are explained.

The study of approximate solution of one class of hypersingular integral equations of first kind

This work presents the method for calculating the approximate solution to the integral equation of the first kind of the Neumann boundary value problems for the Helmholtz equation in two-dimensional space

INVESTIGATION OF THE SOLUTION OF A FREDHOLM INTEGRAL EQUATION OF FIRST KIND WITH RESPECT TO THE FOURIER SERIES

When an integral equation is solved by using the Fourier series then the solution represents a stationary signal. Usually, integral equation is solved by the successive approximation method and the resolvent kernel method in which the solution is not of Fourier series type and this solution does not represent a stationary signal. In this paper, our main goal is to determine the solution of a Fredholm integral equation of first kind by using the Fourier series.

Numerical Approach on Oblique Wave Scattering by a Wide Rectangular Impediment With a Vent Placed Under a Finite Depth Water Body With Ice-Covered Surface

Abstract In this paper, we investigate the scattering of obliquely incident surface water waves on a hurdle as a form of a thick symmetric wall with a gap immersed in a finite depth water body having a cover of a thin ice sheet. In the context of the linear theory of water waves, this two-dimensional problem is formulated as a first-kind integral equation by splitting the velocity potential into symmetric and antisymmetric parts. The integral equation is tackled by using two numerical methods. The first method is the boundary element method where the range of integration is divided into a finite number of small line elements, and choosing the unknown function of the integral equation as constant in each line interval, we reduce the integral equation into a linear system of an algebraic equation. The second method is the multi-term Galerkin Approximation method where the basis functions are chosen as ultraspherical Gegenbauer polynomials to reduce the integral equation to a system of an algebraic equation. These systems of equations are then solved to obtain the unknown function of the integral equation in both methods. Here, very accurate numerical estimates are obtained for the reflection coefficient by both methods which are depicted graphically against the wave number for different parameters involving this problem. Also, there is a good agreement between the results of the reflection coefficient by the two methods.

Legendre spectral projection methods for Fredholm integral equations of first kind

Abstract In this paper, we discuss the Legendre spectral projection method for solving Fredholm integral equations of the first kind using Tikhonov regularization. First, we discuss the convergence analysis under an a priori parameter strategy for the Tikhonov regularization using Legendre polynomial basis functions, and we obtain the optimal convergence rates in the uniform norm. Next, we discuss Arcangeli’s discrepancy principle to find a suitable regularization parameter and obtain the optimal order of convergence in uniform norm. We present numerical examples to illustrate the theoretical results.

Solving System of Fredholm Integral Equations of First Kind By Using Homotopy Perturbation Method

Solving System of Fredholm Integral Equations of First Kind By Using Homotopy Perturbation Method

Interaction among offset parallel cracks in an orthotropic plane under thermo‐mechanical loading

AbstractThe article deals with the problem of three thermally insulated cracks embedded in a fibrous composite material subjected to thermo‐mechanical loading. Orthotropy is ensured by the unidirectional reinforcement of fiber. The considered infinite orthotropic plane is divided mathematically into three parts. The present mathematical model comprises of a central crack parallel to two symmetric collinear cracks at an offset distance to it. All the cracks under consideration are thermally insulated and aligned in the fiber direction. Using the Fourier integral transformed technique, the governing equations are reduced into a pair of singular integral equations. The Chebyshev polynomials are used to solve the integral equations of first kind with Cauchy kernel functions. Numerical values of the mode I stress intensity factors (SIFs) at the tips of the cracks are found for different lengths and distance between those. The effect of cracks' interaction on the SIFs is depicted graphically for the fiber‐reinforced composite (FRC) material constituted by carbon‐based fiber—Graphite and matrix—Epoxy.

Coupled Domain-Boundary Variational Formulations for Hodge\u2013Helmholtz Operators

We couple the mixed variational problem for the generalized Hodge–Helmholtz or Hodge–Laplace equation posed on a bounded 3D Lipschitz domain with the first-kind boundary integral equations arising from the latter when constant coefficients are assumed in the unbounded complement. Recently developed Calderón projectors for the relevant boundary integral operators are used to perform a symmetric coupling. We prove stability of the coupled problem away from resonant frequencies by establishing a generalized Gårding inequality (T-coercivity). The resulting system of equations describes the scattering of monochromatic electromagnetic waves at a bounded inhomogeneous isotropic body possibly having a “rough” surface. The low-frequency robustness of the potential formulation of Maxwell’s equations makes this model a promising starting point for Galerkin discretization.

Exact solution of non-linear volterra integral equation of first kind using Rishi transform

The problems of Engineering and Science can easily represent by developing their mathematical models in the terms of integral equations. Various analytical and numerical methods are available that can be used for solving integral equations of different kinds. In this paper, authors have considered recently developed integral transform “Rishi Transform” for obtaining the exact solution of non-linear Volterra integral equation of first kind (NLVIEFK). Four numerical problems have considered for demonstrating the complete procedure of determining the exact solution. Results of these problems depict that Rishi transform is very effective integral transform and it provides the exact solution of NLVIEFK without doing complicated calculation work.

