Jiang-Ren Chang

A NEW SHOOTING METHOD FOR QUASI-BOUNDARY REGULARIZATION OF MULTI-DIMENSIONAL BACKWARD HEAT CONDUCTION PROBLEMS

Abstract 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.

The Modified Polynomial Expansion Method for Solving the Inverse Heat Source Problems

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.

The Lie-group shooting method for boundary layer equations in fluid mechanics

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.

Applications of dual MRM for determining the natural frequencies and natural modes of a rod using the singular value decomposition method

Abstract In this article, the dual multiple reciprocity method is employed to solve the natural frequencies and natural modes for a rod. The conventional approach using dual MRM is not qualified as a systematic method because of the following two reasons: (1) it needs to distinguish the spurious eigenvalue only after the corresponding eigenmode is obtained; (2) the possible indeterminancy of eigenvector may be encountered when the constraint equations chosen are highly dependent such that the rank of the leading coefficient matrix is insufficient. To construct a systematic way, we propose to consider all constraint equations together instead of using the singular or hypersingular equation alone as the conventional MRM uses. The singular value decomposition method is, then, used to solve the eigenproblem after combining the singular and hypersingular equations. This method can avoid the spurious eigenvalue problem and the possible indeterminancy of boundary eigenvectors at the same time. Three numerical examples are given to verify the validity of the present method.

Recovering A Heat Source and Initial Value by a Lie-Group Differential Algebraic Equations Method

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.

A Novel Approach to Great Circle Sailings: The Great Circle Equation

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.

Applications of the direct Trefftz boundary element method to the free-vibration problem of a membrane

In this paper, the direct Trefftz method is applied to solve the free-vibration problem of a membrane. In the direct Trefftz method, there exists no spurious eigenvalue. However, an ill-posed nature of numerical instability encountered in the direct Trefftz method requires some treatments. The Tikhonovs regularization method and generalized singular-value decomposition method are used to deal with such an ill-posed problem. Numerical results show the validity of the current approach.

An asymmetric indirect Trefftz method for solving free-vibration problems

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, Tikhonovs 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.

A QUASI-BOUNDARY SEMI-ANALYTICAL METHOD FOR BACKWARD HEAT CONDUCTION PROBLEMS

Abstract 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.

The Lie-group shooting method for steady-state burgers equation with high Reynolds number

Abstract 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.