Stability estimate for an inverse boundary coefficient problem in thermal imaging
Mourad Bellassoueda, Jin Cheng and Mourad Choullic
J. Math. Anal. Appl. 343 (2008), 328-336
We establish a stability estimate for an inverse boundary coefficient problem in thermal imaging. The inverse problem under consideration consists in the determination of a boundary coefficient appearing in a boundary value problem for the heat equation with Robin boundary condition (we note here that the initial condition is assumed to be a priori unknown). Our stability estimate is of logarithmic type and it is essentially based on a logarithmic estimate for a Cauchy problem for the Laplace equation.
A quasi Tikhonov regularization for a two-dimensional backward heat problem by a fundamental solution
Jin Cheng and Jijun Liu
Inverse Problems, 24 (2008) 065012 (18pp).
Consider a two-dimensional backward heat conduction problem for a general domain with a boundary. Based on the fundamental solution to the heat equation, we propose to solve this problem by the boundary integral equation method, which generates a coupled ill-posed integral equation. Then the well posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proven. Our regularizing scheme can be considered a quasi Tikhonov regularization, with the advantage of a relatively small amount of computation compared with the classical Tikhonov regularization. Numerical performances are given to show the validity of our inversion method.
Stable determination of a boundary coefficient in an elliptic equation
Jin Cheng, Mourad Choulli and Junshan Lin
Math. Models Methods Appl. Sci. 18 (2008), 107-123.
We prove a logarithmic stability estimate for a Cauchy problem associated with a second order elliptic operator. Our proof is essentially based on a Carleman estimate by A. L. Bukhgeim. This result is applied to establish a stability estimate for the inverse problem of determining a boundary coefficient (or a boundary function) by a single boundary measurement. This kind of inverse problems is motivated by the corrosion detection problem.
Function reconstruction from noisy local averages
Yu Chen, Jianguo Huang and Weimin Han
Inverse Problems, 24(2008), 024003 (14pp).
A regularization method is proposed for the function reconstruction from noisy local averages in any dimension. Error bounds for the approximate solution in L2-norm are derived. A number of numerical examples are provided to show computational performance of the method, with the regularization parameters selected by different strategies.
Integration processes of ordinary differential equations based on Laguerre-Radau interpolations
Benyu Guo, Zhongqing Wang, Hongjiong Tian and Lilian Wang
Math. Comp., 77(2008), 181-199.
In this paper, we propose two integration processes for ordinary differential equations based on modified Laguerre-Radau interpolations, which are very efficient for long-time numerical simulations of dynamical systems. The global convergence of proposed algorithms are proved. Numerical results demonstrate the spectral accuracy of these new approaches and coincide well with theoretical analysis.
On non-isotropic Jacobi Pseudospectral method
Benyu Guo and Keji Zhang
J. Comp. Math., 26(2008), 511-535.
In this paper, a non-isotropic Jacobi pseudospectral method is proposed and its applications are considered. Some results on the multi-dimensional Jacobi-Gauss type interpolation and the related Bernstein-Jackson type inequalities are established, which play an important role in pseudospectral method. The pseudospectral method is applied to a twodimensional singular problem and a problem on axisymmetric domain. The convergence of proposed schemes is established. Numerical results demonstrate the efficiency of the proposed method.
Irrational approximations and their applications to partial differential equations in exterior domains
Benyu Guo and Jie Shen
Adv. Comput. Math., 28(2008), 237-267.
A family of orthogonal systems of irrational functions on the semi-infinite interval is introduced. The proposed orthogonal systems are based on Jacobi polynomials through an irrational coordinate transform. This family of orthogonal systems offers great flexibility to match a wide range of asymptotic behaviors at infinity. Approximation errors by the basic orthogonal projection and various other orthogonal projections related to partial differential equations in unbounded domains are established. As an example of applications, a Galerkin approximation using the proposed irrational functions to an exterior problem is analyzed and implemented. Numerical results in agreement with our theoretical estimates are presented.
Navier-Stokes equations with slip boundary conditions
Mathematical Methods in the Applied Sciences, 31(2008), 607-626.
