Integral Equations, Boundary Value Problems and Related Problems

The following sections are included:

• PREFACE

• ORGANIZING COMMITTEES

• CONTENTS

This paper has two parts. In the first part, we study the boundedness for a kind of singular integral operator, which is related to the Cauchy-type integral of k-monogenic function in Clifford Analysis. In the second part, we show the Hölder continuous of the singular integral operator.

In this paper, the hyper-complex forms of equations for the plane and the sphere in R³ are given. Then, the concept of symmetric points with respect to the plane and the sphere in R³ is introduced, equivalent equations for symmetric points respect to the plane and the sphere are given. Some special Mobius transformations in hyper-complex space Cl₃ are considered. Some properties of the mentioned Mobius transformation are given. For example: The Mobius transformation maps the plane and the sphere in R³ one-to-one onto the plane or the sphere; The Möbius transformation preserves the symmetric points with respect to the plane and the sphere in Cl₃; The cross ratio of four distinct points is also invariant under the Möbius transformation. A relation between the monogenic function and Möbius transformation is shown. Finally, based on the properties of Mobius transformations, integral formulas for change of variables under Möbius transformations are constructed.

The interaction of SH wave with the array of periodic cracks in a piezoelectric substrate bonded to a functionally graded materials (FGJVI) coatings is considered. The governing equations along with permeable crack boundary, regularity and continuity conditions across the interface are reduced to a coupled set of Hilbert singular integral equations which are solved approximately by applying Cheyshev polynomials. Numerical results for the normalized dynamic stress intensive factors (NDSIF) and the normalized electric displacement intensive factors (NEDIF) are presented. The effects of geometric parameters and the physical parameters, and the effects of frequency and angles of SH wave on the NDSIF and the NEDIF are discussed.

A theoretical treatment of antiplane crack problem of two collinear cracks on the two sides of and perpendicular to the interface between a functionally graded orthotropic strip bonded to an orthotropic homogeneous substrate is put forward. The problem is formulated in terms of a singular integral equation with the crack face displacement as the unknown variable. Numerical calculations are carried out, and the influences of the orthotropy and nonhomogeneous parameters on the mode III stress intensity factors are investigated.

In this article, we mainly deal with the boundary value problem for triharmonic function with value in universal Clifford algebra: where (j = 1, … , 5) ∂Ω is a Liapunov surface in Rⁿ, the Dirac operator are unknown functions with values in an universal Clifford algebra Cl(V_n,n). Under some hypotheses, it is proved that the boundary value problem has a unique solution.

A new dynamical systems method is studied for solving nonlinear operator equation . This method consists of the construction of a nonlinear dynamical systems, that is, a Cauchy problem , where Φ is a suitable operator, which has the following properties: i) :∃u(t) ∀t > 0, ii) :∃u(∞), and iii) . Conditions on are given which allow one to chosen Φ such that i), ii), and iii) hold. This method yields also a convergent iterative method for finding true solution y. Stable approximation to a solution of the equation is constructed by this DSM when f is unknown but fδ is known, when ||f − fδ|| ≤ δ.

We discuss a class of generalized retarded nonlinear integral inequalities, which not only include nonlinear compound function of unknown function but also include retard items, and give upper bound estimation of the unknown function by integral inequality technique. This estimation can be used as tool in the study of differential equations with the initial conditions.

In this paper, we study on elastodynamic fracture problems and present a symplectic scheme, combined with the spectral boundary integral method, for solving anti-plane spontaneous rupture problems via constructing a nonautonomous Hamiltonian system by introducing independent variables in extended phase space. Slip-weakening friction law is considered with the approach. Numerical experiments are employed to test the effectiveness of symplectic scheme, and the comparisons of numerical results show that the present method is of higher accuracy compared to the non-symplectic method. The results show that the method is a efficient tool for simulating quasi-static and dynamic earthquake processes.

Base on the integral representations of the solution being derived via Fokas’ transform method, the high-frequency asymptotics for the solution of the modified Helmholtz equation, in a half-plane and subject to the Dirichlet condition is discussed. For the case of piecewise constant boundary data, full asymptotic expansions of the solution are obtained by using Watson’s lemma and the method of steepest descents for definite integrals.

