Integral Equations with Difference Kernels on Finite Intervals

L\'evy walks on finite intervals with absorbing boundaries are studied using analytic and Monte Carlo techniques. The integral equations for L\'evy walks in infinite 1D systems are generalized to treat the evolution of the probability distribution on finite and semi-infinite intervals. In particular the near-boundary behavior of the probability distribution and also its properties at asymptotically large times are studied. The probability distribution is found to be discontinuous near the boundary for L\'evy walks in finite and semi-infinite systems. Previous results for infinite systems, and a previous scaling for semi-infinite systems, are reproduced. The use of linear operator theory to solve the integral equations governing the evolution of the L\'evy walk implies that the probability distribution decays exponentially at large times. For a jump distribution that satisfies $\ensuremath{\psi}(x)\ensuremath{\sim}|x{|}^{\ensuremath{-}\ensuremath{\alpha}}$ for large $|x|,$ the decay constant for the exponential decay is estimated and found to scale at large L as ${L}^{1\ensuremath{-}\ensuremath{\alpha}}$ for $2l\ensuremath{\alpha}l3$ and ${L}^{\ensuremath{-}1}$ for $1l\ensuremath{\alpha}l2,$ in contrast to ${L}^{\ensuremath{-}2}$ for normal diffusion. For $2l\ensuremath{\alpha}l3,$ the ratio of the decay constants of the first and second eigenfunctions is less than 4 for large $L,$ so that the second eigenfunction is relatively more important in describing the system's large time behavior than the corresponding eigenfunction for normal diffusion. For $1l\ensuremath{\alpha}l2$ the ratio of the decay constants may be greater or less than 4. The shapes of the eigenfunctions for the L\'evy processes are obtained numerically and the strong similarity between the first eigenfunction and its normal diffusion counterpart for $2\ensuremath{\lesssim}\ensuremath{\alpha}l3$ indicate that it would be difficult experimentally to distinguish such a L\'evy process on a finite interval from a normal diffusive system by considering only the asymptotic shape of the probability distribution. For $\ensuremath{\alpha}\ensuremath{\lesssim}2$ we observe significant differences between the first and second eigenfunctions and their normal diffusion counterparts. For moderately large intervals, the first eigenfunction is flatter with large boundary discontinuities while the second eigenfunction can differ from its normal diffusion counterpart in both its symmetry properties and number of nodes.

In this thesis a variety of integral equations and partial differential equations which describe two-dimensional and three-dimensional axisymmetric forced convection problems are studied. The correspondence between the integral equation formulation and partial differential equation formulation of such boundary value problems is investigated and this correspondence is used to develop a new technique for the solution of hitherto intractable Fredholm integral equations of the first kind on a finite interval. Basic properties of the physical models are also established by studying fundamental solutions of the partial differential equations. The problems are developed by considering the transport of heat in an inviscid, incompressible, heat conducting fluid whose thermal constants are assumed invariant in temperature, space and time. The transport mechanisms are those of convection in a prescribed velocity field and diffusion, and the temperature field is given by the solution of the partial differential equation (and, less directly, by the integral equation) which is a heat conservation equation and characterises the balance between the diffusion and convection processes. While such problems are highly idealised they have more practical applications in several special cases in which the partial differential equation takes the form of a linear approximation of the Navier - Stokes equations. Circulation is transported by diffusion and convection in a viscous fluid and in certain circumstances the temperature model can be regarded as a model of a viscous fluid. The integral equations also arise in an elastic half-space problem which is described in the first chapter.The two-dimensional problem has been the subject of-many previous investigations. New contributions are made in the domain of the integral equation whose solution has been given explicitly for the first time, and in the dependence of the solutions on the Peclet number, a non-dimensional parameter which characterises the ratio of the diffusive flux to the convective flux, where certain physically motivated results have been confirmed by a more strict mathematical approach. Similar, but considerably more extensive, investigations are made of a family of three-dimensional axisymmetric problems. In these problems, the first order radial derivative term in the partial differential equation has a singular coefficient whose role in the equation is interpreted as that due to a radial component in the forced convection field (‘radial' means 'radial direction in a cylindrical polar coordinate system') The crucial problem here is the realisation of a full understanding of the relationship between the partial differential equation and the integral equation. This is achieved by rewriting the partial differential equation in what is essentially its adjoint form and interpreting the new partial differential equation as a conservation equation which also expresses the balance of diffusion and convection of a quantity called <<-heat in a prescribed convection field. Physically motivated arguments are used to suggest the relationship between the original partial differential equation and the integral equation formulation which is then proved using rigorous mathematical arguments. A Peclet number is defined in this problem and the dependence of the solutions on this parameter is analysed. Representation and uniqueness theorems are also given in classical cases and a discussion of the extension of these results into generalised function spaces is included.

