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In the paper, we solve two Stefan problems. The first problem recovers an unknown moving boundary by specifying the Cauchy boundary conditions on a fixed left-end. The second problem finds a time-d...

When a nonlinear heat equation is subjected to nonlocal boundary conditions, the difficulty might arise because time-dependent integral conditions present in the problem. To overcome this difficult...

For a third-order nonlinear boundary value problem (BVP), we develop two novel methods to find the solutions, satisfying boundary conditions automatically. A boundary shape function (BSF) is created to automatically satisfy the boundary conditions, which is then employed to develop new numerical algorithms by adopting two different roles of the free function in the BSF. In the first type algorithm, we let the BSF be the solution of the BVP and the free function be a new variable. In doing so, the nonlinear BVP is certainly and exactly transformed to an initial value problem for the new variable with its terminal values as unknown parameters, whereas the initial conditions are given. In the second type algorithm, let the free functions be a set of complete basis functions and the corresponding boundary shape functions be the new bases. Since the solution already satisfies the boundary conditions automatically, we can apply a simple collocation technique inside the domain to determine the expansion coefficients and then the solution is obtained. For the general higher-order boundary conditions, the BSF method (BSFM) can easily and quickly find a very accurate solution. Resorting on the BSFM, the existence of solution is proved, under the Lipschitz condition for the ordinary differential equation system of the new variable. Numerical examples, including the singularly perturbed ones, confirm the high performance of the BSF-based numerical algorithms.

For a second-order nonlinear singularly perturbed boundary value problem (SPBVP), we develop two novel algorithms to find the solution, which automatically satisfies the Robin boundary conditions. For the highly singular nonlinear SPBVP the Robin boundary functions are hard to be fulfilled exactly. In the paper we first introduce the new idea of boundary shape function (BSF), whose existence is proven and it can automatically satisfy the Robin boundary conditions. In the BSF, there exists a free function, which leaves us a chance to develop new algorithms by adopting two different roles of the free function. In the first type algorithm we let the free functions be the exponential type bases endowed with fractional powers, which not only satisfy the Robin boundary conditions automatically, but also can capture the singular behavior to find accurate numerical solution by a simple collocation technique. In the second type algorithm we let the BSF be solution and the free function be another variable, such that we can transform the boundary value problem to an initial value problem (IVP) for the new variable, which can quickly find accurate solution for the nonlinear SPBVP through a few iterations.

In the article, we solve the inverse problems to recover unknown space-time dependent functions of heat conductivity and heat source for a nonlinear convective-diffusive equation, without needing o...

We solve a twofold ill-posed inverse problem of a nonlinear convection-diffusion equation, endowed with unknown source term and unknown moving boundary. When the solution is expanded through the superposition of a family of modified homogenization functions in a reduced domain, the unknown space-time-dependent source function can be obtained by solving a small scale linear system, with the help from the over-specified data of left flux, the measured final time data and some inner data, which are very saving. Simultaneously, the unknown moving boundary can be detected by solving nonlinear equations using the Newton iterative method. Numerical examples are given to confirm that the superposition of homogenization functions method (SHFM) can recover the unknown source function and moving boundary very well.

In the article, we solve an extremal long time span backward heat conduction problem (BHCP) of a 3D nonhomogeneous heat conduction equation with nonhomogeneous boundary conditions in a cubo...

This study presents a hybrid method of inverse method and three-dimensional computational fluid dynamics (CFD) commercial software along with experimental data and various flow models to investigate the natural convection heat transfer and fluid flow characteristics of vertical annular finned tube heat exchangers in a small chimney for various fin diameters. First, the inverse method of the finite difference method along with experimental temperature data is used to estimate the heat transfer coefficient on the fins. Afterwards, the CFD along with various flow models and the inverse results of the heat transfer coefficient is used to determine the air temperature and velocity profiles, the fin surface temperature and the heat transfer coefficient on the fins. More accurate results are obtained when the resulting heat transfer coefficient and fin temperature are as close as possible to the inverse results and the experimental temperature measurements, respectively. The results show that the zero-equation turbulence model is more suitable for this problem than other flow models, especially for larger fin spacing. The more grid points are not necessarily able to obtain more accurate results. The choice of the appropriate flow model and the number of grid points may depend on experimental temperature data and reliable heat transfer coefficient. Finally, it is found that the heat transfer coefficient increases with decreasing fin diameter and increasing fin spacing. Fin efficiency decreases with increasing fin diameter and fin spacing. The optimum fin spacing can vary with the D/d0 value. Also, the proposed correlation between the Nusselt number and the Rayleigh number matches the inverse and numerical results obtained.

