Convergent Neumann Series Research Articles

Solvability of a Nonlocal Problem for an Evolution Equation with a Superstable Semigroup

We study a linear nonlocal problem for an evolution equation in a Banach space. The standard semigroup approach is used but integral averaging over time is used instead of the traditional initial condition. It is assumed that the evolution semigroup associated with the abstract differential equation is superstable (quasinilpotent), i.e., has an infinite negative exponential type. A theorem about unique solvability of the posed nonlocal problem is proved. It is shown that the solution can be represented by a convergent Neumann series. Some corollaries are noted. The case in which the semigroup is nilpotent is treated separately. The class of examples of superstable semigroups that are of interest in mathematical physics is outlined.

Quasi-Monte Carlo method for solving Fredholm equations

Abstract A Monte Carlo method used for the estimation of convergent von Neumann series solutions of a Fredholm equation of second kind is considered. The sum z ( d ) ⁢ ( x ) {z^{(d)}(x)} of d initial terms of the von Neumann series estimating the solution z ⁢ ( x ) {z(x)} of the equation is represented as a d-dimensional integral over the unit cube H d {H_{d}} . This note presents three examples calculating z ( d ) ⁢ ( x ) {z^{(d)}(x)} for different kernels with norms ∥ K ∥ < 1 {\lVert K\rVert<1} . We found that z ( d ) ⁢ ( x ) {z^{(d)}(x)} calculated using a quasi-Monte Carlo (QMC) method converges significantly faster than the corresponding Monte Carlo (MC) estimates in the entire range of ∥ K ∥ {\lVert K\rVert} values. We also found that the average dimension d ^ {\hat{d}} of the integrand in all our examples is small, less than 3. We suggest that the average dimensions d ^ {\hat{d}} of our d-dimensional integrands are bounded as d → ∞ {d\to\infty} .

The Solvability of the Inverse Problem for the Evolution Equation with a Superstable Semigroup

For the evolution equation in a Banach space, the linear inverse source problem is studied. It is required to recover an unknown nonhomogeneous term by means of an additional nonlocal condition written in the form of a Riemann-Stieltjes integral. The main assumption is related to the superstability (quasinilpotency) of the evolution semigroup. More precisely, it is assumed that the evolutionary semigroup associated with the abstract differential equation has an infinite negative exponential type. Without other restrictions, a theorem on the solvability of the inverse problem is obtained. It is shown that the solution can be represented by a convergent Neumann series. Exact conditions under which an infinite series becomes a finite sum are found. Here, the algorithm for calculating the solution becomes finite. Model examples are considered, including an important example of the inverse problem with final overdetermination. The above results can be applicated in special parts of mathematical physics related to the theory of elasticity and the linear transport theory. As is customary, our research takes place in the general case with a choice of the complex scalar field, but the main facts are also true in the real case. The created theory allows transfer to nonlocal problems for evolution equations, when instead of the traditional initial condition special time averaging is used to find the solution.

Enhancing the convergence of multipole expansions at intermediate frequency

In this work, we examine near-field acoustic wave scattering from media having a spatial variation in compressibility contrast. Typically, the scattered acoustic pressure can be expressed as a convergent Neumann series when the compressibility contrast is relatively small. However as the magnitude of the compressibility contrast increases a resonant scattering condition ensues, yielding a divergent series. It has been shown that Padé Approximants method can be used in these cases, thereby extending the range of utility of the Neumann series solution. The proposed research explores hybrid methods that combine multipole-methods, and Padé Approximant approaches to calculate the near-field scattered pressure. As part of this work, we will examine the low-frequency breakdown in the plane wave, and monopole-monopole expansion approaches as it affects the numerical stability of spatial translation operations.

A Dissipative Time Reversal Technique for Photoacoustic Tomography in a Cavity

We consider the inverse source problem arising in thermo- and photoacoustic tomography. It consists in reconstructing the initial pressure from the boundary measurements of the acoustic wave. Our goal is to extend versatile time reversal techniques to the case when the boundary of the domain is perfectly reflecting, effectively turning the domain into a reverberant cavity. Standard time reversal works only if the solution of the direct problem decays in time, which does not happen in the setup we consider. We thus propose a novel time reversal technique with a nonstandard boundary condition. The error induced by this time reversal technique satisfies the wave equation with a dissipative boundary condition and, therefore, decays in time. For larger measurement times, this method yields a close approximation; for smaller times, the first approximation can be iteratively refined, resulting in a convergent Neumann series for the approximation.