We use the Wiener-Hopf method to obtain exact solutions of plane deformation problems for an elastic wedge whose lateral sides are stress free and which has rectilinear cracks on its axis of symmetry. In problem 1, a finite crack issues from the wedge apex edge; in problem 2, a half-infinite crack originates at a certain distance from the wedge apex edge; and in problem 3, the wedge contains an internal finite crack. Earlier, many authors obtained approximate solutions to these problems [1–10] and exact solutions to problem 1 [11–17] and homogeneous problem 2 [18–20] (see also [21]). The method of approximate conformal transformation [1, 2] and that of integral equations [3, 4] were used to study a specific case of problem 1 about equilibrium of an elastic half-plane with a boundary crack perpendicular to the half-plane boundary. The solution to problem 1 was obtained in [11] by reducing the problem to the Riemann problem for analytic functions and in [6, 7], by the Wiener-Hopf method. In [11], the problem coefficient was factorized in terms of Cauchy-type integrals, but no further calculations were presented, and in [6, 7], an approximate factorization was performed by approximating the factorized function. In [8], problems 1 and 2 were reduced to Fredholm integral equations of the second kind, which were solved numerically. In problem 1, values of the integral equation density were calculated, and the normal displacements of the crack edges were expressed in terms of this density in the form of Abel integrals; no calculations were given in problem 2. The exact values of the stress intensity factors in problem 1 were obtained in [12–17] by the Wiener-Hopf method. The paper [12] considered the case of linear normal stresses given on the crack edges in the absence of tangential stresses was considered, the papers [13–16], the case of concentrated forces acting on the crack edges, and the paper [17], the case of concentrated moments applied to the wedge apex. Solutions to the homogeneous problem 2, where the crack edges are stress free and the principal vector and the principal moment of stresses are given at infinity, were constructed by the Wiener-Hopf method in [18–20]. Problem 3 was studied in [6, 7, 9] by different approximate methods: using the asymptotic solution of the integral equation, solving the Fredholm integral equation of the second kind by the method of successive approximations after approximating the Fourier transform of the difference kernel of the original integral equation, and using another approximation of the kernel to reduce the problem to a singular integral equation admitting a solution in closed form. Here, the asymptotic method can be used only in the case of a crack relatively distant from the wedge apex, and the approximation of the kernel transform gives satisfactory results if the angle at the wedge apex exceeds π. The authors present the numerical results for the stress intensity factors and the displacement jump at an internal point of the crack in the case where the wedge is a half-plane. In [6–9], the cases of rigid clamping and hinged support of the faces of an elastic wedge were investigated. Problem 3 was numerically solved by the method of singular integral equations in [10], where the case of fixed sides of the wedge was also considered. In what follows, problems 1 and 2 whose integral equations are given on a half-infinite interval and have distinct kernels are solved by the Wiener-Hopf method [22], and problem 3 is solved by the generalized Wiener-Hopf method, developed in [23–25] for solving integral equations with difference kernel on a finite interval. The coefficient of the functional Wiener-Hopf equation is factorized in terms of infinite products. We also present numerical results for the stress intensity factors, the normal stress distribution on the crack continuation line, and the normal displacements of the crack edges.

