Quadrature methods for periodic singular and weakly singular Fredholm integral equations

High-accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are subsequently used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Throughout the development the periodic nature of the problem plays a crucial role. Such periodic equations are used in the solution of planar elliptic boundary value problems such as those that arise in elasticity, potential theory, conformal mapping, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples.     

Quadrature rules for numerical integration based on Haar wavelets and hybrid functions

In this paper Haar wavelets and hybrid functions have been applied for numerical solution of double and triple integrals with variable limits of integration. This approach is the generalization and improvement of the methods (Siraj-ul-Islam et al. (2010) [9]) where the numerical methods are only applicable to the integrals with constant limits. Apart from generalization of the methods [9], the new approach has two major advantages over the classical methods based on quadrature rule: (i) No need of finding optimum weights as the wavelet and hybrid coefficients serve the purpose of optimal weights automatically (ii) Mesh points of the wavelets algorithm are used as nodal values instead of considering the n nodes as unknown roots of polynomial of degree n. The new methods are more efficient. The novel methods are compared with existing methods and applied to a number of benchmark problems. Accuracy of the methods are measured in terms of absolute errors.     

Chebyshev approximation for solving the radon integral equation

A Volterra integral equation, which relates the Fourier coefficients of the projection (in polar coordinates) with the corresponding coefficients of the unknown density function is deduced. The proposed method describes a new technique based on Chebyshev approximation of the integral equation of Abel type which is deduced by using the Radon inversion transformation, the recursive evaluation of certain integrals is calculated. We obtain also the same integral equation by other inversion formulae based on Hankel transformations. Numerical examples are treated and the method compares quite favourably with other known methods.     

A compression technique for the boundary integral equation reduced from Helmholtz equation

Applying the trigonometric wavelets and the multiscale Galerkin method, we investigate the numerical solution of the boundary integral equation reduced from the exterior Dirichlet problem of Helmholtz equation by the potential theory. Consequently, we obtain a matrix compression strategy, which leads us to a fast algorithm. Our truncated treatment is simple, the computational complexity and the condition number of the truncated coefficient matrix are bounded by a constant. Furthermore, the entries of the stiffness matrix can be evaluated from the Fourier coefficients of the kernel of the boundary integral equation. Examples given for demonstrating our numerical method shorten the runtime obviously.     

An integral equation method for inversion of electromagnetic soundings

The analogy of the problem of electromagnetic induction sounding of the earth's conductivity-depth profile using a two perpendicular-loop system to the general remote sensing problem of the atmosphere is pointed out. The induction sounding problem has been reduced to the solution of a new integral equation of Fredholm type. Kernels of the integral equation are evaluated, and their characteristics are investigated. A nonlinear relaxation inversion technique is suggested for the solution. Convergence of the solution, information content of measurements, and the range of possible retrieval of the conductivity profile are discussed.     

Quadrature rules for the integration of the product of two oscillatory functions with different frequencies

We construct quadrature rules for the efficient computation of the integral of a product of two oscillatory functions y₁(x) and y₂(x), where , and the functions f_i,j(x) are smooth. The weights are evaluated by the exponential fitting technique of Ixaru [Comput. Phys. Comm. 105 (1997) 1-19], which is now extended to cover the case of two frequencies. We give a numerical illustration on how the new rules compare for accuracy with the one-frequency dependent rules and with the classical ones.     

On a random Volterra integral equation

Summary Tsokos [12] showed the existence of a unique random solution of the random Volterra integral equation (*)x(t; ) = h(t; ) + _o ^t k(t, ; )f(, x(; )) d, where , the supporting set of a probability measure space (,A, P). It was required thatf must satisfy a Lipschitz condition in a certain subset of a Banach space. By using an extension of Banach's contraction-mapping principle, it is shown here that a unique random solution of (*) exists whenf is (, )-uniformly locally Lipschitz in the same subset of the Banach space considered in [12].     

B-polynomial multiwavelets approach for the solution of Abel's integral equation

A numerical method for solving Abel's integral equation as singular Volterra integral equations is presented. The method is based upon Bernstein polynomial (B-polynomial) multiwavelet basis approximations. The properties of B-polynomial multiwavelets are first presented. These properties are then utilized to reduce the singular Volterra integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

Numerical solution of an integral equation for perpetual Bermudan options

We consider perpetual Bermudan options, which have no expiration and can be exercised every T time units. We use the Green's function approach to write down an integral equation for the value of a perpetual Bermudan call option on an expiration date; this integral equation leads to a Wiener–Hopf problem. We discretize the integral in the integral equation to convert the problem to a linear algebra problem, which is straightforward to solve, and this enables us to find the location of the free boundary and the value of the perpetual Bermudan call. We compare our results to earlier studies which used other numerical methods.     

A structured low-rank wavelet solver for the Ornstein-Zernike integral equation

In this article, we present a new structured wavelet algorithm to solve the Ornstein-Zernike integral equation for simple liquids. This algorithm is based on the discrete wavelet transform of radial distribution functions and different low-rank approximations of the obtained convolution matrices. The fundamental properties of wavelet bases such as the interpolation properties and orthogonality are employed to improve the convergence and speed of the algorithm. In order to solve the integral equation we have applied a combined scheme in which the coarse part of the solution is calculated by the use of wavelets and Newton-Raphson algorithm, while the fine part is solved by the direct iteration. Tests have indicated that the proposed procedure is more effective than the conventional method based on hybrid algorithms.     