In this paper, a inverse boundary value problem involving filtration for elliptic systems of equations is discussed. The problem can be reduced to a discontinuous Riemann-Hilbert boundary value problem for elliptic complex equations. By using the theory of complex equations and Newton imbedding method, we prove the existence and uniqueness of the inverse boundary value problem.

In this paper we introduce extended cone metric spaces, prove some fixed point theorems of contractive mappings in such spaces. The results directly improve and generalize some fixed point results in cone metric spaces. In addition, we use our conclusions to obtain the existence and uniqueness of solution for an ordinary differential equation with periodic boundary condition.

In this paper we investigate the existence of at least one positive solutions and the existence of multiple positive solutions for the nonlinear singular thirdorder three-point boundary value problem. Our nonlinearity f(t, y) may be singular at t = 0, t = 1, y = 0.

The main objective is to give the asymptotic behavior at infinity for the Neumann integral and the modified Neumann integral in the half space of ℝⁿ. The asymptotic behavior hold outside an explicitly defined exceptional set as [10].

In this paper, we proposed a piecewise Tikhonov regularization scheme to overcome the disadvantage of the classical Tikhonov regularization for solving the Radon integral equation whose solution is piecewise continues arising in CT imaging. In our new scheme, we firstly detect the location of discontinuous interface by applying the classical Tikhonov regularization. Then a so-called piecewise Tikhonov regularization method is applied to reconstruction the smooth solution in each piece. Some numerical examples has also be presented to demonstrate the effectiveness of this two-steps scheme.

In this paper we consider the Jacobian conjecture in the view of hypercomplex analysis. For a polynomial mapping P(w) = (p1(w),p2(w)): ℂ² → ℂ² we discuss the left holomorphic quaternionic function f(z1, z2, z3) = p1 (w) + jp2 (w) where w = (x0 + x1 i, x2 + x3,i) and z1 = x1- x0i, z2 = x2 - x0i, z3 = x3 - x0i. Then we give a new approach to the Jacobian conjecture by the use of argument principle for quaternionic holomorphic functions.

In this paper, by means of Riemann-Schwarz’s symmetry priciple integrated with the analysis singularity of complex potentials, The interaction between antiplane cicular inclusion and circular hole of piezoelectric materials under the antiplane line force and inplane line charge is considered, and their complex potential expressions of the electro-elastic field are derived.

In this paper we propose a fundamental numerical method for heat and moisture transfer model, which is a system of coupled ordinary differential equations(ODEs) on temperature, water vapor pressure, water vapor mass flux and condensation rate through parallel pore textiles. By decoupling the coupled ODEs, four explicit formulations are achieved to describe temperature, water vapor pressure, water vapor mass flux and the rate of condensation respectively. We develop a numerical algorithm for the coupled ODEs by finite difference method (FDM) and further prove that the numerical algorithm is convergent with the convergence rate of the first order theoretically. Numerical simulation is implemented for down and polyester material to verify the validity of the algorithm. The numerical results are well matched with the experiment data on the ”Walter” Manikin.

The plane problem for the Haversian cortical bone with a radial microcrack in the interstitial bone under biaxial tension loading is considered. A two-dimensional coated fiber composite model of the Haversian cortical bone is developed correspondingly. By using the solution for the edge dislocation as Green’s functions, the problem of a radial microcrack located in the interstitial bone in the vicinity of the osteon is formulated into a system of singular integral equations with Cauchy kernels. And the interaction effect of various material and geometric parameters of the Haversian cortical bone upon the microcrack tip’s stress intensity factors is studied. The numerical analysis indicates that the stress intensity factor of the microcrack is dominated by the shear modulus ratio among the Haversian canal, the osteon and the interstitial bone, the distance of the left tip of the microcrack from cement line, the length of the microcrack. The numerical results suggest that when the osteon is softer than the interstitial bone the osteon prompts the microcrack propagation while the osteon is stiffer than the interstitial bone the osteon repels microcrack propagation. However, this interaction effect is limited near the osteon. Some of the numerical results are in accordance with the results obtained. And additional predicted numerical results are given, which need to be confirmed via experiment or computer simulation.