The nonlinear integral equations of Hammerstein type arise in problems with a free phase. A polynomial approach is used to solve such equations. A system of transcendental equations is constructed to find complex parameters (polynomial roots). For more general optimization problems, this system is supplemented by a linear integral equation. For the case of solutions vanishing on the finite interval, a system of integralfunctional equations is obtained. The branching of solutions is investigated.

The Gelfand-Levitan, the Marchenko, and the Gopinath-Sondhi integral equations of inverse scattering theory, regarded in the context of inverse impulse-response problems

We summarize a recently developed integral equation (IE) approach to tackling the long-time existence problem for smooth solution v(x, t) to the 3D Navier–Stokes (NS) equation in the context of a periodic box problem with smooth time independent forcing and initial condition v0. Using an inverse-Laplace transform of in 1/t, we arrive at an IE for , where p is inverse-Laplace dual to 1/t and k is the Fourier variable dual to x. The advantage of this formulation is that the solution to the IE is known to exist a priori for and the solution is integrable and exponentially bounded at ∞. Global existence of NS solution in this formulation is reduced to an asymptotics question. If has subexponential bounds as p→∞, then global existence to NS follows. Moreover, if f=0, then the converse is also true in the following sense: if NS has global solution, then there exists n≥1 for which the inverse-Laplace transform of in 1/tn necessarily decays as q→∞, where q is the inverse-Laplace dual to 1/tn. We also present refined estimates of the exponential growth when the solution is known on a finite interval [0, p0]. We also show that for analytic v[0] and f, with finitely many nonzero Fourier-coefficients, the series for in powers of p has a radius of convergence independent of initial condition and forcing; indeed the radius gets bigger for smaller viscosity. We also show that the IE can be solved numerically with controlled errors. Preliminary numerical calculations for Kida (1985 J. Phys. Soc. Japan 54 2132) initial conditions, though far from being optimized, and performed on a modest interval in the accelerated variable q show decay in q.

This book presents a new approach to analyzing initial-boundary value problems for integrable partial differential equations (PDEs) in two dimensions, a method that the author first introduced in 1997 and which is based on ideas of the inverse scattering transform. This method is unique in also yielding novel integral representations for the explicit solution of linear boundary value problems, which include such classical problems as the heat equation on a finite interval and the Helmholtz equation in the interior of an equilateral triangle. The author s thorough introduction allows the interested reader to quickly assimilate the essential results of the book, avoiding many computational details. Several new developments are addressed in the book, including a new transform method for linear evolution equations on the half-line and on the finite interval; analytical inversion of certain integrals such as the attenuated radon transform and the Dirichlet-to-Neumann map for a moving boundary; analytical and numerical methods for elliptic PDEs in a convex polygon; and integrable nonlinear PDEs. An epilogue provides a list of problems on which the author s new approach has been used, offers open problems, and gives a glimpse into how the method might be applied to problems in three dimensions. Audience: A Unified Approach to Boundary Value Problems is appropriate for courses in boundary value problems at the advanced undergraduate and first-year graduate levels. Applied mathematicians, engineers, theoretical physicists, mathematical biologists, and other scholars who use PDEs will also find the book valuable. Contents: Preface; Introduction; Chapter 1: Evolution Equations on the Half-Line; Chapter 2: Evolution Equations on the Finite Interval; Chapter 3: Asymptotics and a Novel Numerical Technique; Chapter 4: From PDEs to Classical Transforms; Chapter 5: Riemann Hilbert and d-Bar Problems; Chapter 6: The Fourier Transform and Its Variations; Chapter 7: The Inversion of the Attenuated Radon Transform and Medical Imaging; Chapter 8: The Dirichlet to Neumann Map for a Moving Boundary; Chapter 9: Divergence Formulation, the Global Relation, and Lax Pairs; Chapter 10: Rederivation of the Integral Representations on the Half-Line and the Finite Interval; Chapter 11: The Basic Elliptic PDEs in a Polygonal Domain; Chapter 12: The New Transform Method for Elliptic PDEs in Simple Polygonal Domains; Chapter 13: Formulation of Riemann Hilbert Problems; Chapter 14: A Collocation Method in the Fourier Plane; Chapter 15: From Linear to Integrable Nonlinear PDEs; Chapter 16: Nonlinear Integrable PDEs on the Half-Line; Chapter 17: Linearizable Boundary Conditions; Chapter 18: The Generalized Dirichlet to Neumann Map; Chapter 19: Asymptotics of Oscillatory Riemann Hilbert Problems; Epilogue; Bibliography; Index.

