Singular Value Decomposition of a Finite Hilbert Transform Defined on Several Intervals and the Interior Problem of Tomography: The Riemann‐Hilbert Problem Approach

We study the asymptotics of singular values and singular functions of a finite Hilbert transform (FHT), which is defined on several intervals. Transforms of this kind arise in the study of the interior problem of tomography. We suggest a novel approach based on the technique of the matrix Riemann‐Hilbert problem (RHP) and the steepest‐descent method of Deift‐Zhou. We obtain a family of matrix RHPs depending on the spectral parameter λ and show that the singular values of the FHT coincide with the values of λ for which the RHP is not solvable. Expressing the leading‐order solution as λ → 0 of the RHP in terms of the Riemann Theta functions, we prove that the asymptotics of the singular values can be obtained by studying the intersections of the locus of zeroes of a certain Theta function with a straight line. This line can be calculated explicitly, and it depends on the geometry of the intervals that define the FHT. The leading‐order asymptotics of the singular functions and singular values are explicitly expressed in terms of the Riemann Theta functions and of the period matrix of the corresponding normalized differentials, respectively. We also obtain the error estimates for our asymptotic results. © 2016 Wiley Periodicals, Inc.     

The singular solutions to a nonsymmetric system of Keyfitz-Kranzer type with initial data of Riemann type

The solutions to the Riemann problem for a nonsymmetric system of Keyfitz-Kranzer type are constructed explicitly when the initial data are located in the quarter phase plane. In particular, some singular hyperbolic waves are discovered when one of the Riemann initial data is located on the boundary of the quarter phase plane, such as the delta shock wave and some composite waves in which the contact discontinuity coincides with the shock wave or the wave back of rarefaction wave. The double Riemann problem for this system with three piecewise constant states is also considered when the delta shock wave is involved. Furthermore, the global solutions to the double Riemann problem are constructed through studying the interaction between the delta shock wave and the other elementary waves by using the method of characteristics. Some interesting nonlinear phenomena are discovered during the process of constructing solutions; for example, a delta shock wave is decomposed into a delta contact discontinuity and a shock wave.     

On the Solvability of Singular Integral Equations with Reflection on the Unit Circle

The solvability of a class of singular integral equations with reflection in weighted Lebesgue spaces is analyzed, and the corresponding solutions are obtained. The main techniques are based on the consideration of certain complementary projections and operator identities. Therefore, the equations under study are associated with systems of pure singular integral equations. These systems will be then analyzed by means of a corresponding Riemann boundary value problem. As a consequence of such a procedure, the solutions of the initial equations are constructed from the solutions of Riemann boundary value problems. In the final part of the paper, the method is also applied to singular integral equations with the so-called commutative and anti-commutative weighted Carleman shifts.     

On the characteristic initial value problem for the wave equation in odd spatial dimensions with radial initial data

Summary The paper obtains an explicit solution of the characteristic initial value problem for the wave equation in odd spatial dimensions with radial initial data via solution of a characteristic boundary value problem involving a singular differential equation. The solution of the latter problem is obtained by a modified Riemann method. It is shown that on the time axis the solution of the original problem reduces to the solution that is obtainable by the use of Asgeirsson’s mean value theorem. Entrata in Redazione il 29 agosto 1971.     

A nonlocal problem for degenerate hyperbolic equation

We consider a nonlocal problem for a degenerate equation in a domain bounded by characteristics of this equation. The boundary-value conditions of the problem include linear combination of operators of fractional integro-differentiation in the Riemann–Liouville sense. The uniqueness of solution of the problem under consideration is proved by means of the modified Tricomi method, and existence is reduced to solvability of either singular integral equation with the Cauchy kernel or Fredholm integral equation of second kind.     

