Generalized Abel type integral equations with localized fractional integrals and derivatives

Generalized Abel type integral equations with Gauss, Kummer's and Humbert's confluent hypergeometric functions in the kernel and generalized Abel type integral equations with localized fractional integrals are considered. The left-hand sides of these equations are inversed by using generalized fractional derivatives. Explicit solutions of the equations in the class of locally summable functions are obtained. They are represented in terms of hypergeometric functions. Asymptotic power exponential type expansions of the generalized and localized fractional integrals are obtained. The base solutions of the generalized Abel type integral equation are given in the form of asymptotic series.     

广义函数法边界积分方程的建立

从广义函数论出发,本文引入一特殊广义函数δθ_P,通过它以及它的各阶导数建立了任一足够光滑函数的各阶导数的边界积分方程。对于由线性偏微分算子定义的问题,只要存在着相应的基本解,问题的偏微分方程总可转换成边界积分方程。     

Method of lines with boundary elements for 1‐D transient diffusion‐reaction problems

Time‐dependent differential equations can be solved using the concept of method of lines (MOL) together with the boundary element (BE) representation for the spatial linear part of the equation. The BE method alleviates the need for spatial discretization and casts the problem in an integral format. Hence errors associated with the numerical approximation of the spatial derivatives are totally eliminated. An element level local cubic approximation is used for the variable at each time step to facilitate the time marching and the nonlinear terms are represented in a semi‐implicit manner by a local linearization at each time step. The accuracy of the method has been illustrated on a number of test problems of engineering significance. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

Local mass conservative Hermite interpolation for the solution of flow problems by a multi-domain boundary element approach

This paper presents a local Hermite radial basis function interpolation scheme for the velocity and pressure fields. The interpolation for velocity satisfies the continuity equation (mass conservative interpolation) while the pressure interpolation obeys the pressure equation. Additionally, the Dual Reciprocity Boundary Element method (DRBEM) is applied to obtain an integral representation of the Navier-Stokes equations. Then, the proposed local interpolation is used to obtain the values of the field variables and their partial derivatives at the boundary of the sub-domains. This interpolation allows one to obtain the boundary values needed for the integral formulas for velocity and pressure at some nodes within the sub-domains. In the proposed approach the boundary elements are merely used to parameterize the geometry, but not for the evaluation of the integrals as it is usually done. The presented multi-domain approach is different from the traditional ones in boundary elements because the resulting integral equations are non singular and the boundary data needed for the boundary integrals are approximated using a local interpolation. Some accurate results for simple Stokes problems and for the Navier-Stokes equations at low Reynolds numbers up to Re = 400 were obtained.     

Nyström discretization of parabolic boundary integral equations

A Nyström method for the discretization of thermal layer potentials is proposed and analyzed. The method is based on considering the potentials as generalized Abel integral operators in time, where the kernel is a time dependent surface integral operator. The time discretization is the trapezoidal rule with a corrected weight at the endpoint to compensate for singularities of the integrand. The spatial discretization is a standard quadrature rule for surface integrals of smooth functions. We will discuss stability and convergence results of this discretization scheme for second-kind boundary integral equations of the heat equation. The method is explicit, does not require the computation of influence coefficients, and can be combined easily with recently developed fast heat solvers.     

Solution of boundary-value problems for a degenerating elliptic equation of the second kind by the method of potentials

In this paper we study basic boundary value problems for one multidimensional degenerating elliptic equation of the second kind. Using the method of potentials we prove the unique solvability of the mentioned problems. We construct a fundamental solution and obtain an integral representation for the solution to the equation. Using this representation we study properties of solutions, in particular, the principle of maximum. We state the basic boundary value problems and prove their unique solvability. We introduce potentials of single and double layers and study their properties. With the help of these potentials we reduce the boundary value problems to the Fredholm integral equations of the second kind and prove their unique solvability.     

Analysis of the stress singularity for a bi-material V-notch by the boundary element method

A new algorithm coupling the boundary element technique with the characteristic expansion method is proposed for the computation of the singular stress field in the V-notched bi-material structure. After the stress asymptotic expansions are introduced into the linear elasticity equilibrium equations, the governing equations at the small sector dug out from the bi-material V-notch tip region are transformed into the ordinary differential eigen-equations. All the parameters in the asymptotic expansions except the combination coefficients can be achieved by solving the established eigen-equations with the interpolating matrix method. Furthermore, the conventional boundary element method is applied to modeling the remaining structure without the notch tip region. The combination coefficients in the asymptotic expansion forms can be computed by the discretized boundary integral equations. Thus, the singular stress field at the V-notch tip and the generalized stress intensity factors of the bi-material notch are successfully calculated. The accurate singular stress field obtained here is very useful in the evaluation of the fracture property and the fatigue life of the V-notched bi-material structure.     

