Determination of inner boundaries in modified Helmholtz inverse geometric problems using the method of fundamental solutions

In this paper, an inverse geometric problem for the modified Helmholtz equation arising in heat conduction in a fin, which consists of determining an unknown inner boundary (rigid inclusion or cavity) of an annular domain from a single pair of boundary Cauchy data is solved numerically using the method of fundamental solutions (MFS). A nonlinear minimisation of the objective function is regularised when noise is added into the input boundary data. The stability of numerical results is investigated for several test examples.     

The method of fundamental solutions applied to boundary value problems on the surface of a sphere

In this work we propose using the method of fundamental solutions (MFS) to solve boundary value problems for the Helmholtz–Beltrami equation on a sphere. We prove density and convergence results that justify the proposed MFS approximation. Several numerical examples are considered to illustrate the good performance of the method.     

Computation of the stresses in a moving reference system in a half-space due to a traversing time-varying concentrated load

An analytical approach is employed to investigate the transient and steady-state stresses in an isotropic, homogeneous half-space subjected to moving concentrated loads with subsonic speeds. Applying the Stokes–Helmholtz resolution to the Navier’s equation of motion for the half-space results in a system of wavetype partial differential equations. Based on the new moving coordinate system, a modified system of partial differential equations is obtained. Applying a concurrent two-sided and one-sided Laplace transformation, this system is modified to a system of ordinary differential equations, the solutions of which are obtained with respect to boundary conditions. The transformed transient stresses can be inverted by the Cagniard–de Hoop method. Special properties of Laplace transformation yield the steady-state stresses through an analytical approach. Numerical examples are presented to illustrate the methodology. Final results revealed the importance of considering the stresses related to the initial stages of the loading.     

On the quadratic integral equations and their applications

The class of quadratic integral equations contains, as a special case, numerous integral equations encountered in the theory of radiative transfer, the queuing theory, the kinetic theory of gases and the theory of neutron transport. As a pursuit of this, in the following pages, sufficient conditions are given for the existence of positive continuous solutions to some possibly singular quadratic integral equations. Meanwhile, we prove the existence of maximal and minimal solutions of our problems. The method used here depends on both Schauder and Schauder–Tychonoff fixed point principles. Unlike all previous contributions of the same type, no assumptions in terms of the measure of noncompactness were imposed on the nonlinearity of the given functions. As far as we know, the approach presented in this paper, in particular, the discussion of the existence of maximal and minimal solutions to the quadratic integral equations was never applied in the field of the quadratic integral equations and so is new.     

Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods

This paper presents new formulations of the boundary–domain integral equation (BDIE) and the boundary–domain integro-differential equation (BDIDE) methods for the numerical solution of the two-dimensional Helmholtz equation with variable coefficients. When the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or BDIDE. However, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

The method of fundamental solutions for oscillatory and porous buoyant flows

Accurate solutions of oscillatory Stokes flows in convection and convective flows in porous media are studied using the method of fundamental solutions (MFS). In the solution procedure, the flows are represented by a series of fundamental solutions where the intensities of these sources are determined by the collocation on the boundary data. The fundamental solutions are derived by transforming the governing equation into the product of harmonic and Helmholtz-type operators, which can be classified into three types depending on the oscillatory frequencies of temperature field. All the velocities, the pressure, and the stresses corresponding to the fundamental solutions are expressed explicitly in tensor forms for all the three cases. Three numerical examples were carried out to validate the proposed fundamental solutions and numerical schemes. Then, the method was also applied to study exterior flows around a sphere. In these studies, we derived the MFS formulas of drag forces. Numerical results were compared accurately with the analytical solutions, indicating the ability of the MFS for obtaining accurate solutions for problems with smooth boundary data. This study can also be treated as a preliminary research for nonlinear convective thermal flows if the particular solutions of the operators can be supplied, which are currently under investigations.     

