Linear Integral Operator Research Articles

Support vector regression for the solution of linear integral equations

The aim of this paper is to establish a support vector regression method for semi-discrete ill-posed problems. We consider the equation Af = g, where a linear integral operator A is known; discrete measurements of the right-hand side are observed and a solution f* is sought. For the reconstruction, instead of a standard square loss function, Vapnik's ϵ-intensive function is used to measure the distance between Af and g. This avoids an overfitting to disturbed data and guarantees additional stability given that the cut-off parameter ϵ is chosen appropriately. The resulting solution procedure can be formulated as a quadratic program. Besides the method, a Sobolev error analysis and a parameter strategy for the regularization parameters are provided. The results are substantiated with a numerical example.

Criteria for univalence of the Dziok–Srivastava and the Srivastava–Wright operators in the class A

Abstract In the geometric function theory (GFT) much attention is paid to various linear integral operators mapping the class S of the univalent functions and its subclasses into themselves. In [12,13] Hohlov obtained sufficient conditions that guarantee such mappings for the operator defined by means of Hadamard product with the Gauss hypergeometric function. In our earlier papers as [20,19,17,18] , etc., we extended his method to the operators of the generalized fractional calculus (GFC, [16] ). These operators have product functions of the forms m +1 F m and m +1 Ψ m and integral representations by means of the Meijer G - and Fox H -functions. Here we propose sufficient conditions that guarantee mapping of the univalent, respectively of the convex functions, into univalent functions in the case of the celebrated Dziok–Srivastava operator ( [8] : J. Dziok, H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 103 , No 1 (1999), pp. 1–13) defined as a Hadamard product with an arbitrary generalized hypergeometric function p F q . Similar conditions are suggested also for its extension involving the Wright p Ψ q -function and called the Srivastava–Wright operator (Srivastava, [36] ). Since the discussed operators include the above-mentioned GFC operators and many their particular cases (operators of the classical FC), from the results proposed here one can derive univalence criteria for many named operators in the GFT, as the operators of Hohlov, Carlson and Shaffer, Saigo, Libera, Bernardi, Erdelyi–Kober, etc., by giving particular values to the orders p ⩽ q + 1 of the generalized hypergeometric functions and to their parameters.

Positive solutions of a2nth-order boundary value problem involving all derivatives via the order reduction method

This paper is mainly concerned with the existence, multiplicity and uniqueness of positive solutions for the 2 n th-order boundary value problem { ( − 1 ) n u ( 2 n ) = f ( t , u , u ′ , … , ( − 1 ) [ i 2 ] u ( i ) , … , ( − 1 ) n − 1 u ( 2 n − 1 ) ) , u ( 2 i ) ( 0 ) = u ( 2 i + 1 ) ( 1 ) = 0 ( i = 0 , 1 , … , n − 1 ) , where n ≥ 2 and f ∈ C ( [ 0 , 1 ] × R + 2 n , R + ) ( R + ≔ [ 0 , ∞ ) ) . We first use the method of order reduction to transform the above problem into an equivalent initial value problem for a first-order integro-differential equation and then use the fixed point index theory to prove the existence, multiplicity, and uniqueness of positive solutions for the resulting problem, based on a priori estimates achieved by developing spectral properties of associated parameterized linear integral operators. Finally, as a byproduct, our main results are applied for establishing the existence, multiplicity and uniqueness of symmetric positive solutions for the Lidstone problem involving all derivatives.

Higher Order Numerical Discretization Methods with Sobolev Norm Minimization

Abstract This paper presents a method of producing higher order discretization weights for linear differential and integral operators using the Minimum Sobolev Norm idea[1][2] in arbitrary geometry and grid configurations. The weight computation involves solving a severely ill-conditioned weighted least-squares system. A method of solving this system to very high-accuracy is also presented, based on the theory of Vavasis et al [3]. An end-to-end planar partial differential equation solver is developed based on the described method and results are presented. Results presented include the solution error, discretization error, condition number and time taken to solve several classes of equations on various geometries. These results are then compared with those obtained using the Matlab's FEM based PDE Solver as well as Dealii[4].

Second order boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions

The authors consider the boundary value problem with a two-parameter nonhomogeneous multi-point boundary condition \gather u''+g(t)f(t,u)=0, \quad t\in (0,1),\nonumber u(0)=\alpha u(\xi )+\lambda ,\quad u(1)=\beta u(\eta )+\mu .\nonumber \endgather Criteria for the existence of nontrivial solutions of the problem are established. The nonlinear term $f(t,x)$ may take negative values and may be unbounded from below. Conditions are determined by the relationship between the behavior of $f(t, x)/x$ for $x$ near $0$ and $\pm \infty $, and the smallest positive characteristic value of an associated linear integral operator. The analysis mainly relies on topological degree theory. This work complements some recent results in the literature. The results are illustrated with examples.