A class of nonlinear singular integral equations of Cauchy type on a finite interval is transformed to an equivalent class of (discontinuous) boundary value problems for holomorphic functions in the complex unit disk. Using recent results on the solvability of explicit Riemann–Hilbert problems, we prove the existence of solutions to the integral equation with bounded piecewise continuous nonlinearities. We discuss the influence of parameters and additional conditions and demonstrate the approach for a free boundary problem arising from seepage near a channel. Copyright © 2001 John Wiley &amp; Sons, Ltd.

We study solutions to the integral equation ω(x)=Γ-x2∫01K(θ)H(ω(xθ))dθ\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\omega (x) = \\Gamma - x^2 \\int \\nolimits _{0}^1 K(\\theta ) \\, H(\\omega (x\\theta )) \\, \\mathrm {d}\\theta \\end{aligned}$$\\end{document}where Gamma >0, K is a weakly degenerate kernel satisfying, among other properties, K(theta ) sim k , (1-theta )^sigma as theta rightarrow 1 for constants k>0 and sigma in (0, log _2 3 -1), H denotes the Heaviside function, and x in [0,infty ). This equation arises from a reaction-diffusion equation describing Liesegang precipitation band patterns under certain simplifying assumptions. We argue that the integral equation is an analytically tractable paradigm for the clustering of precipitation rings observed in the full model. This problem is nontrivial as the right hand side fails a Lipschitz condition so that classical contraction mapping arguments do not apply. Our results are the following. Solutions to the integral equation, which initially feature a sequence of relatively open intervals on which omega is positive (“rings”) or negative (“gaps”) break down beyond a finite interval [0,x^*] in one of two possible ways. Either the sequence of rings accumulates at x^* (“non-degenerate breakdown”) or the solution cannot be continued past one of its zeroes at all (“degenerate breakdown”). Moreover, we show that degenerate breakdown is possible within the class of kernels considered. Finally, we prove existence of generalized solutions which extend the integral equation past the point of breakdown.

The results obtained in this chapter may be of some interest from the point of view of analysis. However, they have an immediate interpretation in terms of certain representation theorems for stationary random processes on a finite time interval, and this provided part of the motivation for the investigation. Our interest is in finite interval translation kernel integral equation eigenvalue problems, that is, in the integral equation 1 $$\int_{ - T}^T {r(t - \tau )\phi \left( \tau \right)d \tau = \lambda \phi \left( t \right)} $$

A new approach to the cumulative operational time for semi-Markov models of repairable systems

Introduction. The closely related problems of prediction and filtering of random time series and of the detection of signals in noise from continuous observations over a finite time interval have received extensive treatment in recent scientific literature. The books by Grenander and Rosenblatt [5] and Laning and Battin [8] contain fairly adequate bibliographies on the subject. The primary reason for the present paper on this same subject is the exposition of a practical, direct method for the solution of a certain restricted class of these problems. An examination of the difficulties involved in the indirect methods for the solution of such problems by means of linear operators which are solutions of linear integral equations should convince the reader of the need for practical, direct methods (see Laning and Battin [81, Chapter 8). The methods of the present paper are not simple. Indeed they may be very tedious in nontrivial problems. The best that can be said is that application of these methods should lead to substantially shorter and easier calculations than does the integral equation method. In most of the existing literature on prediction, filtering, and detection problems that involve random time series, solutions for these problems have been dependent upon the solutions of certain linear integral equations or systems of such equations. Notable exceptions are found in the papers by Mann [9] and Reich and Swerling [12] where integral equations are not used in solving certain special cases of the filtering and detection problems. The methods used by Mann [9] and Mann and Moranda [10] will be simplified in the present paper and extended to cover somewhat more general stochastic processes than were considered by those authors. They treated two noise processes by similar but entirely separate statistical analyses. In the present paper it is shown that an important extension of their second case is reducible to the first case by a simple linear transformation. Thus separate treatments are unnecessary. The standard criterion of minimum error variance will be applied to prediction, filtering, and detection of the mean value function of a stationary stochastic process which has a spectral density function that is the reciprocal of a polynomial. This problem is distinct from the problems of prediction and filtering of a stochastic function considered by Wiener [15]. A criterion alternative to the minimum error variance will be considered in comparison with the latter.