Application of the Galerkin and Least-Squares Finite Element Methods in the solution of 3D Poisson and Helmholtz equations

This paper presents the numerical solution, by the Galerkin and Least Squares Finite Element Methods, of the three-dimensional Poisson and Helmholtz equations, representing heat diffusion in solids. For the two applications proposed, the analytical solutions found in the literature review were used to compare with the numerical solutions. The analysis of results was made from the L₂ norm (average error throughout the domain) and L_∞ norm (maximum error in the entire domain). The results of the two applications (Poisson and Helmholtz equations) are presented and discussed for testing of the efficiency of the methods.     

Numerical investigation on convergence of boundary knot method in the analysis of homogeneous Helmholtz, modified Helmholtz, and convection-diffusion problems

This paper concerns a numerical study of convergence properties of the boundary knot method (BKM) applied to the solution of 2D and 3D homogeneous Helmholtz, modified Helmholtz, and convection-diffusion problems. The BKM is a new boundary-type, meshfree radial function basis collocation technique. The method differentiates from the method of fundamental solutions (MFS) in that it does not need the controversial artificial boundary outside physical domain due to the use of non-singular general solutions instead of the singular fundamental solutions. The BKM is also generally applicable to a variety of inhomogeneous problems in conjunction with the dual reciprocity method (DRM). Therefore, when applied to inhomogeneous problems, the error of the DRM confounds the BKM accuracy in approximation of homogeneous solution, while the latter essentially distinguishes the BKM, MFS, and boundary element method. In order to avoid the interference of the DRM, this study focuses on the investigation of the convergence property of the BKM for homogeneous problems. The given numerical experiments reveal rapid convergence, high accuracy and efficiency, mathematical simplicity of the BKM.     

求解任意波数的三维Helmholtz方程

提出通过Adomian分解法求解任意波数的三维Helmholtz方程。通过Adomian分解法可以把三维Helmholtz微分方程转换成递归代数公式,并进一步把其边界条件转换成适用符号计算的简单代数公式。利用边界条件可以很容易得到方程的解析解表达式。Adomian分解法的主要特点在于计算简单快速,并且不需要进行线性化或离散化。最后通过数值计算以验证Adomian分解法求解任意波数下三维Helmholtz方程的有效性。数值计算结果表明：Adomian分解法的计算结果非常接近精确解,并且该方法在大波数情况下还具有良好的收敛性。     

A method of fundamental solutions for radially symmetric and axisymmetric backward heat conduction problems

We propose and investigate an application of the method of fundamental solutions (MFS) to the radially symmetric and axisymmetric backward heat conduction problem (BHCP) in a solid or hollow cylinder. In the BHCP, the initial temperature is to be determined from the temperature measurements at a later time. This is an inverse and ill-posed problem, and we employ and generalize the MFS regularization approach [B.T. Johansson and D. Lesnic, A method of fundamental solutions for transient heat conduction, Eng. Anal. Boundary Elements 32 (2008), pp. 697–703] for the time-dependent heat equation to obtain a stable and accurate numerical approximation with small computational cost.     

Exact one-way methods for acoustic waveguides

Exact one-way re-formulations of the Helmholtz equation are useful for waveguide problems, since the resulting equations can be efficiently solved as ‘initial’ value problems by range marching methods. Some numerical methods for these re-formulations are reviewed in this paper. This includes a switched method that avoids the singularities of the operators and the large range step methods that give exact solutions for range independent regions and allow large range steps for weakly range dependent regions. For waveguides with curved bottoms, a method based on a local orthogonal transformation is described. As an interesting application, the scattering problem of periodic waveguides is considered.     

An inverse scattering method for a 1-D dissipative Helmholtz equation

In this numerical method for simultaneous reconstruction of permittivity and conductivity in the one-dimensional inverse problem for the Helmholtz equation, a system of Riccati equations with trace formulae is derived, then used to propagate reflection and transmission data into the interior of an interval to obtain inhomogeneous material profiles. Numerical results are examined.     

