Narrow Results

In this study, we introduce a new procedure for the numerical solution of linear and nonlinear integral equations by combining the Monte Carlo simulation and collocation method. One of the major advantages of this method is reducing calculations by converting the integral equation to a system of algebraic equations. The numerical examples of various types of integral equations indicate the efficiency and accuracy of the proposed method. The results show this method is able to solve linear and nonlinear integral equations and also nonlinear weakly singular equations, with remarkable accuracy. Finally, the method is generalized for solving integral-differential equation of the neutron transport for the stationary case.

Image restoration is the theory and practice of processing an image to correct it for distortions caused by the image formation process. The first efforts in image restoration appeared more than 25 years ago. In this article we review the more recent trends in image restoration and discuss the main directions that are expected to influence the continued evolution of this technology.

Using the replica Ornstein–Zernike (ROZ) integral equations, we investigate the adsorption of a simple fluid in a hard sphere quenched matrix made of two species. Our main focus is in the dependencies of the density of fluid species on the chemical potential in matrices with different microporosity and for several compositions. The simple fluid is considered by using a hard sphere model. The fluid-matrix interactions are assumed either solely repulsive or attractive of the Yukawa type. The ROZ equations are supplemented by both the Percus–Yevick (PY) and the hypernetted chain (HNC) closures. The PY closure is used to study the model with solely repulsive forces (reference system) and then the contribution of attractive forces into adsorption is included in the mean field approximation. On the other hand, the HNC approximation is used to get insight into the structure of adsorbed fluid and the fluid-matrix correlations in the presence of attractive forces.

In this paper, we use the decomposition technique of Noor and Noor [M. A. Noor and K. I. Noor, Three-step iterative methods for nonlinear equations, preprint (2006)] to suggest and analyze a new iterative methods for solving the integral equations. Our method of developing this new method is very simple as compared with other methods. Several numerical examples are given to illustrate the efficiency and performance of the new method. Results reveal that the proposed method is very effective and simple. Our method can be considered as an improvement of the existing methods.

In this paper, systems of nonlinear integro-differential Volterra equations are examined that can be represented as feedback interconnections of linear time-invariant block and periodic nonlinearities. The interest in such systems is motivated by their numerous applications in mechanical, electrical and communication engineering; examples include, but are not limited to, models of phase-locked loops, pendulum-like mechanical systems, coupled vibrational units and electric machines. Systems with periodic nonlinearities are usually featured by multistability and have infinite sequences of (locally) stable and unstable equilibria; their trajectories may exhibit nontrivial (e.g. chaotic) behavior. We offer frequency-domain criteria, ensuring convergence of any solution to one of the equilibria points, and this property is referred to as the gradient-like behavior and corresponds to phase locking in synchronization systems. Although it is hard to find explicitly the equilibrium, attracting a given trajectory, we give a constructive estimate for the distance between this limit equilibrium and the initial condition. The relevant estimates are closely related to the analysis of cycle slipping in synchronization systems. In the case where the criterion of gradient-type behavior fails, a natural question arises — which nonconverging solutions may exist in the system and, in particular, how many periodic solutions it has. We show that a relaxation of the frequency-domain convergence criterion ensures the absence of high-frequency periodic orbits. The results obtained in the paper are based on the method of integral quadratic constraints that has arisen in absolute stability theory and stems from Popov’s techniques of “a priori integral indices”. We illustrate the analytic results by numerical simulations.

We consider the evolution problem for the wave equation with nonlinear boundary conditions, modeling the anti-plane shearing on a system of finite faults under a slip-dependent friction in a linear elastic continuum. The associated eigenvalue problem is studied for bounded and unbounded domains. Recent studies showed that this eigenvalue problem plays a key role in earthquake nucleation modeling. We investigate the structure of the positive eigenvalues set and prove the monotone dependence of the first eigenvalue on the friction parameter. For the elastic plane with a collinear system of faults, the eigenvalue problem is reduced to a hyper-singular integral equation. This equation is solved through a semi-analytical technique. Numerical results show a sensible dependence of the first eigenfunction on the geometry and friction heterogeneity.