Determining Fresnel reflection coefficients in 3D half‐space geometry by GPR multistatic data

ABSTRACTThis paper deals with a procedure for retrieving Fresnel reflection coefficients of a half‐space medium starting from measurements collected under a reflection mode multistatic configuration. In particular, the procedure consists in inverting a linear integral operator that links the reflected field measurements and the Fresnel reflection coefficients.This contribution extends our previous work already developed for 2D scalar configurations. Here, we consider a 3D source and accordingly a dyadic representation of the reflection coefficient. Accordingly, the corresponding inverse problem amounts to solving a vector problem with a dyadic unknown. The amount of data necessary to solve the inverse problem is achieved by collecting different Cartesian components of the reflected field. Accordingly, the reconstruction problem is cast as the inversion of several scalar integral equations, which provides the transverse electric and magnetic Fresnel reflection coefficients as functions of the angle of incidence. Synthetic examples dealing with the case of a homogeneous half‐space soil are presented to show the effectiveness of the proposed procedure.

Single Point Gradient Blow‐Up and Nonuniqueness for a Third‐Order Nonlinear Dispersion Equation

A basic mechanism of a formation of shocks via gradient blow‐up from analytic solutions for the third‐order nonlinear dispersion PDE from compacton theoryimageis studied. Various self‐similar solutions exhibiting single point gradient blow‐up in finite time, as t → T−< ∞, with locally bounded final time profiles u(x, T−), are constructed. These are shown to admit infinitely many discontinuous shock‐type similarity extensions for t > T, all of them satisfying generalized Rankine–Hugoniot's condition at shocks. As a consequence, the nonuniqueness of solutions of the Cauchy problem after blow‐up is detected. This is in striking difference with general uniqueness‐entropy theory for the 1D conservation laws such as (a partial differential equation, PDE, Euler's equation from gas dynamics)imagecreated by Oleinik in the middle of the 1950s. Contrary to(1)and not surprisingly, self‐similar gradient blow‐up for(2)is shown to admit a unique continuation.Bearing in mind the classic form(2), the NDE(1)can be written asimagewith the standard linear integral operator (−D2x)−1> 0. However, because(3)is a nonlocal equation, no standard entropy and/or BV‐approaches apply (moreover, the x‐variations of solutions of(3)is increasing for BV data u0(x)).

True amplitude imaging by inverse generalized Radon transform based on Gaussian beam decomposition of the acoustic Green's function

True amplitude migration is one of the most important procedures of seismic data processing. As a rule it is based on the decomposition of the velocity model of the medium into a known macrovelocity component and its sharp local perturbations to be determined. Under this decomposition the wavefield can be considered as the superposition of an incident and reflected/scattered waves. The single scattering approximation introduces the linear integral operator that connects the sharp local perturbations of the macrovelocity model with the multishot/multioffset data formed from reflected/scattered waves. We develop the pseudoinverse of this operator using the Gaussian beam based decomposition of acoustic Green's functions. The computation of this pseudoinverse operator is done pointwise by shooting Gaussian beams from the target area towards the acquisition system. The numerical implementation of the pseudoinverse operator was applied to the synthetic data Sigsbee2A. The results obtained demonstrate the high quality of the true amplitude images computed both in the smooth part of the model and under the salt body.

Applications of certain operators to the classes related with generalized Janowski functions

Recently, Srivastava and Attiya [H.M. Srivastava and A.A. Attiya, An integral operator associated with the Hurwitz–Lerch Zeta function and differential subordination, Integral Transforms Spec. Funct. 18 (2007), pp. 207–216] introduced a linear integral operator on the class of analytic function associated with Hurwitz–Lerch Zeta function. Using this Srivastava–Attiya operator and generalized Janowski functions, certain classes of analytic functions related with the functions of bounded boundary and bounded radius rotation are introduced and studied. Inclusion results, rate of growth of coefficients, radius problem and behaviour of these classes under certain integral operators are investigated.

Approximation with Respect to Goffman–Serrin Variation by Means of Non-Convolution Type Integral Operators

We obtain strong convergence results and order of approximation for a class of linear non-convolution type integral operators in the space , where denotes the space of all the L 1-functions with bounded ℱ-variation in endowed with the norm Here, ℱ is a positive sublinear functional defined on ℝ N and μ f is the derivative (vector) measure associated to f. Some meaningful examples are discussed.

Existence of Nontrivial Global Solution for a Class of Nonlinear Volterra Integral Equations in Banach Space

We get a global existence result for a class of nonlinear integral equations by applying a new fixed point thereom in weaker condition.Because there is a nonlinear integral operator in nonlinear term,compared with linear integral operator,some results in knowed papers are expanded and enriched.

Periodic solutions for functional differential equations with sign-changing nonlinearities

We establish some eigenvalue criteria for the existence of non-trivial T-periodic solutions of a class of first-order functional differential equations with a nonlinearity f(x). The nonlinear term f(x) can take negative values and may be unbounded from below. Conditions are determined by the relationship between the behaviour of the quotient f(x)/x for x near 0 and ±∞ and the smallest positive characteristic value of an associated linear integral operator. This linear operator plays a key role in the proofs of the results and its construction is non-trivial. Applications to related eigenvalue problems are also discussed. The analysis mainly relies on the topological degree theory.