In this paper, we consider incompressible viscous fluid flows with slip boundary conditions. We first prove the existence of solutions of the unsteady Navier-Stokes equations in n-spacial dimensions. Then, we investigate the stability, uniqueness and regularity of solutions in two and three spacial dimensions. In the compactness argument, we construct a special basis fulfilling the incompressibility exactly, which leads to an efficient and convergent spectral method. In particular, we avoid the main difficulty for ensuring the incompressibility of numerical solutions, which occurs in other numerical algorithms. We also derive the vorticity-stream function form with exact boundary conditions, and establish some results on the existence, stability and uniqueness of its solutions.
Mathematical Modelling of prostate tumor growth under intermittent androgen suppression
Qian Guo, Youshan Tao and Aihara Kazuyuki
Int. J. Bifur. Chaos, 18(2008), 3789-3797.
Since most prostate tumors are initially hormone-sensitive, hormonal therapy with androgen suppression is a major treatment for them. In this hormonal therapy, however, a tumor relapse is a crucial problem. Androgen-independent tumor cells are considered to be responsible for such a relapse. These cells are not sensitive to androgen suppression but rather apt to proliferate even in an androgen-poor environment. Bruchovsky et al. proposed intermittent androgen suppression (IAS), which may prolong the relapse time when compared with continuous androgen suppression (CAS). IAS therapy is based on switching of medication through monitoring of the serum prostate-specific antigen (PSA). Namely, the medication is suspended when the PSA concentration falls below the lower threshold during on-treatment periods and it is reinstituted when the concentration exceeds the upper threshold during off-treatment periods. In this paper, we propose a model of partial differential equations (PDE) for IAS therapy, on the basis of our previous model of ordinary differential equations, under the assumption that the prostate tumor is a mixed assembly of androgen-dependent (AD) and androgen-independent (AI) cells. Numerical analysis compares the effect of the IAS therapy with that of the CAS therapy for different growth rates of the AI cells, which suggests an optimal protocol of the IAS therapy.
Vibration analysis of Kirchhoff plates by the Morley element method
Jianguo Huang, Ling Guo, Zhongci Shi
J. Comp. Appl. Math., 213 (2008), 14-34.
Vibration analysis of Kirchhoff plates is of great importance in many engineering fields. The semi-discrete and the fully discrete Morley element methods are proposed to solve such a problem, which are effective even when the region of interest is irregular. The rigorous error estimates in the energy norm for both methods are established. Some reasonable approaches to choosing the initial functions are given to keep the good convergence rate of the fully discrete method. A number of numerical results are provided to illustrate the computational performance of the method in this paper.
Fully discrete Jacobi-spherical harmonic spectral method for Navier-Stokes equations
Wei Huang and Benyu Guo
Appl. Math. Mech. 29(2008), 453-476.
A fully discrete Jacobi-spherical harmonic spectral method is provided for the Navier-Stokes equations in a ball.Its stability and convergence are proved.Numerical results show efficiency of this approach.The proposed method is also applicable to other problems in spherical geometry.
Error analysis of a parameter expansion for corrosion detection in a pipe
Xuehai Huang, Jianguo Huang and Yu Chen
Computers and Mathematics with Applications, 56(2008), 2539-2549
Error analysis is developed for a parameter expansion method of determining the corrosion coefficient in a pipe. For the two approximate solutions proposed, the magnitude of their errors is shown to be O(a) and O(a2), respectively, where a stands for the thickness of the pipe.
Vibration analysis for elastic multi-beam structures by the C0-continuous time-stepping finite element method
Junjiang Lai, Jianguo Huang and Zhongci Shi
Comm. Numer. Methods Eng., 2008
Some C0-continuous time-stepping finite element method is proposed for investigating vibration analysis of elastic multi-beam structures. In the time direction, the C0-continuous Galerkin method is used to discretize the generalized displacement field. In the space directions, the longitudinal displacements and rotational angles on beams are discretized using conforming linear elements, while the transverse displacements on beams are discretized by the Hermite elements of third order. The error of the method in the energy norm is proved to be O(h+k3), where h and k denote the mesh sizes of the subdivisions in the space and time directions, respectively. The finite difference analysis in time is developed to discuss the spectral behavior of the algorithms as well as their dissipation and dispersion properties in the low-frequency regime. The method has also been extended to study some nonlinear problems. A number of numerical tests are included to illustrate the computational performance of the method.