This study presents a hybrid method of inverse method and three-dimensional computational fluid dynamics (CFD) commercial software along with various flow models and experimental data to investigate natural convection in a heated cavity with a horizontal fin on a hot sidewall. The effect of fin position and initial temperature on the results obtained is investigated. First, the inverse method of the finite difference method along with the experimental temperature data is used to estimate the heat transfer coefficient on the heating fin. Afterwards, the CFD along with various flow models and the inverse results of the obtained heat transfer coefficient is used to determine the air temperature and velocity profiles, the fin surface temperature and the heat transfer coefficient on the fins. More accurate results and appropriate flow model and grid points can be obtained if the resulting heat transfer coefficient and fin temperature are as close as possible to the inverse results and the experimental temperature measurements, respectively. The results show that the zero-equation turbulence model is more suitable for this problem than other flow models. More grid points may not necessarily get more accurate results. The present results are new findings and are different from previous 2D results. The heat transfer coefficients on the upper and lower surfaces of the fin can be different and vary with the position of the fin. The average heat transfer coefficient at the 35 mm fin position is 1.70 times that of the 85 mm fin position. An interesting finding is that the vortices at upper two corners of the cavity and near the upper surface of the fin edge vary with fin position. This means that the influence of fin position on the flow field cannot be ignored. The proposed correlation between Nusselt number and Rayleigh number well matches the inverse and numerical results obtained.

The commercial software along with the inverse method and the experimental data is used to obtain the mixed convection heat transfer and fluid flow characteristics of the single-tube vertical annular finned tube heat exchanger with different tube diameters. Various flow models along with near-wall treatments are introduced to investigate their effects on the obtained numerical results. The inverse method of the finite difference method along with the experimental data is first applied to determine the fin temperature and the heat transfer coefficient for a smaller tube. Afterwards, the air temperature and velocity profiles, the fin temperature distribution and the heat transfer coefficient on the fins are determined by the commercial software along with various flow models. More accurate results are obtained if the resulting heat transfer coefficient and the fin temperature at the measurement locations are respectively close to the inverse results and the experimental data. In addition, the comparison between the obtained air velocity pattern and the existing experimental pattern is desirable. An important finding is that the choice of appropriate flow models, near-wall treatments, friction factor, y+ value and the number of grid points varies with air velocity and tube diameter to obtain more accurate results. The heat transfer coefficient increases with increasing air velocity and fin spacing. However, the friction factor increases as the fin spacing increases and the air velocity decreases. The two vortices are symmetrical in the wake region behind the tube and rotate in the opposite direction. Increasing the number of grid points may not necessarily obtain more accurate results. The proposed correlation of the heat transfer coefficient is closer to the obtained inverse and numerical results than the existing correlation. To our knowledge, few researchers have used the similar method to investigate this problem in the open literature.

In this paper, we develop the high-order Lie group scheme to estimate a real-time external force exerted on nonlinear system. For estimating a real-time data, the equations of motion and the supple...

Microspeakers are key components of mobile devices in current consumer products; moreover, the user experience of multimedia products plays an essential role. The objective of this study was to investigate the acoustic performance of microspeakers in different enclosure parameters for mobile devices. Two methodologies are performed. The first model is an electro-mechano-acoustical model (EMA) of microspeaker along with distribution model of enclosure effect; the 2nd model, the details of diaphragms are taken into account in calculating the distribution models of microspeaker with enclosure parameters. They were combined and applied to determine microspeaker performance levels for mobile devices. Two methodologies show acceptable agreement with experimental results. The distribution model can well simulate the high frequency response than lumped model of microspeaker. The results can predict microspeaker system performance before industrial design and mechanism placement processes are executed.