Positive solutions for higher order multi-point boundary value problems with nonhomogeneous boundary conditions

We study nth order boundary value problems with a nonlinear term f ( t , x ) subject to nonhomogeneous multi-point boundary conditions. Criteria for the existence of positive solutions of such problems are established. Conditions are determined by the relationship between the behavior of f ( t , x ) / x near 0 and ∞ when compared with the smallest positive characteristic value of an associated linear integral operator. This work improves and extends some recent results in the literature for the second order problems. The results are illustrated with examples.

Multivalent functions based on a linear integral operator

Multivalent functions based on a linear integral operator

Higher order multi-point boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions

We study classes of nth order boundary value problems consisting of an equation having a sign-changing nonlinearity f(t, x) together with several different sets of nonhomogeneous multi-point boundary conditions. Criteria are established for the existence of nontrivial solutions, positive solutions, and negative solutions of the problems under consideration. Conditions are determined by the behavior of f(t, x)/x near 0 and ±∞ when compared to the smallest positive characteristic values of some associated linear integral operators. This work improves and extends a number of recent results in the literature on this topic. The results are illustrated with examples.

Global dynamics of a class of SEIRS epidemic models in a periodic environment

In this paper, we study a class of periodic SEIRS epidemic models and it is shown that the global dynamics is determined by the basic reproduction number R 0 which is defined through the spectral radius of a linear integral operator. If R 0 < 1 , then the disease free periodic solution is globally asymptotically stable and if R 0 > 1 , then the disease persists. Our results really improve the results in [T. Zhang, Z. Teng, On a nonautonomous SEIRS model in epidemiology Bull. Math. Biol. 69 (8) (2007) 2537–2559] for the periodic case. Moreover, from our results, we see that the eradication policy on the basis of the basic reproduction number of the time-averaged system may overestimate the infectious risk of the periodic disease. Numerical simulations which support our theoretical analysis are also given.

Eigenvalues of semi-positone Hammerstein integral equations and applications to boundary value problems

Eigenvalue intervals and the existence of finitely many positive eigenfunctions for semi-positone Hammerstein integral equations are obtained. The positive characteristic values and their upper and lower bounds of the corresponding linear Hammerstein integral operators are studied. Applications of the results are given to third-order differential equations with three-point boundary conditions.

Can One Bind Three Electrons with a Single Proton?

Of course not for an ideal H– – atom. But with the help of an intense homogeneous magnetic field B, the question deserves to be reconsidered. It is known (see, e.g. Baumgartner et al. in Commun Math Phys 212(3):703–724, 2000; Brummelhuis and Duclos in J Math Phys 47:032103, 2006) that as B → ∞ and in the clamped nucleus approximation, this ion is described by a one-dimensional Hamiltonian $$\sum_{i=1}^N-{\Delta_i\over2}-Z\delta(x_i)+\sum_{1\le i < j\le N}\delta(x_i-x_j)\quad {\rm acting}\,{\rm in} \quad L^2(\mathbb {R}^3),\quad\quad\quad\quad\quad(1)$$ where N = 3, Z = 1 is the charge of the nucleus, and δ stands for the well known “delta” point interaction. We present an extension of the “skeleton method” (Cornean et al. in Few-Body Syst 38(2–4):125–131, 2006; Proc Symp Pure Math AMS 77:657–672, 2008) to the case of three degree of freedom. This is a tool, that we learn from Rosenthal (J Chem Phys 35(5):2474–2483, 1971) for the case N = 2, which reduces the spectral analysis of (1) to determining the kernel a system of linear integral operators acting on the supports of the delta interactions. As an application of this method we present numerical results which indicates that (1) has a bound state for Z = 1 and N = 3.

Vanishing Theorems in Colombeau Algebras of Generalized Functions

AbstractUsing a canonical linear embedding of the algebra of Colombeau generalized functions in the space of -valued ℂ-linear maps on the space of smooth functions with compact support, we give vanishing conditions for functions and linear integral operators of class . These results are then applied to the zeros of holomorphic generalized functions in dimension greater than one.

Imaging Small PEC Spheres by a Linear $\delta$ Approach

The problem of localizing small inhomogeneities from the knowledge of their scattered field is dealt with. In particular, the case of small perfect electric conducting spheres is of concern, with the scattered field data collected under multistatic/multifrequency/single-view or multistatic/single-frequency/multiview far zone configurations. The multiple scattering between the spheres is neglected, and their locations are represented as the supports of the Dirac delta functions. This allows one to cast the problem as the inversion of a linear integral operator, with the delta functions being the unknowns of the problem. The inversion of this linear integral operator is achieved by means of the truncated singular value decomposition. The performance of the linear inversion algorithm against the model error (i.e., for situations where the multiple scattering is not negligible) is investigated by numerical simulations. Furthermore, the effect of noise is also analyzed by corrupting the data by an uncorrelated additive white Gaussian process.