Multiwave imaging in an enclosure with variable wave speed

In this paper we consider the mathematical model of thermo- and photo-acoustic tomography for the recovery of the initial condition of a wave field from knowledge of its boundary values. Unlike the free-space setting, we consider the wave problem in a region enclosed by a surface where an impedance boundary condition is imposed. This condition models the presence of physical boundaries such as interfaces or acoustic mirrors which reflect some of the wave energy back into the enclosed domain. By recognizing that the inverse problem is equivalent to a statement of boundary observability, we use control operators to prove the unique and stable recovery of the initial wave profile from knowledge of boundary measurements. Since our proof is constructive, we explicitly derive a solvable equation for the unknown initial condition. This equation can be solved numerically using the conjugate gradient method. We also propose an alternative approach based on the stabilization of waves. This leads to an exponentially and uniformly convergent Neumann series reconstruction when the impedance coefficient is not identically zero. In both cases, if well-known geometrical conditions are satisfied, our approaches are naturally suited for variable wave speed and for measurements on a subset of the boundary.

Thermoacoustic tomography arising in brain imaging

We study the mathematical model of thermoacoustic and photoacoustic tomography when the sound speed has a jump across a smooth surface. This models the change of the sound speed in the skull when trying to image the human brain. We derive an explicit inversion formula in the form of a convergent Neumann series under the assumptions that all singularities from the support of the source reach the boundary.

Shielding effect of a thin strip

This paper deals with the evaluation of the shielding effect due to the induced current on a thin PEC strip, illuminated by a current line source. The extremely low frequency (ELF) solution is analytically determined, whereas the dynamic case is reformulated in terms of a rapidly convergent Neumann series. The convergence in the case of proximity of the current line can be improved by means of an incremental formulation with respect to the ELF case.

Asymptotic Analysis of an Elasticity Equation for a Finger-Like Hydraulic Fracture

We derive a novel integral equation relating the fluid pressure in a finger-like hydraulic fracture to the fracture width. By means of an asymptotic analysis in the small height to length ratio limit we are able to establish the action of the integral operator for receiving points that lie within three distinct regions: (1) an outer expansion region in which the dimensionless pressure is shown to be equal to the dimensionless width plus a small correction term that involves the second derivative of the width, which accounts for the nonlocal effects of the integral operator. The leading order term in this expansion is the classic local elasticity equation in the PKN model that is widely used in the oil and gas industry; (2) an inner expansion region close to the fracture tip within which the action of the elastic integral operator is shown to be the same as that of a finite Hilbert transform associated with a state of plane strain. This result will enable pressure singularities and stress intensity factors to be incorporated into analytic models of these finger-like fractures in order to model the effect of material toughness; (3) an intermediate region within which the action of the Fredholm integral operator of the first kind is reduced to a second kind operator in which the integral term appears as a small perturbation which is associated with a convergent Neumann series. These results are important for deriving analytic models of finger-like hydraulic fractures that are consistent with linear elastic fracture mechanics.

The Neumann series as a fundamental solution of the two-dimensional convection–diffusion equation with variable velocity

The fundamental solution of the two-dimensional convection–diffusion equation with variable coefficients and its adjoint equation are obtained in complex form in terms of the unknown density of two equivalent uniquely solvable Volterra integral equations of the second kind whose analytical solutions are given explicitly as convergent Neumann series. The Volterra integral equations are obtained by integrating the complex form of the original differential equations, without additional change of variables as proposed by previously authors. In the numerical examples, cases corresponding to non-self-adjoint operators are considered. As a validation, the proposed approach is used to derive the fundamental solution of the adjoint to the convection–diffusion equation with constant velocity. In this case, the series solution can be evaluated analytically. For more general velocity fields, the recursive terms of the series can be evaluated by symbolic computation or numerical integration.

Stress state of a rock mass in the vicinity of seam containing a working. Three-dimensional problem

Formulas are obtained to determine the stresses at any point of a rock mass. The formulas are derived through double integrals of an auxiliary function represented by a convergent Neumann series. Approximations of the kernels are presented by algebraic combinations of elementary functions.

Spectral radius properties for layer potentials associated with the elastostatics and hydrostatics equations in nonsmooth domains

By producing a L2 convergent Neumann series, we prove the invertibility of the elastostatics and hydrostatics boundary layer potentials on arbitrary Lipschitz domains with small Lipschitz character and 3D polyhedra with large dihedral angles.