Vibration analysis of plane elasticity problems by the C0-continuous time stepping finite element method
Junjiang Lai，Jianguo Huang and Chuanmiao Chen
Appl. Numer. Math., 59(2009), 905-919.
This paper proposes a C0-continuous time stepping finite element method to solve vibration problems of plane elasticity. In the time direction, unlike the existing methods [F. Costanzo, H. Huang, Proof of unconditional stability for a single-field discontinuous Galerkin finite element formulation for linear elasto-dynamics, Comput. Methods Appl. Mech. Engrg. 194 (2005) 2059–2076; D.A. French, A space–time finite element method for the wave equation, Comput. Methods Appl. Mech. Engrg. 107 (1993) 145–157; H. Huang, F. Costanzo, On the use of space–time finite elements in the solution of elasto-dynamic problems with strain discontinuities, Comput. Methods Appl. Mech. Engrg. 191 (2002) 5315–5343; T.J.R. Hughes, G. Hulbert, Space–time finite element methods for elastodynamics: Formulations and error estimates, Comput. Methods Appl. Mech. Engrg. 66 (1988) 339–363; G. Hulbert, T.J.R. Hughes, Space–time finite element methods for second-order hyperbolic equations, Comput. Methods Appl. Mech. Engrg. 84 (1990) 327–348; C. Johnson, Discontinuous Galerkin finite element methods for second order hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 107 (1993) 117–129; X.D. Li, N.E. Wiberg, Structural dynamic analysis by a time-discontinous Galerkin finite element method, Int. J. Numer. Methods Engrg. 39 (1996) 2131–2152; X.D. Li, N.E. Wiberg, Implementation and adaptivity of a space–time finite element method for structural dynamics, Comput. Methods Appl. Mech. Engrg. 156 (1998) 211–229], this method does not use the discontinuous Galerkin (DG) method to simultaneously discretize the displacement and velocity fields, but only use the C0-continuous Galerkin method to discretize the displacement field instead. This greatly reduces the size of the linear system to be solved at each time step. The finite element in the space directions is taken as the usual Pr−1-conforming element with r 2. It is proved that the error of the method in the energy norm is O(hr−1+k3), where h and k denote the mesh sizes of the subdivisions in the space and time directions, respectively. Some numerical tests are included to show the computational performance of the method.
P-optimal robust designs for multiresponse approximately linear regression
Xin Liu and Rongxian Yue
Applied Mathematics - A Journal of Chinese Universities, 23(2008), 168-174
This paper deals with the problem of -optimal robust designs for multiresponse approximately linear regression models. Each response is assumed to be only approximately linear in the regressors, and the bias function varies over a given ℒ2-neighbourhood. A kind of bivariate models with two responses is taken as an example to illustrate how to get the expression of the design measure.
Inversion analysis for magnetic resonance elastography
Gen Nakamura, Yu Jiang, Sei Nagayasu and Jin Cheng
Appl. Anal. 87 (2008), 165-179.
We will propose a new reconstruction scheme to identify the viscoelasticity of a living body from MRE measurements. The reconstruction scheme consists of application of the oscillating-decaying solution, Taylor expansion, complex geometric optics solutions, and an iterative method for solving the Cauchy problem for elliptic equations.
Numerical solutions of a three-competition Lotka–Volterra system
Chia Ven Pao and Yuanming Wang
Appl. Math. Comp., 204 (2008), 423–440.
