A Quasi-Boundary Semi-Analytical Approach for Two-Dimensional Backward Advection-Dispersion Equation

In this study, we employ a semi-analytical approach to solve a two-dimensional advection-dispersion equation (ADE) for identifying the contamination problems. First, the Fourier series expansion technique is used to calculate the concentration field C(x, y, t) at any time t < T. Then, we ponder a direct regularization by adding an extra termaC(x, y, 0) on the final time data C(x, y, T), to reach a second-kind Fredholm integral equation. The termwise separable property of kernel function allows us obtaining a closed-form solution of the Fourier coefficients. A strategy to choose the regularization parameter is offered. The solver utilized in this work can retrieve the spatial distribution of the groundwater contaminant concentration. Several numerical examples are scrutinized to display that the new method can recover all the past data very well, and is good enough to deal with heterogeneous parameters, even though the final time data are noised seriously.     

一种基于变尺度积分滑动平均的载荷识别方法

首先,基于积分滑动平均思想构造了变尺度加权积分函数,提出了最小二乘意义下加权积分滑动平均最佳近似响应函数模型。其次,利用形函数方法构造最佳近似载荷模型,组合近似载荷及响应得到实际情况下的Duhamel积分方程,对Duhamel积分离散化得到用于载荷识别的离散线性系统方程。再次,使用正则化方法进行载荷识别。利用正则化方法中最小二乘解构造以正则化参数为自变量的函数,提出了一种选取最优正则化参数的新方法。最后,数值仿真及试验验证将该文提出方法与传统方法进行了比较,结果说明新方法能够得到精度较好的近似稳定解,并且具有较好的抗噪性。     

Fredholm integral equation of the Laser Intensity Modulation Method (LIMM): Solution with the polynomial regularization and L-curve methods

The Laser Intensity Modulation Method (LIMM) is widely used for the determination of the spatial distribution of polarization in polar ceramics and polymers, and space charge in non-polar polymers. The analysis of experimental data requires a solution of a Fredholm integral equation of the 1st kind. This is an ill-posed problem that has multiple and very different solutions. One of the more frequently used methods of solution is based upon Tikhonov regularization. A new method, the Polynomial Regularization Method (PRM), was developed for solving the LIMM equation with an 8th degree polynomial using smoothing to achieve a stable and optimal solution. An algorithm based upon the L-curve method (LCM) was used for the prediction of the best regularization parameter. LIMM data were simulated for an arbitrary polarization distribution and were analyzed using PRM and LCM. The calculated distribution function was in good agreement with the simulated polarization distribution. Experimental polarization distributions in a poorly poled sample of polyvinylidene fluoride (PVDF) and in a LiNbO₃ bimorph, and space charge in polyethylene were analyzed. The new techniques were applied to the analysis of 3-dimensional polarization distributions.     

Direct second kind boundary integral formulation for Stokes flow problems

A direct boundary element method is formulated for the Stokes flow problem based on an integral equation representation for the components of traction. For problems in which the components of velocity are prescribed on the boundary of the domain, this new formulation results in a hypersingular Fredholm integral equation of the second kind. A method of regularization to evaluate the hypersingular integral is discussed. For certain problems involving flows about particles, the integral equation representation for the tractions is not unique because of the existence of rigid body eigenmodes. A method to constrain out these rigid body modes is also discussed. Several example problems are considered in which this new formulation is compared to more traditional boundary element formulations.     

New Fredholm integral equation for multiple crack problem in plane elasticity and antiplane elasticity

A singular integral equation for the multiple crack problem of plane elasticity is formulated in this paper. In the formulation we choose the crack opening displacement (COD) as unknown function and the resultant force as the right hand term of the equation. After using Vekua's regularization procedure or making inversion of the Cauchy singular integral in the equation, a new Fredholm integral equation is obtainable. The obtained Fredholm integral equation is compact in form and easy for computation. After solving the equation, the CODs of the cracks and the stress intensity factors (SIFs) at the crack tips can be derived immediately. Similar formulation for the multiple crack problem of antiplane elasticity is also presented. Finally, numerical examples are given to demonstrate the use of the proposed integral equation approach.     

Elastic Torsion Bar with Arbitrary Cross-Section Using the Fredholm Integral Equations

By using a meshless regularized integral equation method (MRIEM), the solution of elastic torsion problem of a uniform bar with arbitrary cross-section is presented by the first kind Fredholm integral equation on an artificial circle, which just encloses the bar's cross-section. The termwise separable property of kernel function allows us to obtain the semi-analytical solutions of conjugate warping function and shear stresses. A criterion is used to select the regularized parameter according to the minimum principle of Laplace equation. Numerical examples show the effectiveness of the new method in providing very accurate numerical solutions as compared with the exact ones.     

