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 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.     

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.     

A method of fundamental solutions for two-dimensional heat conduction

We investigate an application of the method of fundamental solutions (MFS) to heat conduction in two-dimensional bodies, where the thermal diffusivity is piecewise constant. We extend the MFS proposed in Johansson and Lesnic [A method of fundamental solutions for transient heat conduction, Eng. Anal. Bound. Elem. 32 (2008), pp. 697–703] for one-dimensional heat conduction with the sources placed outside the space domain of interest, to the two-dimensional setting. Theoretical properties of the method, as well as numerical investigations, are included, showing that accurate results can be obtained efficiently with small computational cost.     

A new investigation into regularization techniques for the method of fundamental solutions

This study examines different regularization approaches to investigate the solution stability of the method of fundamental solutions (MFS). We compare three regularization methods in conjunction with two different regularization parameters to find the optimal stable MFS scheme. Meanwhile, we have investigated the relationship among the condition number, the effective condition number, and the MFS solution accuracy. Numerical results show that the damped singular value decomposition under the parameter choice of the generalized cross-validation performs the best in terms of the MFS stability analysis. We also find that the condition number is a superior criterion to the effective condition number.     

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.     

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.     

The method of particular solutions for solving nonlinear Poisson problems

Based on the recent development in the method of particular solutions, we re-exam three approaches using different basis functions for solving nonlinear Poisson problems. We further propose to simplify the solution procedure by removing the insolvency condition when the radial basis functions are augmented with high order polynomial basis functions. We also specify the deficiency of some of these methods and provide necessary remedy. The traditional Picard method is introduced to compare with the recent proposed methods using MATLAB optimization toolbox solver for solving nonlinear Poisson equations. Ranking on these three approaches are given based on the results of numerical experiment.     

A regularization method for a Cauchy problem of Laplace's equation in an annular domain

In this paper, we propose a new regularization method based on a finite-dimensional subspace generated from fundamental solutions for solving a Cauchy problem of Laplace's equation in an annular domain. Based on a conditional stability for the Cauchy problem of Laplace's equation, we obtain a convergence estimate under the suitable choice of a regularization parameter and an a-priori bound assumption on the solution. A numerical example is provided to show the effectiveness of the proposed method from both accuracy and stability.     

Variable-coefficient auxiliary equation method for exact solutions of non-linear evolution equations

In this paper, a variable-coefficient auxiliary equation method is proposed to seek more general exact solutions of non-linear evolution equations. Being concise and straightforward, this method is applied to the Kawahara equation, Sawada–Kotera equation and (2+1)-dimensional Korteweg–de Vries equations. As a result, many new and more general exact solutions are obtained including Jacobi elliptic, hyperbolic and trigonometric function solutions. It is shown that the proposed method provides a straightforward and effective method for non-linear evolution equations in mathematical physics.     

Laminar fluid flow and heat transfer in an internally corrugated tube by means of the method of fundamental solutions and radial basis functions

The paper shows application of the method of fundamental solutions in combination with the radial basis functions for analysis of fluid flow and heat transfer in an internally corrugated tube. Cross-section of such a tube is mathematically described by a cosine function and it can potentially represent a natural duct with internal corrugations, e.g. inside arteries. The boundary value problem is described by two partial differential equations (one for fluid flow problem and one for heat transfer problem) and appropriate boundary conditions. During solving this boundary value problem the average fluid velocity and average fluid temperature are calculated numerically. In the paper the Nusselt number and the product of friction factor and Reynolds number are presented for some selected geometrical parameters (the number and amplitude of corrugations). It is shown that for a given number of corrugations a minimal value of the product of friction factor and Reynolds number can be found. As it was expected the Nusselt number increases with increasing amplitude and number of corrugations.     

Conformal mapping for the efficient MFS solution of Dirichlet boundary value problems

In this work, we use conformal mapping to transform harmonic Dirichlet problems of Laplace’s equation which are defined in simply-connected domains into harmonic Dirichlet problems that are defined in the unit disk. We then solve the resulting harmonic Dirichlet problems efficiently using the method of fundamental solutions (MFS) in conjunction with fast fourier transforms (FFTs). This technique is extended to harmonic Dirichlet problems in doubly-connected domains which are now mapped onto annular domains. The solution of the resulting harmonic Dirichlet problems can be carried out equally efficiently using the MFS with FFTs. Several numerical examples are presented.      

Approximate periodic solutions for the Helmholtz-Duffing equation

Approximate periodic solutions for the Helmholtz-Duffing oscillator are obtained in this paper. He’s Energy Balance Method (HEBM) and He’s Frequency Amplitude Formulation (HFAF) are adopted as the solution methods. Oscillation natural frequencies are analytically analyzed. Error analysis is carried out and accuracy of the solution methods is evaluated.     

On fundamental solutions for multidimensional Helmholtz equation with three singular coefficients

The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with three singular coefficients, which could be expressed in terms of a confluent hypergeometric function of four variables. In addition, the order of the singularity is determined and the properties of the found fundamental solutions that are necessary for solving boundary value problems for degenerate elliptic equations of second order are found.     

Asymptotic behaviour of the solutions of second order functional differential equations

This paper presents integral criteria to determine the asymptotic behaviour of the solutions of second order nonlinear differential equations of the type y^″(x)+q(x)f(y(x))=0, with q(x)>0 and f(y) odd and positive for y>0, as x tends to +∞. It also compares them with the results obtained by Chanturia (1975) in [11] for the same problem.     

A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations

In this study, the decomposition method for solving the linear heat equation and nonlinear Burgers equation is implemented with appropriate initial conditions. The application of the method demonstrated that the partial solution in the x-direction requires more computational work when compared with the partial solution developed in the t-direction but the numerical solution in the x-direction are performed extremely well in terms of accuracy and efficiency.     

A Matrix Decomposition MFS Algorithm for Problems in Hollow Axisymmetric Domains

In this work we apply the Method of Fundamental Solutions (MFS) with fixed singularities and boundary collocation to certain axisymmetric harmonic and biharmonic problems. By exploiting the block circulant structure of the coefficient matrices appearing when the MFS is applied to such problems, we develop efficient matrix decomposition algorithms for their solution. The algorithms are tested on several examplesAMS SUBJECT CLASSIFICATION: Primary 65N12; 65N38; Secondary 65N15; 65T50;35J25      

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.     

Perturbation of solutions of ordinary linear homogeneous differential equations of the second order

In this paper we give a method for peturbation of solutions of linear homogeneous differential equation of the second order.