Unitary integral transforms play an important role in mathematical physics. A primary example is the Fourier transform whose kernel is of the form k(x,y) = k(xy), i.e., of the product type. Here we consider the determination of spectrum for such unitary operators as the issue is important in the solvability of the corresponding inhomogeneous Fredholm integral equation of the second kind. A Main Theorem is proven that characterizes the spectral set. Properties of eigenfunctions and eigenspace dimensions are further derived as consequences of the Main Theorem. Concrete examples are also offered as applications.

Studies have shown that long-period ground motion can be amplified due to basin structure, the amplification effects generate such as the amplitude increased, the duration lengthened. The basin effects of ground motion are extremely detrimental to engineering structures of long natural vibration period, such as high-rise buildings and bridges. The paper carries out the numerical simulation of long-period ground motion on basin effects with far-field earthquake input, which uses previous developed by the applicant structurepreserving algorithm, symplectic discrete singular convolution differentiator (SDSCD). The preliminary test shows that the algorithm is feasible to simulate long-period ground motion on basin effects. It will lay the foundation on proposing evaluation method to meet the needs of engineering for long-period ground motion on basin effects, to provide more reliable scientific basis for the seismic risk management of major projects.

The first fundamental complete plane strain (CPS) problem for an infinite onedimensional hexagonal quasi crystals body containing a doubly-periodic array of cracks in periodical and aperiodical plane is considered. Employing the superposition principle of force, the complete plane strain state, which is a special three-dimensional elastic system, is resolved into two linearly independent two-dimensional (plane) elastic systems, one is the generalized plane strain state, and another is the longitudinal displacement state. Using a technique that based on the complex potential function of Kolosov and Muskhelishvili, solving this problem are transferred into seeking analytic functions which fit certain boundary value problems. Furthermore, the general representation for the solution is constructed, under some general restrictions the boundary value problem is reduced to a normal type singular integral equation with a Weierstrass zeta kernel along the boundary of cracks, and the unique solvability of which is proved.

In this paper, by introducing the appropriate conformal transformation and using the complex variable function method, the problem about an elliptic hole with III asymmetry cracks in one-dimensional hexagonal piezoelectric quasicrystals is sovlved and the SIFs at the crack tip are obtained in the case of impermeable and permeable boundary conditions. In some special onditions, the cracks can be turned into a Griffith crack. By imitation computation, we can get the SIFs at the tip of the Griffith crack. especially, when the quasicrystals become vestigial the crystal, we can also get the SIFs at the tip of the Griffith crack, which is in accordance with the know results.

In this paper, we disscuss one-dimensional hexagonal quasicrystals and infinite elasticity plane with single periodic of second fundamental problem by the condition basical hypothesis. We use the plane elasticity of complex variables method and Hilbert kernel integral formula proving the onedimensional hexagonal quasicrystals and infinite elasticity plane with single periodic of second fundamental problem, and obtaining the solution is not only existence but also unique. And in the proving process, we provide the specific arithmetic. At last, we offer an example to verify.

In this paper we derive some optimal convex combination bounds related to Centroidal mean. We find the greatest values α and the least values β such that the double inequalities and hold for all a,b > 0 with a ≠ b. Here A(a,b), C(a,b), H(a,b), T(a,b) and T₁(a,b) denote the arithmetic, contraharmonic, harmonic, the second Seiffert’s and Centroidal means of two positive numbers a and b, respectively.

The method of fundamental solution (MFS) has high accuracy and fast speed in solving a class of elliptical partial differential equations (PDEs). The coordinate transformation technique is used to convert the elliptical PDEs to biharmonic equations, which can extend the application areas of MFS. In the symmetrical problem, the fundamental functions are reconstructed to increase the computational efficiency. From the examples, it is concluded that the numerical results have high accuracy in the bending problem.

This paper presents the computational techniques for fractional Fredholm-Volterra integral equations. Various rationalized wavelet functions approximation together with collocation method is utilized to reduce this form of integral equations into a system of algebraic equations. Moreover, through illustrative example, a comparison of numerical solutions by using Harr wavelets, Legendre wavelets, Chebyshev wavelets, confirms the expected accuracy. Specially, this method is computationally attractive while the equations have not been solved analytically.