An extrapolation method for a class of boundary integral equations

Boundary value problems of the third kind are converted into boundary integral equations of the second kind with periodic logarithmic kernels by using Green's formulas. For solving the induced boundary integral equations, a Nyström scheme and its extrapolation method are derived for periodic Fredholm integral equations of the second kind with logarithmic singularity. Asymptotic expansions for the approximate solutions obtained by the Nyström scheme are developed to analyze the extrapolation method. Some computational aspects of the methods are considered, and two numerical examples are given to illustrate the acceleration of convergence.

     

Generalized solutions of initial-boundary value problems for second-order hyperbolic systems

The method of boundary integral equations is developed as applied to initial-boundary value problems for strictly hyperbolic systems of second-order equations characteristic of anisotropic media dynamics. Based on the theory of distributions (generalized functions), solutions are constructed in the space of generalized functions followed by passing to integral representations and classical solutions. Solutions are considered in the class of singular functions with discontinuous derivatives, which are typical of physical problems describing shock waves. The uniqueness of the solutions to the initial-boundary value problems is proved under certain smoothness conditions imposed on the boundary functions. The Green’s matrix of the system and new fundamental matrices based on it are used to derive integral analogues of the Gauss, Kirchhoff, and Green formulas for solutions and solving singular boundary integral equations.     

Potential-based numerical solution of Dirichlet problems for the Helmholtz equation

Three-dimensional Dirichlet problems for the Helmholtz equation are considered in generalized formulations. By applying single-layer potentials, they are reduced to Fredholm boundary integral equations of the first kind. The equations are discretized using a special averaging method for integral operators with weak singularities in the kernels. As a result, the integral equations are approximated by systems of linear algebraic equations with easy-to-compute coefficients, which are solved numerically by applying the generalized minimal residual method. A modification of the method is proposed that yields solutions in the spectra of interior Dirichlet problems and integral operators when the integral equations are not equivalent to the original differential problems and are not well-posed. Numerical results are presented for assessing the capabilities of the approach.     

Wall effects on the slow steady motion of a particle in a viscous incompressible fluid

In this paper, asymptotic expansions with respect to small Reynolds numbers are proved for the slow steady motion of an arbitrary particle in a viscous, incompressible fluid past a single plane wall. The flow problem is modelled by a certain boundary value problem for the stationary, nonlinear Navier-Stokes equations. The coefficients of these expansions are the solutions of various, linear Stokes problems which can be constructed by single layer potentials and corresponding boundary integral equations on the boundary surface of the particle. Furthermore, some asymptotic estimates at small Reynolds numbers are presented for the slow steady motion of an arbitrary particle in a viscous, incompressible fluid between two parallel, plane walls and in an infinitely long, rectilinear cylinder of arbitrary cross section. In the case of the flow problem with a single plane wall, the paper is based on a thesis, which the author has written under the guidance of Professor Dr. Wolfgang L. Wendland.     

Localized boundary‐domain singular integral equations of Dirichlet problem for self‐adjoint second‐order strongly elliptic PDE systems

The paper deals with the three‐dimensional Dirichlet boundary value problem (BVP) for a second‐order strongly elliptic self‐adjoint system of partial differential equations in the divergence form with variable coefficients and develops the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of localized boundary‐domain integral equations. The equivalence between the Dirichlet BVP and the corresponding localized boundary‐domain integral equation system is studied. We establish that the obtained localized boundary‐domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener–Hopf factorization method, we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces. Copyright © 2016 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons, Ltd.     

Method of boundary integral equations as applied to the numerical solution of the three-dimensional Dirichlet problem for the laplace equation in a piecewise homogeneous medium

A Dirichlet problem is considered in a three-dimensional domain filled with a piecewise homogeneous medium. The uniqueness of its solution is proved. A system of Fredholm boundary integral equations of the second kind is constructed using the method of surface potentials, and a system of boundary integral equations of the first kind is derived directly from Green’s identity. A technique for the numerical solution of integral equations is proposed, and results of numerical experiments are presented.     

