Initial Boundary Value Problem of an Equation from Mathematical Finance

Consider the initial boundary value problem of the strong degenerate parabolic equation ?_(xx)u + u?_yu-?_tu = f(x, y, t, u),(x, y, t) ∈ Q_T = Ω×(0, T)with a homogeneous boundary condition. By introducing a new kind of entropy solution, according to Oleinik rules, the partial boundary condition is given to assure the well-posedness of the problem. By the parabolic regularization method, the uniform estimate of the gradient is obtained, and by using Kolmogoroff 's theorem, the solvability of the equation is obtained in BV(Q_T) sense. The stability of the solutions is obtained by Kruzkov's double variables method.     

Dynamic boundary value problems of the second-order: Bernstein–Nagumo conditions and solvability

This article analyzes the solvability of second-order, nonlinear dynamic boundary value problems (BVPs) on time scales. New Bernstein–Nagumo conditions are developed that guarantee an a priori bound on the delta derivative of potential solutions to the BVPs under consideration. Topological methods are then employed to gain solvability.     

具有脉冲的分数阶Bagley-Torvik 模型边值问题

将具有脉冲的分数阶Bagley-Torvik微分方程边值问题巧妙地转化为积分方程,定义加权Banach空间及全连续算子,运用不动点定理获得该边值问题解的存在性定理.举例说明了定理的应用.最后提出有趣的研究问题.     

On the Vekua Pair in Spheroidal Geometry and its Role in Solving Boundary Value Problems

The Vekua pair forms a transformation between the kernel of the Laplace's and the kernel of the Helmholtz's operator. In fact, it provides an interior solution of the Helmholtz's equation once an interior harmonic function is given, and conversely, given an interior solution of the Helmhotz's equation an interior harmonic function is constructed. Consequently, it seems that the Vekua connection offers the perfect ground to obtain solutions of boundary value problems connected with Helmholtz operator. Vekua expressed his transformation in spherical coordinates. Nevertheless, when a change of coordinates is applied, the transformation assumes a much more complicated form, but it still remains a very useful technique for dealing with solutions of the equations of Laplace and Helmholtz. Here we extend the Vekua theory to a new integral transformation pair concerning solutions of the aforementioned operators in exterior domains. In addition, the form of the Vekua transformation is analyzed in spheroidal coordinates and its implication to boundary value problems is investigated.     

On solvability of singular integral-differential equations with convolution

In this paper, we study a class of singular integral-different equations of convolution type with Cauchy kernel. By means of the classical boundary value theory, of the theory of Fourier analysis, and of the principle of analytic continuation, we transform the equations into the Riemann-Hilbert problems with discontinuous coefficients and obtain the general solutions and conditions of solvability in class $\{0\}$. Thus, the result in this paper generalizes the classical theory of integral equations and boundary value problems.     

The Schwarz problem for analytic functions in torus-related domains

The Cauchy kernel is one of the two significant tools for solving the Riemann boundary value problem for analytic functions. For poly-domains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in poly-domains. This symmetry is lost if the classical counterpart of the one-dimensional form of the Cauchy kernel is applied. It is also decisive for the establishment of connection between the Riemann–Hilbert problem and the Riemann problem. Thus, not only the Schwarz problem for holomorphic functions in poly-domains is solved, but also the basis is established for solving some other problems. The boundary values of functions, holomorphic in poly-domains, are classified in the Wiener algebra. The general integral representation formulas for these functions, the solvability conditions and the solutions of the corresponding Schwarz problems are given explicitly. A necessary and sufficient condition for the boundary values of a holomorphic function for arbitrary poly-domains is given. At the end, well-posed formulations of the torus-related problems are considered.     

Finite difference schemes for the fourth-order parabolic equations with different boundary value conditions

In this paper, the fourth-order parabolic equations with different boundary value conditions are studied. Six kinds of boundary value conditions are proposed. Several numerical differential formulae for the fourth-order derivative are established by the quartic interpolation polynomials and their truncation errors are given with the aid of the Taylor expansion with the integral remainders. Effective difference schemes are presented for the third Dirichlet boundary value problem, the first Neumann boundary value problem and the third Neumann boundary value problem, respectively. Some new embedding inequalities on the discrete function spaces are presented and proved. With the method of energy analysis, the unique solvability, unconditional stability and unconditional convergence of the difference schemes are proved. The convergence orders of derived difference schemes are all O(τ² + h²) in appropriate norms. Finally, some numerical examples are provided to confirm the theoretical results.     

A singular integral equation with the Cauchy kernel on a closed interval in a class of distributions

In the present paper, we introduce the notion of a singular integral with the Cauchy kernel for distributions and consider a singular integral equation with the Cauchy kernel on a closed interval for the case in which the right-hand side is a distribution that admits a representation in the form of the sum of a distribution vanishing in neighborhoods of the endpoints and an ordinary function satisfying the Hölder condition. The solution is also sought in the form of a distribution. Distributions are treated as linear functionals on some test functions. We analyze the solvability of the equation in the class of distributions and obtain explicit formulas for the inversion of this equation, similar to formulas for ordinary solutions. To analyze the solvability of the singular integral equation, we use an approach based on the consideration of the Riemann boundary value problem for analytic functions with a generalized boundary condition. When stating and studying this problem, we use the results in [1, 2].Translated from Differentsialnye Uravneniya, Vol. 40, No. 9, 2004, pp. 1208–1218.Original Russian Text Copyright © 2004 by Setukha.     

