Some non‐local problems for the parabolic‐hyperbolic type equation with complex spectral parameter

In the present paper the unique solvability of two non‐local problems for the mixed parabolic‐hyperbolic type equation with complex spectral parameter is proved. Sectors for values of the spectral parameter where these problems have unique solutions are shown. Uniqueness of the solution is proved by the method of energy integral and existence is proved by the method of integral equations. In particular cases, eigenvalues and corresponding eigenfunctions of the studied problems are found. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary value problems with continuous and special gluing conditions for parabolic-hyperbolic type equations

In the present paper we study the unique solvability of two non-local boundary value problems with continuous and special gluing conditions for parabolic-hyperbolic type equations. The uniqueness of the solutions of the considered problems are proven by the “abc” method. Existence theorems for the solutions of these problems are proven by the method of integral equations. The obtained results can be used for studying local and non-local boundary-value problems for mixed-hyperbolic type equations with two and three lines of changing type.      

带非局部积分项常微分方程的讨论及其应用

研究带非局部积分项的二阶线性常微分方程及其在金融保险上的应用.首先讨论带非局部积分项的二阶常微分方程解的存在唯一性,通过变量代换和累次积分交换积分顺序将非局部项简化,将方程化为方程组,然后完成了对方程组解的存在唯一性的证明.接着分析了带非局部项的二阶常微分方程解的结构,给出了方程解的形式.最后通过推导,指出带非局部项的线性常微分方程在保险公司的破产概率研究中的应用,重点放在二阶方程的应用上,并且在某一特定情况下,举出了一个可以给出解析解的例子.     

On existence problems for the optimal control of certain nonlinear integral equations of Urysohn type

In Refs. 1–3, existence results have been obtained for optimal control problems whose state equations are described by certain nonlinear integral equations of Urysohn type. We generalize and synthesize these results by formulating a general lower closure result from which the results of Refs. 1–3 are shown to follow. In the course of this, we also present a novel and rather abstract treatment of existence problems for variable-time optimal control, quite in the spirit of Ref. 4.     

Initial-boundary value problem for a parabolic-hyperbolic equation with power-law degeneration on the type change line

We consider the first boundary value problem for equations of mixed type in a rectangular domain. A criterion for the solution uniqueness is proved by the spectral expansion method. The solution is constructed in the form of a series in the eigenfunctions of the corresponding one-dimensional spectral problem. The stability of the solution with respect to the initial function is proved.     

Inverse problems for the Sturm-Liouville equation with the discontinuous coefficient

In this study we derive the Gelfand-Levitan-Marchenko type main integral equation of the inverse problem for the boundary value problem $L$ and prove the uniquely solvability of the main integral equation. Further, we give the solution of the inverse problem by the spectral data and by two spectrum.     

Modified equations of the first kind for the Helmholtz equation

Integral equations of the first kind for exterior problems arising in the study of the three‐dimensional Helmholtz equation are considered. These equations are derived by seeking solutions in the form of layer potentials with modified fundamental solutions. For each first kind equation, existence and uniqueness of solution are proved with the aid of composition relations involving associated modified boundary integral operators. For the Dirichlet problem, an optimal choice of the modification coefficients is considered in order to minimize the condition number of the resulting integral operator. Copyright © 2014 John Wiley & Sons, Ltd.     

A FEAST algorithm with oblique projection for generalized eigenvalue problems

The contour integral‐based eigensolvers are the recent efforts for computing the eigenvalues inside a given region in the complex plane. The best‐known members are the Sakurai–Sugiura method, its stable version CIRR, and the FEAST algorithm. An attractive computational advantage of these methods is that they are easily parallelizable. The FEAST algorithm was developed for the generalized Hermitian eigenvalue problems. It is stable and accurate. However, it may fail when applied to non‐Hermitian problems. Recently, a dual subspace FEAST algorithm was proposed to extend the FEAST algorithm to non‐Hermitian problems. In this paper, we instead use the oblique projection technique to extend FEAST to the non‐Hermitian problems. Our approach can be summarized as follows: (a) construct a particular contour integral to form a search subspace containing the desired eigenspace and (b) use the oblique projection technique to extract desired eigenpairs with appropriately chosen test subspace. The related mathematical framework is established. Comparing to the dual subspace FEAST algorithm, we can save the computational cost roughly by a half if only the eigenvalues or the eigenvalues together with their right eigenvectors are needed. We also address some implementation issues such as how to choose a suitable starting matrix and design‐efficient stopping criteria. Numerical experiments are provided to illustrate that our method is stable and efficient.     

