An integral equation in conformal geometry

Motivated by Carleman's proof of the isoperimetric inequality in the plane, we study the problem of finding a metric with zero scalar curvature maximizing the isoperimetric ratio among all zero scalar curvature metrics in a fixed conformal class on a compact manifold with boundary. We derive a criterion for the existence and make a related conjecture.     

The completed double layer boundary integral equation method for two-dimensional Stokes flow

Power and Miranda (1987) explained how integral equations ofthe second kind can be obtained for general exterior three-dimensionalStokes flows. They observed that, although the double layerrepresentation that leads to an integral equation of the secondkind coming from the jump property of its velocity field acrossthe density carrying surface can represent only those flow fieldsthat correspond to a force and torque free surface, the representationmay be completed by adding terms that give arbitrary total forceand torque in suitable linear combinations, precisely a Stokesletand a Rotlet located in the interior of the three-dimensionalparticles. Karrila and Kim (1989) called Power and Miranda'snew method the completed double layer boundary integral equationmethod, since it involves the idea of completing the deficientrange of the double layer operator. The main objective of thispaper is to extend Power and Miranda's completed method to theproblem of multiple cylinders in twodimensional bounded andunbounded domains. This extension is not trivial, owing to theunbounded behaviour at infinity of the fundamental solutionof the Stokes equation in two dimensions and the associatedparadoxes arising from this unbounded behaviour.     

Note on an integral equation occurring in stagnation point flow

Zusammenfassung Die Differentialgleichung der Staupunktströmung wird nach einer Methode vonWeyl behandelt, und es wird gezeigt, dass das Weylsche Verfahren für elektronische Rechenautomaten besonders geeignet ist.     

An integral equation method in conformal mapping

This paper describes an integral equation method for computing the conformal mapping of a given simply-connected domain onto the interior of the unit circle. Simple error estimates are available when the given domain possesses symmetry, and results are presented for a number of trial problems of this form.This work was supported by the National Science Foundation, under grant GP. 4213.On leave from Mathematics Division, National Physical Laboratory, Teddington, Middlesex, England.     

Boundary integral equation method in micropolar elasticity

This paper presents the integral representations of the displacement and rotation fields of the micropolar continuum, and the regularized integral representations of the traction and couple vector. Thus the boundary integral equations and the boundary integro-differential equations are derived. In addition, the formulation for crack problems is given by both the boundary integral equations and the boundary integro-differential equations.     

Meshless solution of two-dimensional incompressible flow problems using the radial basis integral equation method

The two-dimensional incompressible fluid flow problems governed by the velocity–vorticity formulation of the Navier–Stokes equations were solved using the radial basis integral (RBIE) equation method. The RBIE is a meshless method based on the multi-domain boundary element method with overlapping subdomains. It solves at each node for the potential and its spatial derivatives. This feature of the RBIE is advantageous in solving the velocity–vorticity formulation of the Navier–Stokes equations since the calculated velocity gradients can be used to compute the vorticity that is prescribed as a boundary condition to the vorticity transport equation. The accuracy of the numerical solution was examined by solving the test problem with known analytical solution. Two benchmark problems, i.e. the lid driven cavity flow and the thermally driven cavity flow were also solved. The numerical results obtained using the RBIE showed very good agreement with the benchmark solutions.     

Solutionof an integral equation by consistency method

We consider the problem to invert an integral equation obtained from integral geometry. We apply a consistencymethodsuggestedbytheauthortosolvetheproblemandfindarepresentationforthegeneratingdensityofcentrally symmetrical convex bodies. Finally, we check the representation.     

Galerkin method for the linear-radiator integral equation

The article examines the general Galerkin method scheme for the singular integral equation in the problem of radiation from a finite-thickness linear radiator. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 150–158.     