This study applies three-dimensional computational fluid dynamics (CFD) commercial software along with the inverse method and experimental data to determine the mixed convection heat transfer and fluid flow characteristics of a plate-fin heat sink in a wind tunnel. The inverse method of the finite difference method along with the experimental temperature data is applied to determine the unknown heat transfer coefficient on the fin. Commercial software combined with various flow models is used to obtain air temperature and velocity profiles, heat transfer coefficient on fins, fin surface temperature and pressure drop. More accurate heat transfer and fluid flow characteristics can be obtained by the appropriate flow model and the number of grid points, if the resulting heat transfer coefficient and the fin temperature at each measurement location are close to the inverse results of the heat transfer coefficient and the experimental temperature data, respectively. The interesting finding is that the results obtained by the RNG k-e turbulence model are more accurate than those by the laminar flow model. FLUENT 4 has better accuracy than FLUENT 15 along with standard wall functions and enhanced wall treatment. In addition, the total number of grid points needs to be increased with increasing air velocity and fin spacing. The dimensionless wall distance can vary with air velocity. The pressure drop has a large variation in the specific range of the fin spacing, and the secondary vortices can be found at both corners of the wind tunnel. It is worth mentioning that the strength of the secondary flow decreases with decreasing fin spacing. The effect of the flow model, near-wall treatment, FLUENT version and grid points on the results obtained cannot be ignored. To our knowledge, few researchers have used similar methods to investigate this problem in the open literature. The two proposed correlations are closer to the obtained inverse and numerical results than the existing results.

This study uses three-dimensional computational fluid dynamics (CFD) commercial package along with the inverse method and experimental data to investigate the natural convection heat transfer and fluid flow characteristics of the single-tube vertical annular finned tube heat exchanger for various values of fin spacing and tube diameter. The air temperature and velocity distributions, the fin surface temperature and the heat transfer coefficient on the fins are determined by the FLUENT along with various flow models in the box with the upward opening. The inverse method along with the finite difference method and the experimental temperature data is applied to determine the fin temperature and heat transfer coefficient for the smaller tube. More accurate results are obtained by the appropriate flow model, if the heat transfer coefficient obtained is close to the inverse results and matches existing correlations. The fin temperature obtained at the selected measurement locations coincides with the experimental data. The results show that the zero-equation turbulence model is more suitable for this problem than laminar and RNG k-e turbulence models. The grid independence assumption cannot be appropriate. The proposed correlations between the Nusselt number and the Rayleigh number are close to FLUENT and inverse results.

The main purpose of this study is to use the three-dimensional commercial package along with experimental data and the inverse result of the heat transfer coefficient to investigate the effect of the computational domain boundary set on the natural convection heat transfer and fluid flow characteristics of single-tube vertical annular fin and tube heat exchanger in a box with an upward opening. Various flow models are introduced. The air temperature and velocity profiles, fin temperature and heat transfer coefficient on the fins are determined by the commercial package. More accurate numerical results and an appropriate flow model can be obtained if the resulting heat transfer coefficient and the fin temperature are close to the inverse result and experimental fin temperature data, respectively. However, differences in boundary condition set at the side walls of the box can affect air velocity and temperature profiles. The choice of the appropriate flow model can also be related to the boundary set of the computational domain. Thus, the suitability of the boundary condition set must be checked at the side walls of the box.

In this study, we address a new and simple non-iterative method to solve Cauchy problems of non-linear evolution equations without initial data. To start with, these ill-posed problems are analysed by utilizing a semi-discretization numerical scheme. Then, the resulting ordinary differential equations at the discretized times are numerically integrated towards the spatial direction by the group-preserving scheme (GPS). After that, we apply a two-stage GPS to integrate the semi-discretized equations. We reveal that the accuracy and stability of the new approach is very good from several numerical experiments even under a large random noisy effect and a very large time span.

This study uses three-dimensional computational fluid dynamics commercial package along with experimental data and various flow models to investigate the natural convection heat transfer and fluid flow characteristics of a single-tube vertical plate fin and tube heat exchangers for various values of fin spacing and tube diameter. Temperature and velocity distributions of air between the two fins, fin temperature and heat transfer coefficient on the fins are determined using FLUENT along with various flow models. The inverse method in conjunction with the finite difference method and the experimental temperature data is applied to determine the fin temperature and heat transfer coefficient for the smaller tube. More accurate results can be obtained, if the heat transfer coefficient obtained is close to the inverse results and matches existing correlations. The numerical results of the fin temperature at the selected measurement locations also coincide with the experimental temperature data. The results show that RNG k–e turbulence model is more suitable for this problem than laminar flow model. Three proposed new correlations between the Nusselt number and the Rayleigh number are in good agreement with the inverse results and numerical results obtained.