A modified boundary integral evolution formulation for the wave equation

We apply a modified boundary integral formulation otherwise known as the Green element method (GEM) to the solution of the two-dimensional scalar wave equation.GEM essentially combines three techniques namely: (a) finite difference approximation of the time term (b) finite element discretization of the problem domain and (c) boundary integral replication of the governing equation. These unique and advantageous characteristics of GEM facilitates a direct numerical approximation of the governing equation and obviate the need for converting the governing partial differential equation to a Helmholtz-type Laplace operator equation for an easier boundary element manipulation. C₁ continuity of the computed solutions is established by using Overhauser elements. Numerical tests show a reasonably close agreement with analytical results. Though in the case of the Overhauser GEM solutions, the level of accuracy obtained does not in all cases justify the extra numerical rigor.     

Solution of Hallen's integral equation using multiwavelets

An effective method based upon Alpert multiwavelets is proposed for the solution of Hallen's integral equation. The properties of Alpert multiwavelets are first given. These wavelets are utilized to reduce the solution of Hallen's integral equation to the solution of sparse algebraic equations. In order to save memory requirement and computation time, a threshold procedure is applied to obtain algebraic equations. Through numerical examples, performance of the present method is investigated concerning the convergence and the sparseness of resulted matrix equation.     

On a weakly singular Volterra integral equation

The need for providing reliable numerical methods for the solution of weakly singular Volterra integral equations ofI ^st Kind stems from the fact that they are connected to important problems in the theory and applications of stochastic processes. In the first section are briefly sketched the above problems and some peculiarities of such equations. Section 2 described the method for obtaining an approximate solution whose properties are described in section 3: such properties guarantee that our approximate solution always oscillates around the rigorous one. Section 4 discusses the applicability to our case of some classical bounds on the errors. The remaining sections are all devoted to the construction of upper bounds on the oscillating error in order to reach a high degree of reliability for our solution. All the bounds are independent on the numerical method which is employed for obtaining the numerical solution. In section 5 is derived a Volterra II Kind integral equation by subtracting to the original kernel the weak singularity, while in section 6 is given an upper bound to the error in the case of Wiener and Ornstein-Ühlenbeck kernels with constant barriers. Such a bound is generalized to other kinds of barriers in section 7 while in section 8 is suggested an approximation of the Kernel for the O. Ü. case with constant barriers and by means of it is given an explicit bound for the error in terms of Abel's transform of the known term in the original integral equation. A rough estimation of the error is also given under the assumption that \(y(t) - \int\limits_0^t {K(t,\tau )\tilde x(\tau )d_\tau [\tilde x(\tau )} \) denotes any approximate solution of (1a) obtained by any method] can be approximated by means of a sinusoidal function. In section 9 is derived another kind of bound, for constant barriers, by using the approximate Kernel of section 7 and classical results.     

Approximation of solutions to fractional integral equation

In this paper we shall study a fractional integral equation in an arbitrary Banach space $X$. We used the analytic semigroups theory of linear operators and the fixed point method to establish the existence and uniqueness of solutions of the given problem. We also prove the existence of global solution. The existence and convergence of the Faedo–Galerkin solution to the given problem is also proved in a separable Hilbert space with some additional assumptions on the operator $A$. Finally we give an example to illustrate the applications of the abstract results.     

The Trefftz method as an integral equation

The Trefftz method may be described in terms of an integral equation formulation based on eigenfunction expansions rather than the usual element (BEM) solution method. Using an ‘Embedding Integral Equation’ approach, the Trefftz method may not only be seen to be related to integral equation methods but it is also seen to be amenable to partitioning, allowing more rapid convergence.     

A structural equation modeling approach to generate explanations for induced rules

The performance of an expert system depends on the quality and validity of the domain-specific knowledge built into the system. In most cases, however, domain knowledge (e.g. stock market behavior knowledge) is unstructured and differs from one domain expert to another. So, in order to acquire domain knowledge, expert system developers often take an induction approach in which a set of general rules is constructed from past examples. Expert systems based upon the induced rules were reported to perform quite well in the hold-out sample test.

However, these systems hardly provide users with an explanation which would clarify the results of a reasoning process. For this reason, users would remain unsure about whether to accept the system conclusion or not. This paper presents an approach in which explanations about the induced rules are constructed. Our approach applies the structural equation model to the quantitative data, the qualitative format of which was originally used in rule induction. This approach was implemented with Korean stock market data to show that a plausible explanation about the induced rule can be constructed.     

Application of singular value decomposition for integral equation procedures in MIMICAD

An efficient moment method procedure which utilizes a point matching scheme together with a generalized inverse is formulated and implemented with the use of the singular value decomposition. The integral representation of Green's functions and a combination of semi-infinite and subdomain expansion functions are used to formulate the numerical technique required to solve for the scattering parameters of planar microstrip multiports. Scattering parameters of typical microstrip discontinuities computed by utilizing this procedure are presented and compared with known published results. © 1992 John Wiley & Sons, Inc.     

Chebyshev collocation method for solving singular integral equation with cosecant kernel

A collocation method based on Chebyshev polynomials is proposed for solving cosecant-type singular integral equations (SIE). For solving SIE, difficulties lie in its singular term. In order to remove singular term, we introduce Gauss–Legendre integration and integral properties of the cosecant kernel. An advantage of this method is to approximate the best uniform approximation by the best square approximation to obtain the unknown coefficients in the method. On the other hand, the convergence is fast and the accuracy is high, which is verified by the final numerical experiments compared with the existing references.     

Interval arithmetic error estimation for the solution of Fredholm integral equation

Finite element method with a posteriori estimation using interval arithmetic is discussed for a Fredholm integral equation of the second kind. This approach is general. It leads to the guaranteed L ^∞ asymptotically exact estimate without the usual overestimation when interval arithmetic is used. An algorithm is provided for determination of an approximate solution such that the computed error bound between the exact solution and its approximation in L ^∞ is less than the given tolerance ?. Numerical solution for the equation with only C ¹ kernel illustrates the approach.