An installment option is a derivative financial security where the price is paid in installments instead of as a lump sum at the time of purchase. The valuation of these options involves a free boundary problem in that at each installment date, the holder of the derivative has the option of continuing to pay the premiums or allowing the contract to lapse, and the decision will depend upon whether the present value of the expected pay-off is greater or less than the present value of the remaining premiums. Using a model installment option where the premiums are paid continuously rather than on discrete dates, an integral equation is derived for the position of this free boundary by applying a partial Laplace transform to the underlying partial differential equation for the value of the security. Asymptotic analysis of this integral equation allows us to deduce the behavior of the free boundary close to expiry.

The solution of acoustic problems via integral equations (IEs) is considered. Symmetrical evaluation of the resulting integrals is proposed. The possible reduction of the system matrix to complex symmetric form is considered. Applications of the proposed method to scattering by a soft as well as a hard sphere are presented. Another application to radiation from a uniformly vibrating sphere is also considered. To overcome the nonuniqueness problem, CHIEF method is used and Lagrange multipliers are employed to deal with the resulting overdetermined system to preserve the symmetry of the system matrix. The application of CHIEF via a close interior surface of a hard sphere with ka = 22.602185 illustrates the success of CHIEF at high frequencies (HF).

A usual approach to monitoring an autocorrelated process is to apply a control chart to the process residuals. In this paper, we study the statistical behavior of the residuals of a controlled second-order autoregressive (AR(2)) cutting process when a special-cause shift occurs to the process mean. Shewhart, exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) control charts are applied to the residuals to monitor the cutting process. Formulas, integral equations and recursive procedures for computing the average run lengths (ARLs) of the charts are derived. Numerical results are presented and the relative performance of the charts is investigated.

In this paper, an integral equation approach used for pricing American-style Parisian options is extended to pricing Parasian options, after overcoming an additional difficulty of losing the “reset” feature of the latter in addition to still dealing with the high nonlinearity associated with American-style exercises. More specifically, compared with an American Parisian option, the accumulative feature of the “tracking clock” time results in a pair of coupled three-dimensional (3D) Partial Differential Equation (PDE) systems instead of coupled two-dimensional (2D) and 3D PDE systems. Upon successfully establishing the couple integral equation systems, we have successfully computed the option price and the optimal exercise price and compared the results with those of its Parisian counterpart. Through the comparison, we can directly observe how the accumulativeness of the “tracking clock” time affects the option and the optimal exercise prices and provide some meaningful financial explanations.

In order to obtain the geometry of a global monopole without cosmological constant and electric charge in $2+1$ dimensions, we make use of the broken $O(2)$ symmetry. In the absence of an exact solution, we determine the series solutions for both the metric and monopole functions in a consistent manner that satisfies all equations in appropriate powers. The new expansion elements are of the form $\frac{1}{r^n}{(lnr)}^m,$ for the radial distance $r$ and positive integers $m$ and $n$ constrained by $m\le n$. To the lowest order of expansion, we find that in analogy with the negative cosmological constant the geometry of the global monopole acts repulsively, i.e. in the absence of a cosmological constant the global monopole plays at large distances the role of a negative cosmological constant.

In this work, a unified scheme for computing the fundamental solutions of a three-dimensional homogeneous elliptic partial differential operator is presented. The scheme is based on the Rayleigh expansion and on the Fourier representation of a homogeneous function. The scheme has the advantage of expressing the fundamental solutions and their derivatives up to the desired order without any term-by-term differentiation. Moreover, the coefficients of the series need to be computed only once, thus making the presented scheme attractive for numerical implementation. The scheme is employed to compute the fundamental solution of isotropic elasticity showing that the spherical harmonics expansions provide the exact expressions. Then, the accuracy of the scheme is assessed by computing the fundamental solutions of a generally anisotropic magneto-electro-elastic material.