For the recovery of unknown external force in the inverse vibration problems, we first transform the linear ordinary differential equation of motion into a linear parabolic-type partial differential equation and then use the Green second identity to derive a boundary integral equation in terms of the adjoint Trefftz test functions. It results in a weak-form integral method to recover the external force for nonlinear structures, of which only the data of displacements and two velocities at initial and final times are specified. The advantages gained in this transformation to a weak-form integral formulation can be seen when we test some nonlinear inverse vibration problems within a long time span and under a large noise.

In this paper, the modified polynomial expansion method is developed to solve problems of identifying the time dependent heat source, in which an inverse problem is encountered. Aimed at this problem, the variation of variables is adopted to eliminate the unknown heat source and obtain a six-line boundary value problem. As compared with the conventional four-line boundary value problem, the six-line boundary value problem is quite hard to be dealt with. After the unknown non-homogeneous term being eliminated, the polynomial expansion method is introduced to discretize the time and space fields, respectively. Then, the distribution of temperature is expressed as a linear superposition of polynomial functions. After that, a characteristic length concept is adopted to resolve the ill-posed matrix problems arising in those conventional polynomial expansion methods. The desired heat source function can be obtained by putting the solution of the six-line boundary value problem into differential operations. Several numerical experiments with designed examples are included to validate the accuracy and effectiveness of the proposed approach.

In this article, the solutions for the Helmholtz equation for forward problems with high wave number and ill-posed inverse problems using the multiple scales Trefftz collocation method are investigated. The resulting linear algebraic systems for these problems are ill-posed and therefore require special treatments. The equilibrated matrix concept is adopted to determine the scales and to construct an equivalent linear algebraic problem with a leading matrix less ill-posed such that standard solver like the conjugate gradient method (CGM) can be adopted. Five examples, including two forward problems with the high wave number and three inverse Cauchy problems, are given to show the validity for the approach. Results show that the equilibrated matrix concept can yield a less ill-posed leading matrix such that the conventional linear algebraic solver like CGM can be successfully adopted. This approach has a very good noise resistance.

This study applies the Laplace transform method in conjunction with the sequential-in-time concept and experimental temperature data to determine the unknown surface temperature and surface heat flux on a hot surface of the test material. The advantage of the present inverse scheme is that the unknown surface temperature and surface heat flux on a hot surface are determined without any iterative process and the initial guess. More importantly, their functional forms do not need to be assumed in advance. However, the whole time domain may be divided into several sub-time intervals. Later, the appropriate polynomial function in each sub-time interval is selected to fit the experimental temperature data at each measurement location. The results show that the present results are not very sensitive to the measurement location. In order to validate the accuracy of the present method, two experimental examples and an analytical example are illustrated. The comparison between the present results, exact re...

We consider an inverse problem of a nonlinear heat conduction equation for recovering unknown space-dependent heat source and initial condition under Cauchy-type boundary conditions, which is known as a sideways heat equation. With the aid of two extra measurements of temperature and heat flux which are being polluted by noisy disturbances, we can develop a Lie-group differential algebraic equations (LGDAE) method to solve the resulting differential algebraic equations, and to quickly recover the unknown heat source and initial condition simultaneously. Also, we provide a simple LGDAE method, without needing extra measurement of heat flux, to recover the above two unknown functions. The estimated results are quite promising and robust enough against large random noise.

In this paper, we combine a novel finite difference method (FDM) with a manifold-based exponentially convergent algorithm (MBECA) to solve a nonlinear elliptic boundary-value problem defined in an arbitrary plane domain. It is very difficult to solve nonlinear and geometric complexity problems by conventional FDM. To overcome these problems, the concepts of internal residual and boundary residual in a fictitious rectangular domain are introduced. Besides, by adding a fictitious time coordinate, we avoid the need to treat complex boundary conditions but only use shape functions. Also, it is not necessary to solve an inverse matrix of algebraic equations when by using the MBECA. Moreover, in order to increase the numerical stability of the MBECA, we introduce a group-preserving scheme (GPS) to address fictitious time integration. Given the cone structure of the GPS and MBECA and their manifold properties, we can preserve the manifold path on the cone structure by a weighting factor such that the MBECA must also exhibit a cone construction, Lie algebra and group properties at each fictitious time step. Finally, the accuracy and convergence behaviour of this present method are demonstrated in several examples.