This paper is concerned with finite difference solutions of a Lotka–Volterra reaction–diffusion system with three-competing species. The reaction–diffusion system is discretized by the finite difference method, and the investigation is devoted to the finite difference system for the time-dependent solution and its asymptotic behavior in relation to the corresponding steady-state problem. Three monotone iterative schemes for the computation of the time-dependent solution are presented, and the sequences of iterations are shown to converge monotonically to a unique positive solution. Also discussed is the asymptotic behavior of the time-dependent solution in relation to various steady-state solutions. A simple condition on the competing rate constants is obtained, which ensures that for every nontrivial nonnegative initial function the corresponding time-dependent solution converges either to a unique positive steady-state solution or to one of the semitrivial steady-state solutions. The above results lead to the coexistence and permanence of the competing system as well as computational algorithms for numerical solutions. Some numerical results from these computational algorithms are given. All the conclusions for the reaction–diffusion equations are directly applicable to the finite difference solution of the corresponding ordinary differential system.
同济大学学报（自然科学版）, 36(2008), 849-853.
On spectral methods for Volterra type integral equations and the convergence analysis
Tao Tang, Xiang Xu and Jin Cheng
J. Comput. Math. 26 (2008), 825-837.
The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical results confirm the theoretical prediction of the exponential rate of convergence. The result in this work seems to be the first successful spectral approach (with theoretical justification) for the Volterra type equations.
A mathematical model of combined therapies against cancer using viruses and inhibitors
Youshan Tao and Qian Guo
Science in China Series A: Mathematics, 51(2008), 2315-2329.
This paper deals with a procedure for combined therapies against cancer using oncolytic viruses and inhibitors. Replicating genetically modified adenoviruses infect cancer cells, reproduce inside them and eventually cause their death (lysis). As infected cells die, the viruses inside them are released and then proceed to infect other tumor cells. The successful entry of virus into cancer cells is related to the presence of the coxsackie-adenovirus receptor (CAR). Mitogen-activated protein kinase kinase (known as MEK) inhibitors can promote CAR expression, resulting in enhanced adenovirus entry into cancer cells. However, MEK inhibitors can also cause G1 cell-cycle arrest, inhibiting reproduction of the virus. To design an effective synergistic therapy, the promotion of virus infection must be optimally balanced with inhibition of virus production. We introduce a mathematical model to describe the effects of MEK inhibitors and viruses on tumor cells, and use it to explore the reduction of the tumor size that can be achieved by the combined therapies. Furthermore, we find an optimal dose of inhibitor: Poptimal = 1 − μ/δ for a certain initial density of cells (where μ is the removal rate of the dead cells and δ is the death rate of the infected cells). The optimal timing of MEK inhibitors is also numerically studied.
Asymptotic stability of numerical methods for linear delay parabolic differential equations
Comp. Math. Appl., 56 (2008) 1758–1765
This paper is concerned with the asymptotic stability property of some numerical processes by discretization of parabolic differential equations with a constant delay. These numerical processes include forward and backward Euler difference schemes and Crank–Nicolson difference scheme which are obtained by applying step-by-step methods to the resulting systems of delay differential equations. Sufficient and necessary conditions for these difference schemes to be delay-independently asymptotically stable are established. It reveals that an additional restriction on time and spatial stepsizes of the forward Euler difference scheme is required to preserve the delay-independent asymptotic stability due to the existence of the delay term. Numerical experiments have been implemented to confirm the asymptotic stability of these numerical methods.
Asymptotic stability, contractivity and dissipativity of one-leg θ-method for non-autonomous delay functional differential equations
Hongjiong Tian and Ni Guo
Appl. Math. Comp., 203(2008), 333-342.
This paper focuses on asymptotic stability, contractivity and dissipativity of non-autonomous nonlinear delay functional differential equations with bounded lag, and the corresponding dynamical properties of one-leg -method. Sufficient conditions for these delay functional differential equations to be dissipative, asymptotically stable and contractive are established. One-leg -method is constructed to solve such equations numerically. An important result on the growth of solution of a class of difference inequalities with variable coefficients is obtained. Finally, it is proved that the one-leg -method is asymptotically stable, contractive and dissipative if and only if = 1. Numerical examples are given to confirm our theoretical results.