An adaptive regularization algorithm for recovering the rate constant distribution from biosensor data

We present here the theoretical results and numerical analysis of a regularization method for the inverse problem of determining the rate constant distribution from biosensor data. The rate constant distribution method is a modern technique to study binding equilibrium and kinetics for chemical reactions. Finding a rate constant distribution from biosensor data can be described as a multidimensional Fredholm integral equation of the first kind, which is a typical ill-posed problem in the sense of J. Hadamard. By combining regularization theory and the goal-oriented adaptive discretization technique, we develop an Adaptive Interaction Distribution Algorithm (AIDA) for the reconstruction of rate constant distributions. The mesh refinement criteria are proposed based on the a posteriori error estimation of the finite element approximation. The stability of the obtained approximate solution with respect to data noise is proven. Finally, numerical tests for both synthetic and real data are given to show the robustness of the AIDA.     

A Special Solution-Selection Principle in Using a Tikhonov Regularizing Algorithm for Processing Multiwave Lidar Data

An algorithm is considered for recovering the aerosol size distribution and complex refractive index from optical data measured with a certain error δ. The size distribution and the optical data are related by a linear integral Fredholm equation of the first kind with an inaccurately specified kernel, which is solved by Tikhonov regularization. A new principle is proposed for selecting solutions, which is based on not one solution but a certain set of them. Averaging on that set results in a stable conclusion on the recovery of the aerosol parameters. __________ Translated from Izmeritel'naya Tekhnika, No. 10, pp. 14–19, October, 2005.     

A Quasi-Boundary Semi-Analytical Approach for Two-Dimensional Backward Heat Conduction Problems

In this article, we propose a semi-analytical method to tackle the two-dimensional backward heat conduction problem (BHCP) by using a quasi-boundary idea. First, the Fourier series expansion technique is employed to calculate the temperature field u(x, y, t) at any time t < T. Second, we consider a direct regularization by adding an extra termau(x, y, 0) to reach a second-kind Fredholm integral equation for u(x, y, 0). The termwise separable property of the kernel function permits us to obtain a closed-form regularized solution. Besides, a strategy to choose the regularization parameter is suggested. When several numerical examples were tested, we find that the proposed scheme is robust and applicable to the two-dimensional BHCP.     

An initial value method for dual integral equations

An initial value method is derived for a set of dual integral equations encountered in solving mixed boundary value problems in mathematical physics with a circular line of separation of boundary conditions. It is shown that the solution itself, not just a transform of the solution, of the dual integral equations satisfies a Fredholm integral equation. The initial value problem is derived from this Fredholm equation.     

Statistical approach to the solution of first-kind integral equations arising in the study of materials and their properties

There are a number of problems arising when studying the properties of materials, which require for their solution the inversion of a Fredholm first-kind integral equation. Examples include the determination of the distribution of adsorption energies on the surface of a solid and the evaluation of the distribution of pore radii of a solid from diffusion data. Such equations are, in practice, notoriously difficult to solve. This paper describes a general methodology for solving equations of this type. The method combines the ideas of regularization with a quadratic programming algorithm for minimizing quadratic expressions subject to non-negativity constraints. The condition of non-negativity is essential if we are to recover distribution functions for physical attributes of a solid. The method proposed is tested on simulated data for which the true solution to the equation is already known and on real data arising in each of the two situations described above. The method is shown to perform well in recovering the true solution for the simulated data and to produce results in the real data situations that are consistent with the data observed and with observations of related physical quantities.     

A New Quasi-Boundary Scheme for Three-Dimensional Backward Heat Conduction Problems

In this study, we employ a semi-analytical scheme to resolve the three-dimensional backward heat conduction problem (BHCP) by utilizing a quasi-bound -ary concept. First, the Fourier series expansion method is used to estimate the temperature field u(x, y, z, t) at any time t < T. Second, we ponder a direct regularization by adding an extra term a(x, y, z, 0) to transform a second-kind Fredholm integral equation for u(x, y, z, 0). The termwise separable property of the kernel function allows us to acquire a closed-form regularized solution. In addition, a tactic to determine the regularization parameter is recommended. We find that the proposed method is robust and applicable to the three-dimensional BHCP when several numerical experiments are examined.     

Numerical solution of the integral equations for optimal sequential filtering

Inhomogeneous Fredholm integral equations occur frequently in communication theory where it is desired to determine optimal receivers and filters for signal detection and estimation. In this paper an initial value method is utilized to determine the Fredholm resolvent and the solution of the integral equation. Numerical results are given for a simple example. The method is of particular interest where sequential solutions are desired.     

A Tikhonov Regularizing Algorithm for Processing Multiwave Lidar Data in the Absence of Information on the Measurement Error

An algorithm is considered for recovering the aerosol size distribution and also the mean (effective) radius and quantitative concentration from optical data: back-scattering and total attenuation coefficients as measured with a certain error δ at various wavelengths. These quantities are connected by a linear integral Fredholm equation of the first kind. A solution is provided by the modified discrepancy, which enables one to estimate the regularization parameter in the absence of information on the error δ. __________ Translated from Izmeritel'naya Tekhnika, No. 10, pp. 8–14, October, 2005.     