Inverse spectral problems for differential pencils on the half-line with turning points

The inverse spectral problem of recovering pencils of second-order differential operators on the half-line with turning points is studied. We establish properties of the spectral characteristics, give a formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem.     

Boundary value problems for mixed type equations and applications

In this paper we outline a general method for finding well-posed boundary value problems for linear equations of mixed elliptic and hyperbolic type, which extends previous techniques of Berezanskii, Didenko, and Friedrichs. This method is then used to study a particular class of fully nonlinear mixed type equations which arise in applications to differential geometry.     

Analog of the Gellerstedt problem for a loaded equation of the third order

In this paper, we study an analog of the Gellerstedt problem for the third‐order loaded equation with boundary conditions on nonparallel characteristics in the hyperbolic domain of the equation.     

Cauchy-type singular integral equation with constant coefficients on the real line

In this paper, exact solutions of the characteristic and dominant equation with Cauchy kernel on the real line are presented. Next, trigonometric polynomials are used to derive approximate solutions of these equations. Moreover, estimations of errors of the approximated solutions are presented and proved.     

The solvability of some kinds of singular integral equations of convolution type with variable integral limits

In this paper, we discuss several classes of convolution type singular integral equations with variable integral limits in class $ H^*_1 $. By means of the theory of complex analysis, Fourier analysis and integral transforms, we can transform singular integral equations with variable integral limits into the Riemann boundary value problems with discontinuous coefficients. Under the solvability conditions, the existence and uniqueness of the general solutions can be obtained. Further, we analyze the asymptotic properties of the solutions at the nodes. Our work improves the Noether theory of singular integral equations and boundary value problems, and develops the knowledge architecture of complex analysis.     

A solution for antiplane‐strain local problems using elliptic integrals of Cauchy type

Quasi‐periodic piecewise analytic solutions, without poles, are found for the local antiplane‐strain problems. Such problems arise from applying the asymptotic homogenization method to an elastic problem in a parallel fiber‐reinforced periodic composite that presents an imperfect contact of spring type between the fiber and the matrix. Our methodology consists of rewriting the contact conditions in a complex appropriate form that allow us to use the elliptic integrals of Cauchy type. Several general conditions are assumed including that the fibers are disposed of arbitrary manner in the unit cell, that all fibers present imperfect contact with different constants of imperfection, and that their cross section is smooth closed arbitrary curves. Finally, we obtain a family of piecewise analytic solutions for the local antiplane‐strain problems that depend of a real parameter. When we vary this parameter, it is possible to improve classic bounds for the effective coefficients. Copyright © 2017 John Wiley & Sons, Ltd.     

A strong solution of a high-order mixed type partial differential equation with integral conditions

In this article, we study a mixed problem with integral boundary conditions for a high-order partial differential equation of mixed type. We prove the existence and uniqueness of a strong solution. The proof is based on energy inequality and on the density of the range of the operator generated by the considered problem.     

On the Riemann Hilbert type problems in Clifford analysis

In this paper boundary value problems combining Jump — Riemann and Hilbert problems for monogenic functions in Ahlfors-David regular surfaces and in the upper half space respectively are investigated. The explicit formula of the solution is obtained.     

A nonlocal problem with integral gluing condition for a third-order loaded equation with parabolic-hyperbolic operator involving fractional derivatives

This work devoted to prove of uniqueness and existence of solution of the nonlocal problem with integral gluing condition for the loaded parabolic-hyperbolic type equation involving Caputo derivatives which loaded parts in Riemann-Liouville integral-differential operator. Using the method of integral energy, the uniqueness of solution have been proved. Existence of solution was proved by the method of integral equations.