This article is aimed to reconstruct a time-dependent heat source for a one-dimensional heat conduction equation. The extra measurement data are used to transform the original equation into a homogeneous equation with three-point boundary conditions. Then the modified polynomial expansion method is developed to deal with the resulting three-point boundary-value problem. By considering the characteristic length, the modified polynomial expansion method can obtain a convergent series solution and improve the stability of the algorithm. The accuracy and efficiency of the present method are validated by comparing the estimating results with those of designed examples even under noisy measurement data.

Here, we develop a multi-scaling Trefftz method (MSTM) for the Laplace equation associated with the group-preserving scheme (GPS) to describe nonlinear sloshing behavior of the incompressible, non-viscous, and irrotational fluid. Chen et al. [29] proposed that the characteristic length of the Trefftz method and the concept of controlled volume could be used to overcome numerical errors and dissipation in the simulation of the sloshing problem. However, the nonlinear dependence of the characteristic length on initial conditions was neglected in the numerical development. In addition, this study presents a numerical method with automatically adaptive computational steps for describing the nonlinear sloshing behavior as well as for satisfying the conservation of mass at each time step. The method developed here presents a simple and stable way to cope with the nonlinear sloshing problem.

This paper develops a modified Trefftz method (MTM) associated with the group-preserving scheme (GPS) to deal with the two-dimensional non-linear sloshing problem. For non-linear sloshing phenomena, the conventional numerical method may encounter numerical instabilities and numerical dissipation, as the coefficient matrix obtained from the conventional numerical method is ill-posed, and the free surface exhibits non-linear kinematics. To overcome these problems, we introduce the concept of controlled volume into the numerical procedure to prevent the elevation from vanishing, and we take the characteristic length into account to maintain the stability of the new numerical method. Moreover, the use of the GPS eliminates the need to use second-order derivatives, thus increasing the numerical efficiency, and weighting factors can be used to observe the vanishing of the velocity potential. Comparisons of results of the present study with those in the literature show that the numerical results are better than those obtained using previous methods. The method developed here is a simple and stable way to cope with the non-linear sloshing problem.

In this paper, we propose a semi‐analytical method to deal with the backward heat conduction problem due to a quasi‐boundary idea. First of all, the Fourier series expansion technique is used to calculate the temperature field u(x, t) at any time t < T. Second, we consider a direct regularization by adding the term αu(x, 0) into the final time condition to obtain a second kind Fredholm integral equation for u(x, 0). The termwise separable property of the kernel function allows us to transform the backward problem into a two‐point boundary value problem and therefore, a closed‐form solution is derived. The uniform convergence and error estimation of the regularized solution uα (x, t) are provided and a tactic to choose the regularization parameter is suggested. When several numerical examples are amenable, we discover that the present approach can retrieve all the past data very well and is robust even for seriously noised final data.

In this article, we propose a backward group preserving scheme (BGPS) on the advection-dispersion equation (ADE) for tackling of the contamination problems. The BGPS has been successfully applied on the backward heat conduction problems as well as the backward in time Burgers equation, but it has never been applied to solve the ADE. The BGPS is able to recover the spatial distribution of groundwater contaminant concentration in this work. Several numerical examples are worked out, and we show, based on those numerical examples, that the BGPS is applicable to the ADE and the method can also handle the ADE with piecewise constant dispersion coefficients. When a steep gradient is appeared in the solution, several steps of the BGPS can be used to retrieve the desired initial data and its result is better than the marching-jury backward beam equation (MJBBE) method as far as our examples are concerned. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

We employ a quasi‐boundary regularization to construct a two‐point boundary value problem for multi‐dimensional backward heat conduction equations. The multidimensional backward heat conduction problem (BHCP) is renowned as severely ill‐posed because the solution does not fullly depend on the data. In order to numerically tackle the multi‐dimensional BHCP, we propose a Lie‐group shooting method (LGSM) in the time direction to find the unknown initial conditions. The pivot point is based on the establishment of a one‐step Lie group element G(T) and the construction of a generalized mid‐point Lie group element G(r). Then, by imposing G(T) = G(r) we can search for the missing initial conditions through a minimum discrepancy to the real targets by the numerical ones, in terms of the weighting factor r ? (0, 1). When numerical examples are tested, we find that the LGSM is applicable to the BHCP. Even with noisy final data, the LGSM is also robust against disturbance.