Composite generalized Laguerre-Legendre pseudospectral method for Fokker--Planck equation in an infinite channel
Tianjun Wang and Benyu Guo
Applied Numerical Mathematics, 58(2008), 1448-1466
In this paper, we propose a composite generalized Laguerre-Legendre pseudospectral method for the Fokker-Planck equation in an infinite channel, which behaves like a parabolic equation in one direction, and behaves like a hyperbolic equation in other direction. We establish some approximation results on the composite generalized Laguerre-Legendre-Gauss-Radau interpolation, with which the convergence of proposed composite scheme follows. An efficient implementation is provided. Numerical results show the spectral accuracy in space of this approach and coincide well with theoretical analysis. The approximation results and techniques developed in this paper are also very appropriate for many other problems on multiple-dimensional unbounded domains, which are not of standard types.
Fourth-order compact finite difference method for fourth-order nonlinear elliptic boundary value problems
Yuanming Wang and Benyu Guo
J. Comp. Appl. Math., 221 (2008), 76–97.
A compact finite difference method with non-isotropic mesh is proposed for a two-dimensional fourth-order nonlinear elliptic boundary value problem. The existence and uniqueness of its solutions are investigated by the method of upper and lower solutions, without any requirement of the monotonicity of the nonlinear term. Three monotone and convergent iterations are provided for resolving the resulting discrete systems efficiently. The convergence and the fourth-order accuracy of the proposed method are proved. Numerical results demonstrate the high efficiency and advantages of this new approach.
A fourth-order compact finite difference method for higher-order
Lidstone boundary value problems
Yuanming Wang, Haiyun Jiang and Ravi P. Agarwal
Comp. Math. Appl., 56 (2008), 499–521.
A compact finite difference method is proposed for a general class of 2nth-order Lidstone boundary value problems. The existence and uniqueness of the finite difference solution is investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. A monotone iteration process is provided for solving the resulting discrete system efficiently, and a simple and easily verified condition is obtained to guarantee a geometric convergence of the iterations. The convergence of the finite difference solution and the fourth-order accuracy of the proposed method are proved. Numerical results demonstrate the high efficiency and advantages of this new approach.
A monotone compact implicit scheme for nonlinear reaction-diffusion equations
Yuanming Wang and Benyu Guo
J. Comp. Math., 26(2008), 123–148.
A monotone compact implicit finite difference scheme with fourth-order accuracy in space and second-order in time is proposed for solving nonlinear reaction-diffusion equations. An accelerated monotone iterative method for the resulting discrete problem ispresented. The sequence of iteration converges monotonically to the unique solution of the discrete problem, and the convergence rate is either quadratic or nearly quadratic, depending on the property of the nonlinear reaction. The numerical results illustrate the high accuracy of the proposed scheme and the rapid convergence rate of the iteration.
Global asymptotic stability of 3-species Lotka–Volterra models with diffusion and time delays
Appl. Math. Comp., 195 (2008), 34–48.
This paper is concerned with three 3-species time-delayed Lotka–Volterra reaction–diffusion models with homogeneous Neumann boundary condition. Some simple conditions are obtained for the global asymptotic stability of the nonnegative semitrivial constant steady-state solutions. These conditions are explicit and easily verifiable, and they involve only the reaction rate constants and are independent of the diffusion and time delays. The result of global asymptotic stability not only implies the nonexistence of positive steady-state solution but also gives some extinction results of the models in the ecological sense. The instability of some nonnegative semitrivial constant steady-state solutions is also shown. The conclusions for the reaction–diffusion systems are directly applicable to the corresponding ordinary differential systems.
Jacobi rational approximation and spectral method for differential equations of degenerate type
Zhongqing Wang and Benyu Guo
Math. Comp., 77(2008), 883-907.
We introduce an orthogonal system on the half line, induced by Jacobi polynomials. Some results on the Jacobi rational approximation are established, which play important roles in designing and analyzing the Jacobi rational spectral method for various differential equations, with the coefficients degenerating at certain points and growing up at infinity. The Jacobi rational spectral method is proposed for a model problem appearing frequently in finance. Its convergence is proved. Numerical results demonstrate the efficiency of this new approach.