一类分数阶种群扩散模型解的研究

从推广的Fick扩散定律出发研究了一类时间分数阶Fisher单种群扩散模型。利用变分迭代法求解了三种不同情况下的近似解,对结果进行了讨论和数值模拟。     

等参谱元方法的研究

1.引言 在求解偏微分方程的数值模拟中,主要有以下几种方法:有限差分法、有限元方法、有限分析法、谱方法等. 随着有限元方法成熟研究和谱方法[l]的飞速发展,Patera(1984年)提出了谱     

An adaptive method of fundamental solutions for solving the Laplace equation

In this paper, we propose a residual-type adaptive method of fundamental solutions (AMFS) for solving the two-dimensional Laplace equation. An error estimator is defined only on the boundary of the domain. Initial distributions of source points and collocation points are determined by using approaches proposed in Chen et al. (2006). The adding, removing, and stopping strategies are designed so that the required accuracy can be satisfied within finite steps. Numerical experiments reveal that AMFS improves the accuracy of the MFS approximation obtained from uniformly distributed sources and collocation points, which makes the MFS more practical for non-harmonic and non-smooth boundary conditions. Moreover, it is shown that the error estimator becomes equidistributed after an adaptive iteration. A detailed comparison between AMFS and MFS using uniformly distributed points is also presented for each numerical example.     

Estimation of the Helmholtz–Kohlrausch effect for natural images

The Helmholtz–Kohlrausch (H‐K) effect is the phenomenon in which two color stimuli have the same luminance but different chroma in a certain hue, so the perceived brightness induced by the two stimuli are different. In expanding gamut, it is necessary to consider the H‐K effect. A quantification of the H‐K effect is required in order to evaluate and develop display devices for which the change of perceived brightness of gamut expansion must be considered. For quantification of the H‐K effect, prediction equations that can derive the equivalent luminance in a single color image have been proposed in previous studies. However, these equations have not been applied to natural images that are important. Therefore, the purpose of this study is to quantify the H‐K effect by deriving calculated values for natural images. For this purpose, first, we conducted the quantification of the H‐K effect in natural images by deriving the equivalent luminance as calculated values expanding the three prediction equations proposed in previous studies. Next, we carried out a subjective evaluation experiment by varying image's chroma and luminance. We then verified the effectiveness of the calculated values by comparing them with the result from the experiment.     

New complex solutions for some special nonlinear partial differential systems

In this present work, we explore new applications of direct algebraic method for some special nonlinear partial differential systems and equations. Then new types of complex travelling wave solutions are obtained to the Davey-Stewartson system, the coupled Higgs system and the perturbed nonlinear Schrodinger’s equation; the balance number of it is not a positive integer.     

A collocation method based on the influence matrix technique for Navier-Stokes problems in annular domains

A spectral collocation method is proposed for the solution of the time-dependent Navier-Stokes and energy equations of a Boussinesq fluid inside an annular cavity. The time integration is based on the Adams-Bashforth scheme and on the second order backward differentiation formula. The influence matrix technique results in the resolution of Helmholtz and Poisson equations with Dirichlet boundary conditions. The solutions are validated with respect to former spectral Tau-Chebyshev solutions. Preliminary results concern the simulation of axisymmetric flows submitted to the buoyancy force, to the rotation and to source-sink fluxes.     

Two-dimensional numerical wavelet homogenization for obtaining effective characteristics of composite materials

A new method of homogenization of elliptic differential equations is considered based on wavelet transformation and the finite element method for predicting the effective characteristics and analysis of averaged solutions of equations for composite materials with specified structure and properties of their components.     

Parallel solution of tridiagonal systems for the Poisson equation

A method is described to solve the systems of tridiagonal linear equations that result from discrete approximations of the Poisson or Helmholtz equation with either periodic, Dirichlet, Neumann, or shear-periodic boundary conditions. The problem is partitioned into a set of smaller Dirichlet problems which can be solved simultaneously on parallel or vector computers leaving a smaller tridiagonal system to be solved on one of the processors.