ON HASEMAN BOUNDARY VALUE PROBLEM FOR A CLASS OF META-ANALYTIC FUNCTIONS

In this article, Haseman boundary value problem for a class of meta-analytic functions is studied. The expression of solution and the condition of solvability for Haseman boundary value problem are obtained by changing the problem discussed into the equivalent Haseman boundary value problem of bi-analytic function. And the expression of solution and the condition of solvability depend on the canonical matrix.     

An inverse spectral problem for integro-differential Dirac operators with general convolution kernels

ABSTRACT

An integro-differential Dirac system with a convolution kernel consisting of four independent functions is considered. We prove that the kernel is uniquely determined by specifying the spectra of two boundary value problems with one common boundary condition. The proof is based on the reduction of this nonlinear inverse problem to solving some nonlinear integral equation, which we solve globally. On this basis we also obtain a constructive procedure for solving the inverse problem along with necessary and sufficient conditions for its solvability in an appropriate class of kernels.     

A multipoint boundary value problem for systems of linear generalized differential equations with singularities

We study a multipoint boundary value problem for systems of Kurzweil generalized linear differential equations with singularities on a finite closed interval of the real line. We assume that the offdiagonal entries of the matrix function corresponding to the system, as well as the elements of the right-hand side of the system, have bounded variation on the entire interval; however, the diagonal entries of the matrix function are not assumed to have bounded variation on the entire interval. This is what we mean by saying that the system is singular. We study the unique solvability of the problem. We prove a general theorem and use it to we obtain efficient optimal (in particular, spectral) solvability conditions for the problem.     

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}.     

Boundary value problems with operator right-hand side

For a second-order boundary value problem with operator right-hand side and with functional boundary conditions, we prove solvability theorems with mixed and Dirichlet boundary conditions assuming the existence of a lower and an upper function. These theorems are analogs of theorems for the corresponding boundary value problems for an ordinary second-order differential equation with right-hand side satisfying the Carathéodory conditions.     

On an integro-differential equation with almost difference kernel on the half-line

We study the solvability of a class of integro-differential equations with almost difference kernel on the positive half-line. Using a special three-factor decomposition of the original integro-differential operator, we obtain sufficient conditions for the solvability of this equation in the class of tempered absolutely continuous functions. Under additional conditions on the kernel of the corresponding homogeneous equation with some value of the parameter occurring in it, we prove the existence of a nontrivial absolutely continuous solution, which, depending on the sign of the first moment of the kernel, is either a bounded function or has the asymptotics O(x), x → ∞.     

边界过原点的任意半平面中的Hilbert边值问题

给出了边界过原点的任意半平面中的Hilbert边值问题的提法,定义了函数的一种对称扩张,并利用这种对称扩张将此Hilbert边值问题转化为无穷直线上的Riemann边值问题,得到了该问题的一般解和可解性定理.     

On mixed boundary value problems for pseudoparabolic systems

Solvability analysis of mixed boundary value problems for pseudoparabolic systems in a special scale of weighted Sobolev spaces is presented. The class under consideration contains the linearized Navier-Stokes system. It is proved that, choosing the power weight, one can diminish the number of solvability conditions and in some cases obtain unconditional solvability of the boundary value problems.     

A Boundary Integral Formulation for Poroelastic Materials

Wave propagation in porous media is an important topic, e.g. in geomechanics or the oil-industry. We formulate a linear system of coupled partial differential equations based on Biot's theory with the solid displacements and the pore pressure as the primary unknowns. To solve this system of coupled partial differential equations in a semi-infinite homogeneous domain the BEM (Boundary element method) is especially suitable. Starting from a representation formula a system of two boundary integral equations is derived. These boundary integral equations are used to solve related boundary value problems via a direct approach. Coercivity of the resulting bilinear form is shown, from which unique solvability of the variational formulation follows from injectivity. Using these results we derive the unique solvability of the related boundary integral equations. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Neumann and Robin problems in a cracked domain with jump conditions on cracks

The boundary value problem for the Laplace equation is studied on a domain with smooth compact boundary and with smooth internal cracks. The Neumann or the Robin condition is given on the boundary of the domain. The jump of the function and the jump of its normal derivative is prescribed on the cracks. The solution is looked for in the form of the sum of a single layer potential and a double layer potential. The solvability of the corresponding integral equation is determined and the explicit solution of this equation is given in the form of the Neumann series. Estimates for the absolute value of the solution of the boundary value problem and for the absolute value of the gradient of the solution are presented.     

Existence and Non-existence of Positive Solutions for a Discrete Fractional Boundary Value Problem

In this work, we deal with two-point boundary problem for a finite nabla fractional difference equation. First, we establish an associated Green''s function and state some of its properties. Under suitable conditions, we deduce the existence and non-existence of positive solutions to the considered problem. Finally, we construct a few examples to illustrate the established results.     

上半平面中含参变未知函数的Hilbert边值问题

给出了上半平面中的含参变未知函数的Hilbert边值问题的提法,利用函数的对称扩张,将其转化为无穷直线上含参变未知函数的Riemann边值问题,得到了该问题的一般解和可解性定理.