In this paper, the Laplace problems are solved by using the modified Trefftz method. For discontinuous boundary problem, singular problem, and degenerate scale problem, the conventional Trefftz method encounters the numerical instability owing to rank deficiency and high-order T-complete functions. To overcome these problems, the characteristic length of problem domain and high-order T-complete functions is introduced to obtain a modified Trefftz method, equipping with a characteristic length factor to make sure that this method is stable. Besides, the high-order T-complete functions can be clearly described for the discontinuous boundary conditions. Comparing with the solutions in the previous literature, the present method is powerful even for the problem with complex boundary and with adding random noise on the boundary data. It is also successfully solving the degenerate scale problem, resulting to a highly accurate result never seen before.

For the Falkner–Skan equation, including the Blasius equation as a special case, we develop a new numerical technique, transforming the governing equation into a non-linear second-order boundary value problem by a new transformation technique, and then solve it by the Lie-group shooting method. The second-order ordinary differential equation is singular, which is, however, much saving computational cost than the original third-order equation defined in a semi-infinite range. In order to overcome the singularity we consider a perturbed equation. The newly developed Lie-group shooting method allows us to search a missing initial slope at the left-end in a compact space of t ∈ [ 0 , 1 ] , and moreover, the initial slope can be expressed as a closed-form function of r ∈ ( 0 , 1 ) , where the best r is determined by matching the right-end boundary condition. All that makes the new method much superior than the conventional shooting method used in the boundary layer equation under imposed boundary conditions. When the initial slope is available we can apply the fourth-order Runge–Kutta method to calculate the solution, which is highly accurate. The present method is very effective for searching the multiple-solutions under very complex boundary conditions of suction or injection, and also allowing the motion of plate.

A quasi-boundary regularization leads to a two-point boundary value problem of the backward heat conduction equation. The ill-posed problem is analyzed by using the semi-discretization numerical schemes. Then the resulting ordinary differential equations in the discretized space are numerically integrated towards the time direction by the Lie-group shooting method to find the unknown initial conditions. The key point is based on the erection of a one-step Lie group element G ( T ) and the formation of a generalized mid-point Lie group element G ( r ). Then, by imposing G ( T ) = G ( r ) we can search for the missing initial conditions through a minimum discrepancy of the targets in terms of the weighting factor r ∈ ( 0 , 1 ) . Several numerical examples were worked out to persuade that this novel approach has good efficiency and accuracy. Although the final temperature is almost undetectable and/or is disturbed by large noise, the Lie group shooting method is stable to recover the initial temperature very well.

The first step in investigation the dynamics of a continuous time system described by ordinary differential equations is to integrate them to obtain trajectories. In this paper, we convert the group-preserving scheme (GPS) developed by Liu [International Journal of Non-Linear Mechanics 36 (2001) 1047–1068] to a time step-size adaptive scheme, x l + 1 = x l + h f ( x l , t l ) , where x ∈ R n is the system variables we are concerned with, and f ( x , t ) ∈ R n is a time-varying vector field. The scheme has the form similar to the Euler scheme, x l + 1 = x l + Δ t f ( x l , t l ) , but our step-size h is adaptive automatically. Very interestingly, the ratio h / Δ t , which we call the adaptive factor, can forecast the appearance of chaos if the considered dynamical system becomes chaotical. The numerical examples of the Duffing equation, the Lorenz equation and the Rossler equation, which may exhibit chaotic behaviors under certain parameters values, are used to demonstrate these phenomena. Two other non-chaotic examples are included to compare the performance of the GPS and the adaptive one.

The steady-state Burgers equation with high Reynolds number is a singularly perturbed boundary value problem. In order to depress the singularity we consider a coordinate transformation from the z-domain to the t-domain. Then we construct a very effective Lie-group shooting method to search a missing initial condition of slope through a weighting factor r ∈(0,1). Furthermore, a closed-form formula is derived to calculate the unknown slope in terms of r in a more refined range identified. Numerical examples were examined to show that the new approach has high efficiency and high accuracy.

In this paper, we propose a Lie-group shooting method to tackle two famous boundary layer equations in fluid mechanics, namely, the Falkner-Skan and the Blasius equations. We can employ this method to find unknown initial conditions. The pivotal point is based on the erection of a one-step Lie group element G(T) and the formation of a generalized mid-point Lie group element G(r). Then, by imposing G(T) = G(r) we can seek the missing initial conditions through a minimum discrepancy of the target in terms of the weighting factor r ∊ (0,1). It is the first time that we can apply the Lie-group shooting method to solve the boundary layer equations. Numerical examples are worked out to persuade that this novel approach has good efficiency and accuracy with a fast convergence speed by searching r with the minimum norm to fit two targets.