Modelling of the reflectance profile of axisymmetrical radiolocation objects using the integral equation method

Based on the first-kind integral equation method for the electric field, the procedure and software for calculating the radar cross-section of axisymmetrical objects, bodies of revolution, are developed. Algorithms are proposed for computation of the matrix of mutual impedances and Green's function of a ring source providing the computation accuracy required for obtaining a stable solution. The method of solution approximation accuracy evaluation by azimuthal harmonics is proposed. Comparison with test examples is carried out and the applicability for solving real-world problems is shown.

Taylor approximation for solving linear and nonlinear Ill-Posed Volterra equations via an iteration method

In this paper, we present two algorithms for the approximate or exact solution of a class of Volterra integral equations of first kind. As well known, this is an ill posed problem, but we convert it to well-posedness of the second kind Volterra problems, then we apply the variational iteration method. Finally, we present two examples which show the performance and effciency of our method.

Computation of distribution of relaxation times by Tikhonov regularization for Li ion batteries: usage of L-curve method

In this paper, the distribution of relaxation times (DRTs) functions are calculated numerically in Matlab for synthetic impedance data from single parallel RC circuit and two parallel RC circuits connected in series, experimental impedance data from supercapacitors and α-LiFeO2 anode based Li ion batteries. The quality of the impedance data is checked with the Kramers–Krönig (KK) relations. The DRTs are calculated within the KK compatible regime for all the systems using Tikhonov regularization (TR) method. Here we use a fast and simple L-curve method to estimate the TR parameter (λ) for regularization of the Fredholm integral equations of first kind in impedance. Estimation of the regularization parameters are performed effectively from the offset of the global corner of the L-curve rather than simply using the global corner. The physical significances of DRT peaks are also discussed by calculating the effective resistances and capacitances coupled with peak fitting program. For instance, two peaks in the DRTs justify the electrical double layer capacitance and ion diffusion phenomena for supercapacitors in low to intermediate frequencies respectively. Moreover, the surface film effect, Li/electrolyte and electrode/electrolyte charge transfer related processes are identified for α-LiFeO2 anode based Li-ion batteries. This estimation of the offset of the global corner extends the L-curve approach coupled with the Tikhonov regularization in the field of electrochemistry and can also be applied in similar process detection methods.

Wave propagation over a rectangular trench in the presence of a partially immersed barrier

The problem of water wave propagation over a rectangular submarine trench in the presence of a thin vertical partially immersed barrier for its three different positions are investigated here assuming linear theory. For each position of the barrier, the problem is reduced to solving three weakly singular integral equations of first kind involving horizontal component of velocity above the two edges of the trench and below the barrier. The integral equations are solved employing Galerkin approximation in terms of simple polynomials multiplied by appropriate weight functions whose forms are dictated by the edge conditions at the corners of the trench and at the submerged end of the barrier. The numerical estimates for reflection and transmission coefficients are depicted graphically against the wave number for different lengths and positions of the barrier.

New preconditioners for the Laplace and Helmholtz integral equations on open curves: analytical framework and numerical results

Helmholtz wave scattering by open screens in 2D can be formulated as first-kind integral equations which lead to ill-conditioned linear systems after discretization. We introduce two new preconditioners in the form of square-roots of on-curve differential operators both for the Dirichlet and Neumann boundary conditions on the screen. They generalize the so-called “analytical” preconditioners available for Lipschitz scatterers. We introduce a functional setting adapted to the singularity of the problem and enabling the analysis of those preconditioners. The efficiency of the method is demonstrated on several numerical examples.

SOLUTION OF INTEGRAL EQUATIONS BY BESSEL WAVELET TRANSFORM

The integral equations of the first kind arise in many areas of science and engineering fields such as image processing and electromagnetic theory. The wavelet transform technique to solve integral equation allows the creation of very fast algorithms when compared with known algorithms. Various wavelet methods are used to solve certain type of integral equations. To find the most accurate and stable solution of the integral equation Bessel wavelet is the appropriate method. To study the properties of solution of integral equations on distribution spaces Bessel wavelet transform is also used. In this paper, we accomplished the concept of Hankel convolution and continuous Bessel wavelet transform to solve certain types of integral equations (Volterra integral equation of first kind, Volterra integral equation of second kind and Abel integral equation). Also distributional wavelet transform and generalized convolution will be applied to find the solution of certain Integral equations.

Water wave propagation over an infinite step in the presence of a thin vertical barrier

Problems of water wave propagation over an infinite step in the presence of a thin vertical barrier of four different geometrical configurations are investigated in this paper. For each configuration of the barrier, the problem is reduced to solving an integral equation or a coupled integral equation of first kind involving horizontal component of velocity below or above the barrier and above the step. The integral equations are solved employing Galerkin approximation in terms of simple polynomials multiplied by appropriate weight functions whose forms are dictated by the edge conditions at the corner of the step and at the submerged end(s) of the barrier. The reflection and transmission coefficients are then computed and depicted graphically against the wave number.