On the use of potential fields in fluid mechanics

In classical fluid mechanics, potential fields have been employed to enable the integration of the equations of motion. As is well known, Bernoulli's equation is obtained as a first integral of Euler's equations in the absence of vorticity and viscosity if the velocity vector is perceived as the gradient of a scalar potential. The so-called Clebsch transformation [1] involving three scalar potentials allows for a further extension to flows with non-vanishing vorticity; the resulting equations turn out to be self-adjoint, allowing for a variational formulation. All attempts in classic literature, however, are restricted to inviscid flows and the finding of a potential representation enabling the integration of the Navier-Stokes equations remains desirable. Progress on this topic was reported by [3, 4] who constructed a first integral of the two-dimensional incompressible Navier-Stokes equations by making use of an auxiliary potential field and a representation of the fields in terms of complex coordinates. The new formulation proved to be useful in numerical applications and moreover, replacing the scalar potential by a tensor potential, the theory can be successfully generalised to encompass three-dimensional Navier-Stokes flow. Related to the first integral a finite element method was presented in [2] based on a formulation involving the velocities and the first order derivatives of the introduced potential. This way the dynamic boundary condition could be incorporated elegantly and the system of equations fitted into the first order system least-squares methodology. However, a promising alternative approach results if one considers the streamfunction and a slightly modified potential field as independent variables. This new approach involves Laplacian operators rather than mixed derivatives and allows for a convenient embodiment of the Neumann conditions on the streamfunction that is in contrast to the original stream function / potential formulation [4]. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Series solutions of confluent Heun equations in terms of incomplete Gamma-functions

We present a simple systematic algorithm for construction of expansions of the solutions of ordinary differential equations with rational coefficients in terms of mathematical functions having indefinite integral representation. The approach employs an auxiliary equation involving only the derivatives of a solution of the equation under consideration. Using power-series expansions of the solutions of this auxiliary equation, we construct several expansions of the four confluent Heun equations'' solutions in terms of the incomplete Gamma-functions. In the cases of single- and double-confluent Heun equations the coefficients of the expansions obey four-term recurrence relations, while for the bi- and tri-confluent Heun equations the recurrence relations in general involve five terms. Other expansions for which the expansion coefficients obey recurrence relations involving more terms are also possible. The particular cases when these relations reduce to ones involving less number of terms are identified. The conditions for deriving closed-form finite-sum solutions via right-hand side termination of the constructed series are discussed.     

Ginzburg-Landau方程的非齐次初边值问题

研究具非线性边界条件的一类广义Ginzburg-Landau方程解的整体存在性．推导了Ginzburg-Landau方程的非齐次初边值问题光滑解的几个积分恒等式,由此得到了解的法向导数在边界上的平方模以及解的平方模和导数的平方模估计;通过逼近技巧、先验估计和取极限方法证明了Ginzburg-Landau方程的非齐次初边值问题整体弱解的存在性．     

Integral equations for the Fourier-transformed boundary values for the transmission problems for right-angled wedges and octants

A 4×4-system of integral equations for the Fourier transformed boundary values of the normal derivatives of the wave functions defined in the four quadrants of R ²-space is derived. This system results from the scalar transmission problem with continuous passage of the boundary values of the total wave-fields and of the weighted normal derivatives corresponding to the case of magnetically polarized fields. Several equivalent systems of integral equations are deduced then which show that Banach's fixed point principle may be applied at least for slightly differing media in the four quadrants. The method, which is equivalent to a compatibility condition for holomorphic functions, may be generalized to the case of the scalar transmission problem for octants in R ³-space. There a 12 × 12-system of integral equations for the Fourier transformed normal derivatives on the quarter-plane faces is established.     

A COLLOCATION METHOD FOR THE CONDUCTIVITY PROBLEM WITH DISCONTINUOUS COEFFICIENT

In this paper, a new collocation BEM for the Robin boundary value problem of the conductivity equation ▽(γ▽u) = 0 is discussed, where the 7 is a piecewise constant function. By the integral representation formula of the solution of the conductivity equation on the boundary and interface, the boundary integral equations are obtained. We discuss the properties of these integral equations and propose a collocation method for solving these boundary integral equations. Both the theoretical analysis and the error analysis are presented and a numerical example is given.     

Some Riemann boundary value problems in Clifford analysis

In this paper, we mainly study the R_m (m>0) Riemann boundary value problems for functions with values in a Clifford algebra C?(V_{3, 3}). We prove a generalized Liouville‐type theorem for harmonic functions and biharmonic functions by combining the growth behaviour estimates with the series expansions for k‐monogenic functions. We obtain the result under only one growth condition at infinity by using the integral representation formulas for harmonic functions and biharmonic functions. By using the Plemelj formula and the integral representation formulas, a more generalized Liouville theorem for harmonic functions and biharmonic functions are presented. Combining the Plemelj formula and the integral representation formulas with the above generalized Liouville theorem, we prove that the R_m (m>0) Riemann boundary value problems for monogenic functions, harmonic functions and biharmonic functions are solvable. Explicit representation formulas of the solutions are given. Copyright © 2009 John Wiley & Sons, Ltd.     

广义双正则函数向量的非线性边值问题

利用Schauder不动点定理和积分方程的方法，讨论了实Clifford分析中广义双正则函数向量的非线性边值问题解的存在性及其解的积分表达式。