Mixed spectral method for three-dimensional exterior problems using spherical harmonic and generalized Laguerre functions
Zhongqing Wang, Benyu Guo and Wei Zhang
J. Comp. Appl. Math., 217(2008), 277-298.
In this paper, we develop the mixed spectral method for three-dimensional exterior problems, using spherical harmonic and generalized Laguerre functions. Some basic approximation results are established. The mixed spectral schemes are proposed for two model problems. Their convergences are proved. Numerical results demonstrate the efficiency of this new approach.
Modified Laguerre spectral and pseudospectral methods for nonlinear partial differential equations in multiple dimensions
Chenglong Xu and Benyu Guo
Appl. Math. Mech., 29(2008), 311-331.
The Laguerre spectral and pseudospectral methods are investigated for multidimensional nonlinear partial difierential equations.Some results on the modified Laguerre orthogonal approximation and interpolation are established,which play important roles in the related numerical methods for unbounded domains.As an example,the modified Laguerre spectral and pseudospectral methods are proposed for two-dimensional Logistic equation.The stability and convergence of the suggested schemes are proved.Numerical results demonstrate the high accuracy of these approaches.
Bayesian Design Comparison for Accelerated Life Tests
Haiyan Xu, Yincai Tang and Heliang Fei
14th ISSAT International Conference on Reliability and Quality in Design, Florida, USA, 29-33, 2008.
With Bayesian methods, there are two popular criteria on which accelerated life tests (ALT) design is based. In this article, the theoretical basis of the two criteria is unmasked, and relationships between the two criteria are derived. An example is also presented to illustrate the application of the two criteria in constant-stress ALT situation with log-normal distribution and Type-I censoring.
Heat transfer in composite materials with Stefan-Boltzmann interface conditions
Gufan Yang, Masahiro Yamamoto and Jin Cheng
Math. Methods Appl. Sci. 31 (2008), 1297-1314.
In this paper, we discuss nonstationary heat transfer problems in composite materials. This problem can be formulated as the parabolic equation with Stefan-Boltzmann interface conditions. It is proved that there exists a unique global classical solution to one-dimensional problems. Moreover, we propose a numerical algorithm by the finite difference method for this nonlinear transmission problem.
Bayesian robust designs for linear models with possible bias and correlated errors
Rongxian Yue and Xiaodong Zhou
Consider the design problem for the approximately linear model with serially correlated errors. The correlated structure is the qth degree moving average process, MA(q), especially for q = 1, 2. The optimal design is derived by using Bayesian approach. The Bayesian designs derived with various priors are compared with the classical designs with respect to some specific correlated structures. The results show that any prior knowledge about the sign of the MA(q) process parameters leads to designs that are considerately more efficient than the classical ones based on homoscedastic assumptions.
Minimax Designs for Approximately Linear Multiresponse Models
Rongxian Yue and Xin Liu
Proceedings of Fifth International Conference on Fuzzy Systems and Knowledge Discovery, 2008, 1 (2008), 324-328.
This paper considers the minimax design problem in approximately linear multiresponse regression models. The difference between the fitted linear models and the true responses is a vector of non-linear contaminations restricted by a bound on its L_2 norm in the design region. We derive a theory to guide in the construction of designs which minimize the maximum loss. The minimax design is found for a bivariate regression model with two responses.
Mixed Fourier-Laguerre spectral and pseudospectral methods for exterior problems using generalized Laguerre functions
Rong Zhang, Zhongqing Wang and Benyu Guo
J. Scientific Computing, 36(2008), 263-283.
In this paper, we develop the mixed spectral and pseudospectral methods for two-dimensional exterior problems, by using the scaled generalized Laguerre functions. Some basic results on the mixed Fourier-Laguerre orthogonal approximation and Gauss-type interpolation are established, which play important roles in the related spectral and pseudospectral methods. As an example, we propose the mixed spectral and pseudospectral schemes for a model problem. The convergence of proposed schemes are proved. Numerical results demonstrate the spectral accuracy efficiency of this new approach.
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