数值求解Symm积分方程的正则化MINRES方法

Symm积分方程在位势理论中具有重要应用,它是Hadamard意义下的不适定问题。离散该方程将产生对称线性不适定系统。基于GCV准则,并应用截断奇异值分解,本文提出数值求解Symm积分方程的正则化MINRES方法。与Tikhonov正则化方法相比,在数据出现噪声的情况下,新方法能有效地求得Symm积分方程的数值解。     

Numerical expansion-iterative method for analysis of integral equation models arising in one- and two-dimensional electromagnetic scattering

Most integral equations of the first kind are ill-posed, and obtaining their numerical solution needs often to solve a linear system of algebraic equations of large condition number. So, solving this system may be difficult or impossible. Since many problems in one- and two-dimensional scattering from perfectly conducting bodies can be modeled by Fredholm integral equations of the first kind, this paper presents an effective numerical expansion-iterative method for solving them. This method is based on vector forms of block-pulse functions. By using this approach, solving the first kind integral equation reduces to solve a recurrence relation. The approximate solution is most easily produced iteratively via the recurrence relation. Therefore, computing the numerical solution does not need to directly solve any linear system of algebraic equations and to use any matrix inversion. Also, the method practically transforms solving of the first kind Fredholm integral equation which is inherently ill-posed into solving second kind Fredholm integral equation. Another advantage is low cost of setting up the equations without applying any projection method such as collocation, Galerkin, etc. To show convergence and stability of the method, some computable error bounds are obtained. Test problems are provided to illustrate its accuracy and computational efficiency, and some practical one- and two-dimensional scatterers are analyzed by it.     

Computational methods for solving static field and eddy current problems via Fredholm integral equations

Two-dimensional static field problems can be solved by a method based on Fredholm integral equations (equations of the second kind). This has numerical advantages over the mote commonly used integral equation of the first kind. The method is applicable to both magnetostatic and electrostatic problems formulated in terms of either vector or scalar potentials. It has been extended to the solution of eddy current problems with sinusoidal driving functions. The application of the classical Fredholm equation has been extended to problems containing boundary conditions: 1) potential value, 2) normal derivative value, and 3) an interface condition, all in the same problem. The solutions to the Fredholm equations are single or double (dipole) layers of sources on the problem boundaries and interfaces. This method has been developed into computer codes which use piecewise quadratic approximations to the solutions to the integral equations. Exact integrations are used to replace the integral equations by a matrix equation. The solution to this matrix equation can then be used to directly calculate the field anywhere.     

Isoparametric,finite element,variational solution of integral equations for three-dimensional fields

An improved numerical method, based on a variational approach with isoparametric finite elements, is presented for the solution of the boundary integral equation formulation of three-dimensional fields. The technique provides higher-order approximation of the unknown function over a bounding surface described by two-parameter, non-planar elements. The integral equation is discretized through the Rayleigh–Ritz procedure. Convergence to the solution for operators having a positive-definite component is guaranteed. Kernel singularities are treated by removing them from the relevant integrals and dealing with them analytically. A successive element iterative process, which produces the solution of the large dense matrix of the complete structure, is described. The discretization and equation solution take place one element at a time resulting in storage and computational savings. Results obtained for classical test models, involving scalar electrostatic potential and vector elastostatic displacement fields, demonstrate the technique for the solution of the Fredholm integral equation of the first kind. Solution of the Fredholm equation of the second kind is to be reported subsequently.     

A Quasi-Boundary Semi-Analytical Method for Backward in Time Advection-Dispersion Equation

In this paper, we take the advantage of an analytical method to solve the advection-dispersion equation (ADE) for identifying the contamination problems. First, the Fourier series expansion technique is employed to calculate the concentration field C(x, t) at any time t< T. Then, we consider a direct regularization by adding an extra term αC(x,0) on the final condition to carry off a second kind Fredholm integral equation. The termwise separable property of the kernel function permits us to transform itinto a two-point boundary value problem. The uniform convergence and error estimate of the regularized solution C^α(x,t) are provided and a strategy to select the regularized parameter is suggested. The solver used in this work can recover the spatial distribution of the groundwater contaminant concentration. Several numerical examples are examined to show that the new approach can retrieve all past data very well and is good enough to cope with heterogeneous parameters’ problems, even though the final data are noised seriously.     

Thermal stresses near a penny-shaped crack in an elastic sphere embedded in an infinite elastic space

This paper gives an analysis of the distribution of thermal stresses in a sphere which is bonded to an infinite elastic medium. The thermal and the elastic properties of the sphere and the elastic infinite medium are assumed to be different. The penny-shaped crack lies on the diametral plane of the sphere and the centre of the crack is the centre of the sphere. By making a suitable representation of the temperature function, the heat conduction problem is reduced to the solution of a Fredholm integral equation of the second kind. Using suitable solution of the thermoelastic displacement differential equation, the problem is then reduced to the solution of a Fredholm integral equation, in which the solution of the earlier integral equation arising from heat conduction problem occurs as a known function. Numerical solutions of these two Fredholm integral equations are obtained. These solutions are used to evaluate numerical values for the stress intensity factors. These values are displayed graphically.