In this research, we give details of a new numerical method for the approximate solution of a general two-dimensional Volterra integral equation, using the discontinuous wavelet packets e.g. Walsh functions. The double Walsh approximation we have adopted utilizes a simple robust numerical scheme for approximate solution of the equations. The two-dimensional operational matrix of integration for each subinterval is explicitly constructed, where m is a power of 2. Finally the reliability and efficiency of the proposed scheme are demonstrated by some numerical results.

This work provides an efficient method for solving system of nonlinear Fredholm–Hammerstein integral equations. The propose method approximates the unknown function with hybrid Bernstein Block-Pulse functions (HBPF). In order to do this, we apply these functions and then using the collocation method for the numerical solutions of this system. Furthermore, we implement this method in conjunction with the quadrature rule for converting the problem to a system of algebraic equations that can be solved easily by applying mathematical programming techniques. The merits of this method lie in the fact that, on the one hand, the problem will be reduced to a system of algebraic equations and on the other hand, the efficiency and accuracy of the hybrid Bernstein Block-Pulse functions with the Gauss quadrature rule (HBPF-GQR) for solving this system are remarkable. The existence and uniqueness of solution have been presented. Moreover, the convergence of this algorithm will be shown by preparing some theorems. Several numerical examples are presented to show the superiority and efficiency of current method in comparison with some other well-known methods.

In this paper, a mathematical model of the eye corneal shape is introduced which is a nonlinear boundary value problem. This boundary value problem is converted to a nonlinear integral equation and by means of the collocation method, the problem is reduced into a system of the nonlinear algebraic equations. The approximate solutions are obtained by solving this nonlinear system and numerical results are provided to show that introduced method is efficient. Also, comparisons with other works are made to confirm the reliability of the method.

In this paper we propose a mathematical model for the electromagnetic behaviour of the shield of a new type of Cryogenic Current Comparator that leads to an integral equation for the induced current density. A numerical treatment for such an equation is presented and studied.

The formulation and numerical solution of the dynamics of suspended vesicles in a bounded Stokes flow is outlined in this paper. We propose a semi-implicit time marching scheme, which is based on a boundary integral formulation of the corresponding hydrodynamic flow equations and vesicle deformation dynamics. We restrict our attention to the case in which the fluids inside and outside the vesicle have the same density and viscosity. We use a boundary integral formulation for the fluid that results in a set of nonlinear integro-differential equations for the vesicle dynamics. The motion of the vesicles is governed by the interplay between hydrodynamic and elastic forces. Fluid-structure interaction problems of this type are challenging to simulate. On one hand, explicit time-stepping schemes suffer from a severe stability constraint due to the stiffness related to high-order spatial derivatives. On the other hand, implicit time-stepping schemes can be expensive because they require the solution of a set of nonlinear equations at each time step. To circumvent these difficulties, we employ a semi-implicit scheme that does not have severe stability constraints and whose computational cost per time step is comparable to that of an explicit scheme. We discretize the equations by using a spectral method in space, and a first-order time stepping scheme in time. We report numerical experiments that demonstrate the convergence properties of our scheme.

In this paper we present an overview of a multiresolution approach for solving an integral equation arising in an applied electromagnetics problem. The problem under consideration is to find the electric field radiated by an antenna, given the expression for the excitation. The structures analyzed are assumed to be made of a Perfect Electric Conductor, i.e. they must satisfy Dirichlet boundary conditions. The problem is considered in the integral formulation, numerically discretized with boundary elements. After defining a set of encapsulated spaces for the discretized solution, a multiresolution treatment is addressed. Each space is first separated into a solenoidal and not solenoidal component and then a vector wavelet basis is obtained for each of them, starting from a scalar decomposition, via a proper scalar to vector mapping. Preliminary results are presented in order to show the actual sparsity of the arising linear system and the fast behaviour of iterative solvers.

Asymptotic expansions are derived for the Laplace convolution (f*g)(x) as x → ∞, where f and g have asymptotic power series representation in descending powers of t. Bounds are also constructed for the error terms associated with these expansions. Similar results are given for the convolution integrals

as x → ∞. These results can be used in the study of asymptotic solutions to the renewal equation and the Wiener-Hopf equations.