In this paper, a simplified and direct computation method formulated by the fixed coordinate system and relative meridian concept in conjunction with vector algebra is developed to deal with the classical problems of celestial navigation. It is found that the proposed approach, the Simultaneous Equal-altitude Equation Method (SEEM), can directly calculate the Astronomical Vessel Position (AVP) without an additional graphical procedure. The SEEM is not only simpler than the matrix method but is also more straightforward than the Spherical Triangle Method (STM). Due to tedious computation procedures existing in the commonly used methods for determining the AVP, a set of optimal computation procedures for the STM is also suggested. In addition, aimed at drawbacks of the intercept method, an improved approach with a new computation procedure is also presented to plot the celestial line of position without the intercept. The improved approach with iteration scheme is used to solve the AVP and validate the SEEM successfully. Methods of solving AVP problems are also discussed in detail. Finally, a benchmark example is included to demonstrate these proposed methods.

In this paper, a new asymmetric indirect Trefftz method (AITM) has been developed to solve free-vibration problems. The proposed method is categorized into a regular type of boundary element methods (BEMs) such that no singular or hypersingular integration is necessary. However, like other regular BEMs the proposed approach encounters the numerical instability as the number of elements increases. To deal with such an ill-posed behavior, Tikhonov's regularization method in conjunction with the generalized singular-value decomposition (GSVD) is adopted. It is proved that the degeneracy of the proposed indirect Trefftz method has the same mathematical structure as the direct Trefftz method. Thus, no special effort should be paid in programming. Besides, such an equivalency indicates that the current method does not have spurious eigensolutions. Furthermore, the proposed approach can easily treat a multiply connected domain of genus 1 as shown in Fig. 1. Due to its indirect nature, the present approach can also represent the mode shapes within its own mathematical formulations. Several numerical examples are given to show the validity of the proposed approach.

In this paper, a novel approach using a Great Circle Equation (GCE) formulated by vector algebra is proposed to solve the problems of Great Circle Sailings (GCS). It is found that Great Circle Equation Method (GCEM) can calculate the latitude and longitude of the waypoints more effectively than conventional approaches. The methods of solving the waypoints of GCS are discussed and a technique using minimum error propagation in every step of the calculating procedure is suggested. Comparisons of the GCEM and the conventional computation approach are also conducted for further validation. Numerical results show that the GCEM is simpler and can solve the problems directly without requiring judgments from the solver.

In this paper, the false degeneracy of the Helmholtz boundary integral equations is examined. A new theory to explain the false degeneracy of the Helmholtz boundary integral equations is given. In this proposed theory, a false degeneracy of the boundary integral equation is explained as finding a non-trivial source distribution such that it results in trivial field quantities inside the domain interested and non-trivial field quantities for its counter part, i.e., outside the domain interested. It is clearly explained that such a false degeneracy is independent of prescribed boundary conditions but dependent on the integral equation one selects. Moreover, the false degeneracy of the integral equation for the interested domain relates to the eigenproblem for its counter part. Under such a unified theory, the fictitious eigenvalue, spurious eigenvalue and pseudo-fictitious eigenvalue can be explained in a simple mathematical frame. It is concluded from our theoretical analysis that a multiply connected domain results in the pseudo-fictitious eigenvalue even the complex-valued formulations are used. In order to eliminate various kinds of false degeneracy, two methods are employed according to the previous research. A unified view of these two methods is provided such that they can be thought to be equivalent from mathematical point of view. Several numerical examples are given to show the validity of current approach.

In this paper, the dual boundary element method is used in conjunction with the domain partition to solve the vibration problem for a rod subjected to a time harmonic loading. It is shown that when the constraint equations are all included in the solver and the real part singular or real part hypersingular equation is used alone, the spurious resonance that may arise can be eliminated by using the singular value decomposition method. Even when the complex singular or hypersingular equation is used, if the structure is complicated or heterogeneous, domain partitioning and a subsequent parallel treatment become necessary, this gives rise to both spurious and “pseudo‐spurious” resonance. Three iteration schemes that are often used in parallel computation are shown to be inadequate here, and a new “mixed method” approach is proposed. Three benchmark examples are then used to show how the proposed approaches can be applied to eliminate various kinds of spurious and pseudo‐spurious resonance.