Analytic investigation of features of stresses in plane nonstationary contact problems with moving boundaries

We propose a technique for the analytic investigation of features of contact stresses in the vicinity of the nonstationary moving boundary of a contact region in plane nonstationary contact problems with moving boundaries, which is based on the reduction of a boundary two-dimensional singular integral equation resolving the problem to a system of two one-dimensional singular equations. As tools of research, a method for the reduction of singular integral equations to an equivalent Riemann type problem for piecewise analytic functions and a technique of fractional integro-differentiation are used. It is shown that, on the moving boundary of the contact region, a power singularity, the order of which depends on the velocity of the boundary, takes place.     

Boundary-Value Problems for Second-Order Equations and for Systems of First-Order Equations

The oblique derivative problem for harmonic functions under violation of the Shapiro–Lopatinsky condition is considered as well as some multi-dimensional analogues of the Cauchy–Riemann system. These problems are reduced to nonelliptic pseudo-differential equations. A method generalizing the regularization of singular integral equations is also presented.     

Delta and singular delta locus for one‐dimensional systems of conservation laws

This work gives a condition for existence of singular and delta shock wave solutions to Riemann problem for 2×2 systems of conservation laws. For a fixed left‐hand side value of Riemann data, the condition obtained in the paper describes a set of possible right‐hand side values. The procedure is similar to the standard one of finding the Hugoniot locus. Fluxes of the considered systems are globally Lipschitz with respect to one of the dependent variables. The association in a Colombeau‐type algebra is used as a solution concept. Copyright © 2004 John Wiley &Sons, Ltd.     

Existence of solutions of boundary value problems for singular fractional <Emphasis Type="Italic">q</Emphasis>-difference equations

In this paper, we study the existence and uniqueness of solutions for a class of singular three-point boundary value problems of fractional q-difference equations invovling fractional q-derivative of Riemann–Liouville type. Based on the generalization of Banach contraction principle, we obtain a sufficient condition for existence and uniqueness of solutions of the problem. By applying the Krasnoselskii’s fixed point theorem, we establish a sufficient condition for the existence of at least one solution of the problem. As applications, two examples are presented to illustrate our main results.     

Asymptotic solution of the Cauchy problem for the nonlinear Schrödinger equation with boundary conditions of finite density type

With the help of the dressing procedure the singular Riemann problem corresponding to the Cauchy problem for the nonlinear Schrödinger equation with boundary conditions of finite density type is reduced to a regular Riemann problem. From the asymptotic analysis of the regular Riemann problem we get the principal term of the asymptotic solution of the Cauchy problem in the domain of superpolynomial decrease, which is described in terms of the scattering data corresponding to the initial condition.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 150, pp. 181–195, 1986.The author sincerely thanks A. R. Its for many helpful discussions.     

SINGULAR INTEGRAL DIFFERENTIAL EQUATIONS WITH BOTH CAUCHY KERNEL AND CONVOLUTION KERNEL AND RIEMANN BOUNDARY VALUE PROBLEM

In this paper, we set up and discuss a kind of singular integral differential equation with convolution kernel and Cauchy kernel. By Fourier transform and some lemmas,we turn this class of equations into Riemann boundary value problems, and obtain the general solution and the condition of solvability in class {0}.     

A singular characteristic boundary value problem for the Euler-Poisson-Darboux equation

Summary The paper considers a singular characteristic boundary value problem for the well known EPD equation. By using a modified Riemann method, a formula is obtained which is shown to provide a weak solution of the problem. The weak solution is then shown to be a solution in the classical sense by investigating its differentiability properties. The research of the second author has been supported by NSF research grant GP-11543. Entrata in Redazione il 10 novembre 1971.     

Transplantation and Jacobians of Sunada isospectral Riemann surfaces

We consider relations among the Jacobians of isospectral compact Riemann surfaces constructed using Sunada's theorem. We use a simple algebraic formulation of “transplantation” of holomorphic 1-forms and singular 1-cycles to obtain two main results. First, we obtain a geometric proof of a result of Prasad and Rajan that Sunada isospectral Riemann surfaces have isogenous Jacobians. Second, we determine a relationship (weaker than isogeny) that holds among the Jacobians of Sunada isospectral Riemann surfaces when the Jacobians’ extra structure as principally polarized abelian varieties is taken into account. We also show all Sunada isospectral manifolds have isomorphic real cohomology algebras. Finally, we exhibit transplantation of cycles explicitly in a concrete example of a pair of isospectral Riemann surfaces constructed by Brooks and Tse.     