An iterative procedure for determining an unknown spacewise-dependent coefficient in a parabolic equation

An inverse problem for the determination of an unknown spacewise-dependent coefficient in a parabolic equation is considered. The problem is reformulated as a nonclassical parabolic equation along with the initial and boundary conditions. The iterative fixed point projection method is applied to solve the reformulated problem. The comparison analysis of proposed method with a least square method and some numerical examples are presented.     

System of nonlinear integral equations for the unknown functions in a functional-differential hyperbolic equation

We consider the inverse problem for a functional-differential equation in which the delay function and a function occurring in the source are unknown. The values of the solution and its derivative at x = 0 are given as additional information. We derive a system of nonlinear integral equations for the unknown functions. This system is used to prove a uniqueness theorem for the inverse problem.     

Uniqueness in an integral geometry problem and an inverse problem for the kinetic equation

In this paper, we discuss the uniqueness in an integral geometry problem along the straight lines in a strongly convex domain. Our problem is related with the problem of finding a Riemannian metric by the distances between all pairs of the boundary points. For the proof, the problem is reduced to an inverse source problem for a kinetic equation and then the uniqueness theorem is proved using the tools of Fourier analysis.     

Transient Bernoulli flow in multi-port fluid devices with arbitrary geometry

We present a method for the solution of transient flow in a multi-port fluid device with arbitrary geometry. The method is applicable to fluid devices where the fluid motion is primarily inviscid throughout the volume, but locally near a device port some accommodation to viscous flow is introduced. The internal flow is characterized by an array of purely geometrical factors between ports, essentially a set of generalized impedances; the state variables elicited are the average volume flow rates through the device ports. The method creates a set of coupled non-linear time-dependent ordinary differential equations. The solution to this set of equations is much faster, typically by orders of magnitude, than a single run of a transient CFD model. We demonstrate our method with a simple example; we show that the results of the method agree well with a full CFD calculation.     

Analytical formulation of meshless local integral equation method

In this paper, the exact forms of integrals in the meshless local boundary integral equation method are derived and implemented for elastostatic problems. A weak form for a set of governing equations with a unit test function or polynomial test functions is transformed into local integral equations. Each node has its own support domain and is surrounded by a local integral domain with different shapes of boundaries. The meshless approximation based on the radial basis function (RBF) is employed for the implementation of displacements. A completed set of closed forms of the local boundary integrals are obtained. As there are no numerical integrations to be carried out the computational time is significantly reduced. Three examples are presented to demonstrate the application of this approach in solid mechanics.     

Boundary integral equation method of higher computational accuracy

The boundary integral equation method presented in the paper features the following: (1) no singular kernels, strong or weak, are involved, and computationally no local “element” approximations are needed; (2) the integral equations are well conditioned, including the cases of bounded and multiply connected regions, and no iterative approximations are involved; (3) no post-solution differentiation is involved. These features provide for a higher computational efficiency. The method solves in full a number of engineering problems, and can be used for the stiffness matrix formulation in more complex situations.     

Collocation method for the natural boundary integral equation

In this work, we construct the Legendre wavelet and apply it to investigate the numerical solution of the natural boundary integral equation of the Laplace equation in the upper half-plane by the collocation method. In our algorithm the coefficient matrix of the linear algebraic system is sparse when the order of the matrix is large. Two test examples show that our algorithm yields very accurate results at less computational cost.     

On the integral equation method for buckling loads

An initial value method for the integral equation of the column is presented for determining the buckling load of columns. The differential equation of the column is reduced to a Fredholm integral equation. An initial value problem is derived for this integral equation, which is reduced to a set of ordinary differential equations with prescribed initial conditions in order to find the Fredholm resolvent. The singularities of the resolvent occur at the eigenvalues. Integration of the equations proceeds until the integrals become excessively large, indicating that a critical load has been reached. To check this method, numerical results are given for two examples, for which the critical load is well known. One is the Euler load of a simply supported beam, and the other case is the buckling load of a cantilever beam under its own weight. The advantage of this initial value method is that it can be applied easily to solve other nonlinear problems for which the critical loads are unknown. This approach will be illustrated in future papers.