Modeling space-time domain acoustic wave fields in media with attenuation: The symbolic manipulation approach

A method is presented for the determination of the space-time domain acoustic wave field, and in particular the space-time domain Green’s function, in homogeneous, isotropic, lossy equivalent fluid media that represent solids with a complicated viscoelastic behavior. The loss properties of the equivalent fluids are modeled with the aid of arbitrarily intricate compliance memory functions and, eventually, inertia memory functions. The presented integral transformation-type method consists of three steps. First, a temporal Laplace transformation and a horizontal spatial Fourier transformation are performed. Due to the application of the temporal Laplace transformation, causality of the acoustic wave field is automatically dealt with. Second, the resulting transform domain problem is solved using a convergent Neumann series solution. Analytical expressions for the terms of this Neumann series solution are obtained by means of a recurrence scheme that can ideally be evaluated with the aid of symbolic manipulation. Third, the transformation back to the space-time domain is performed analytically using the Cagniard–De Hoop method. No numerically imposed limitation of the bandwidth of the wave field quantities or the Green’s function shows up. In principle, the method offers the possibility of generating space-time domain results, both for the acoustic wave field and the Green’s function, for any time instant with any desired degree of accuracy. After deriving the general theory, numerical results for several lossy equivalent fluids with an almost constant-Q behavior, as shown by many types of rock, are presented.

ANALYSIS OF TIME-DOMAIN ACOUSTIC WAVEFIELDS IN HORIZONTALLY CONTINUOUSLY LAYERED CONFIGURATIONS BASED ON THE MODIFIED CAGNIARD METHOD

A combination of the Neumann series and the modified Cagniard method is used to derive a theoretically exact space-time domain solution for the 3-D acoustic wave propagation problem in horizontally continuously layered media. Firstly, integral transformations (among which is a one-sided temporal Laplace transformation) are applied to the space-time domain basic acoustic equations. Secondly, a system of transform domain integral equations is derived, which is solved using a convergent Neumann series. Finally, the individual terms of this series are transformed back to the space-time .domain using the modified Cagniard method. The space-time domain series turns out to be convergent for every continuously layered configuration and at any time instant. In contrast to the standard angular wave number-frequency domain analysis, in the present analysis, difficulties due to turning points, etc., are avoided.

Low Reynolds number deformation of compound drops in shear flow

AbstractThe method developed by Rallison and Acrivos (1978) to solve the deformation of a single drop due to exterior shear flow for a small Reynolds number is extended to the problem of the deformation of a compound drop. The result is found in terms of a Fredholm integral equation of the second kind, which is shown to have a unique solution and the solution is given by means of a uniformly convergent Neumann series.

An iterative solution to acoustic scattering by rigid bodies

The scattering problem is formulated as an integral equation over the scattering surface. A solution to the equation is given in a convergent Neumann series where the convergence depends on a parameter α. The boundary element method is used to obtain an approximation to the solution, and the resulting matrix equation is solved by an iterative technique. Results are given for several different geometries and frequencies. A comparison of solution time and accuracy to direct methods such as Gaussian elimination is presented, showing the iteration to be more economical in most cases of practical interest. Finally, there is a short discussion on determining the iteration parameter α and its effects on the rate of convergence.

On constructing the solution to the exterior Dirichlet problem for the Helmholtz equation

In this paper a method is given for constructing the solution to the exterior Dirichlet problem for the Helmholtz equation in three dimensions. This method is modeled after the procedure of Colton and Kleinman (Proc. Roy. Soc. Edin. 86A(1980) , 29–42) for solving the corresponding two-dimensional problem. The scattering problem is reformulated as an integral equation and it is shown that its solution can be represented as a convergent Neumann series for small values of the wave number. Comparisons are made between the present method and known results. Examples are given which illustrate the method.

Alternative to Padé technique for iterative solution of integral equations

It is well known that when the kernel of Fredholm integral equations of the second kind has the absolute magnitude of one or more eigenvalues greater than unity the Neumann series of such equations diverge. Pade technique is a practical way for constructing the correct solution from the divergent Neumann series. Here we propose an alternative method for the iterative solution of such equations, which relies on writing auxiliary integral equations with “weaker” kernels so that the auxiliary equations have a convergent Neumann series. Then the solution of the original equation is expressed in terms of the (convergent Neumann series) solution of the auxiliary equations. The kernel of the auxiliary equations can systematically be made weaker and the rate of convergence of the Neumann series of the auxiliary equations improved in order to make the method more useful. The method is illustrated numerically in certain cases.

The exterior Robin problem for the Helmholtz equation

A new method is given for solving the exterior Robin problem for the Helmholtz equation. The problem is reformulated as a new integral equation which is continuous as the field point approaches the boundary. It is shown that its solution can be represented as a convergent Neumann series for convex surfaces, for small values of the wave number. Examples are included which illustrate the method.

Integral equations and the interior Dirichlet potential problem

The two standard approaches for reformulating the interior Dirichlet potential problem as a boundary integral equation of the second kind are discussed. The integral equation derived from the representation of the solution as a double layer is shown to be more general than the one derived from Green's theorem. The boundary integral equation of the latter method, however, has definite analytical and numerical value. From it a new integral equation is derived whose solution can be represented as a convergent Neumann series and it is shown that the Green's function of the first kind can be obtained from it. An example is supplied to illustrate the method.