Existence of global bounded weak solutions to a symmetric system of Keyfitz-Kranzer type

In this paper, we use the compensated compactness method with BV estimates on the Riemann invariants to obtain the global existence of bounded entropy weak solutions for the Cauchy problem of a symmetric system of Keyfitz-Kranzer type.     

A note on Riemann problems for single‐periodic polyanalytic functions

In this article, a new canonical function has been established to deal with Riemann boundary‐value problem of periodic analytic functions discussed in 16 . In comparison with the corresponding result in 16 , the expression of solution obtained here is much simpler. Then, we demonstrate the equivalence of solutions for the homogeneous Riemann problem. What's more, we obtain the precise rank of matrix of coefficients for the system of linear algebraic equations (4.35) in 16 . Those results can simplify the discussion of Riemann problem of single‐periodic polyanalytic functions in 16 .     

Wave front sets of the Riemann function of elastic interface problems

We obtain an inner and an outer estimates of wave front sets and analytic wave front sets of the Riemann function of elastic interface problems by using the localization method due to Wakabayashi. In our problem the outer estimate of wave front sets and analytic wave front sets of the Riemann function coincides with the inner estimate of those. The strong point of our results is to catch the lateral wave as well as the incident, the reflected, and the refracted waves. Copyright © 2004 John Wiley & Sons, Ltd.     

The contact problem for a piecewise-homogeneous plane with a finite inclusion

A piecewise-homogeneous elastic orthotropic plate, reinforced with a finite inclusion, which meets the interface at a right angle and is loaded with shear forces, is considered. The contact stresses along the contact line are determined, and the behaviour of the contact stresses in the neighbourhood of singular points is established. By using the methods of the theory of analytic functions, the problem is reduced to a singular integro-differential equation in a finite section. Using an integral transformation, a Riemann problem is obtained, the solution of which is presented in explicit form.     

EXTENSION OF SOLUTIONS TO RIEMANN BOUNDARY VALUE PROBLEMS AND ITS APPLICATION

In this paper,solutions of Riemann boundary value problems with nodes are extended to the case where they may have singularties of high order at the nodes.Moreover, further extension is discussed when the free term of the problem involved also possesses singularities at the nodes.As an application,certain singular integral equation is discussed.     

Existence and convergence of solutions of singular boundary value problems for second order ordinary differential equations and applications

In this paper,we consider the singular boundary value problems for second order quasilinear ordinary differential equations and prove existence and convergence on second order perturbation terms.The result is applied to solve the Riemann problem for 2×2 hyperbolic conservation laws,which is a partial differential equation arising in applied mathematical area.     

Nonlinear Riemann - Hilbert Problems without Transversality

Nonlinear Riemann - Hilbert problems (RHP) generalize two fundamental classical problems for complex analytic functions, namely: 1. the conformal mapping problem, and 2. the linear Riemann - Hilbert problem. This paper presents new results on global existence for the nonlinear (RHP) in doubly connected domains with nonclosed restriction curves for the boundary data. More precisely, our nonlinear (RHP) is required to become ?at infinity”?, i.e., for solutions having large moduli, a linear (RHP) with variable coefficients. Global existence for q-connected domains was already obtained in [9] for the special case that the restriction curves for the boundary data ?at infinity”? coincide with straight lines corresponding to linear (RHP)-s with special so-called constant - coefficient transversality boundary conditions. In this paper, the boundary conditions are much more general including highly nonlinear conditions for bounded solutions in the context of nontransversality. In order to prove global existence, we reduce the problem to nonlinear singular integral equations which can be treated by a degree theory of Fredholm - quasiruled mappings specifically constructed for mappings defined by nonlinar pseudodifferential operators.