In this work, we examine the receding contact between a homogeneous elastic layer and a half-plane substrate reinforced by a functionally graded coating. The material properties of the coating are allowed to vary exponentially along its thickness. A distributed traction load applied over a finite segment of the layer surface presses the layer and the coated substrate against each other. It is further assumed that the receding contact between the layer and the coated substrate is frictionless. In the absence of body forces, Fourier integral transforms are used to convert the governing equations and boundary conditions of the plane receding contact problem into a singular integral equation with the contact pressure and contact size as unknowns. Gauss–Chebyshev quadrature is subsequently employed to discretize both the singular integral equation and the force equilibrium condition at the contact interface. An iterative algorithm based on the method of steepest descent has been proposed to numerically solve the system of algebraic equations, which is linear for the contact pressure but nonlinear for the contact size. Extensive case studies are performed with respect to the coating inhomogeneity parameter, geometric parameters, material properties, and the extent of the indentation load. As a result of the indentation, the elastic layer remains in contact with the coated substrate over only a finite interval. Exterior to this region, the layer and the coated substrate lose contact. Nonetheless, the receding contact size is always larger than that of the indentation traction. To validate the theoretical solution, we have also developed a finite-element model to solve the same receding contact problem. Numerical results of finite-element modeling and theoretical development are compared in detail for a number of parametric studies and are found to agree very well with each other.

We present some results on the solvability of an integral equation of the second kind with a difference kernel on a finite interval, construct a counterexample to an assertion, earlier believed to have been proved, on the solvability of this equation, and pose an open problem.

In this paper, we consider the problem of finding positive axially symmetric solutions to boundary value problems for nonlinear elliptic differential equations using the method of two-sided approximations. The first boundary value problem, or Dirichlet problem, is solved. The nonlinearity involved is of an anti-monotonic type, characterized by a power-law dependence with an exponent ranging from –1 to 0. By transitioning to a polar coordinate system and exploiting the axial symmetry of the solution, the original boundary value problem for an elliptic equation is reduced to a boundary value problem for an ordinary differential equation on a finite interval. The solution depends solely on the radial coordinate, eliminating the dependence on the angular variable. In this case, the pole of the polar coordinate system becomes a singular point, where it is necessary to impose a boundedness condition on the solution. For the boundary value problem, the Green’s function is constructed, after which the problem is reduced to a Hammerstein integral equation. This integral equation is treated as a nonlinear operator equation in a Banach space of continuous functions on a finite interval, partially ordered by the cone of non-negative functions on that interval. The corresponding operator is studied with respect to properties such as anti-monotonicity (antitonicity), positivity, boundedness, and pseudo-concavity. The next stage involves determining the initial approximation as the endpoints of a strongly invariant conical segment for the antitone operator, in a way that ensures the highest possible convergence rate of the iterative process. Two iterative sequences of two-sided approximations are then constructed. The first sequence is non-decreasing with respect to the cone, while the second is non-increasing. At each iteration, the arithmetic mean of the upper and lower approximations is chosen as the current estimate. The iterative process continues until the error estimate for the solution meets the prescribed accuracy. The theoretical results obtained in this work were verified through computational experiments. The dependence of the solution and the convergence rate of the iterative process on the parameters in the right-hand side was analyzed and illustrated with corresponding graphs.