Finite difference methods for a class of two-point boundary value problems with mixed boundary conditions

We discuss the construction of three-point finite difference aproximations for the class of two-point boundary value problems: [p(x)y′]′ = f(x, y), α₀y(a) - α₁y′(a) = A, β₀y(b) + β₁y′(b) = B.We first establish an identity from which general three-point finite difference approximations of various orders can be obtained. We then consider in detail obtaining fourth-order methods based on three evaluations of f. We obtain a family of fourth-order discretizations for the differential equations; appropriate discretizations for the boundary conditions are also obtained for use with fourth-order methods. We select the free parameters available in this discretizations which lead to a “simplest” fourth-order method. This method is described and its convergence is established; numerical examples are given to illustrate this new fourth-order method.     

Construction of supersaturated design with large number of factors by the complementary design method

Supersaturated designs (SSDs) have been widely used in factor screening experiments. The present paper aims to prove that the maximal balanced designs are a kind of special optimal SSDs under the E(f _NOD) criterion. We also propose a new method, called the complementary design method, for constructing E(f _NOD) optimal SSDs. The basic principle of this method is that for any existing E(f _NOD) optimal SSD whose E(fNOD) value reaches its lower bound, its complementary design in the corresponding maximal balanced design is also E(f _NOD) optimal. This method applies to both symmetrical and asymmetrical (mixed-level) cases. It provides a convenient and efficient way to construct many new designs with relatively large numbers of factors. Some newly constructed designs are given as examples.     

Low-dose spectral CT reconstruction using image gradient ℓ0–norm and tensor dictionary

Spectral computed tomography (CT) has a great superiority in lesion detection, tissue characterization and material decomposition. To further extend its potential clinical applications, in this work, we propose an improved tensor dictionary learning method for low-dose spectral CT reconstruction with a constraint of image gradient ℓ₀-norm, which is named as ℓ₀TDL. The ℓ₀TDL method inherits the advantages of tensor dictionary learning (TDL) by employing the similarity of spectral CT images. On the other hand, by introducing the ℓ₀-norm constraint in gradient image domain, the proposed method emphasizes the spatial sparsity to overcome the weakness of TDL on preserving edge information. The split-bregman method is employed to solve the proposed method. Both numerical simulations and real mouse studies are perform to evaluate the proposed method. The results show that the proposed ℓ₀TDL method outperforms other competing methods, such as total variation (TV) minimization, TV with low rank (TV+LR), and TDL methods.     

A New Noninterior Predictor–Corrector Method for the P0 LCP

In this paper a new predictor-corrector noninterior method for LCP is presented, in which the predictor step is generated by the Levenberg-Marquadt method, which is new in the predictor-corrector-type methods, and the corrector step is generated as in [3]. The method has the following merits: (i) any cluster point of the iteration sequence is a solution of the P₀ LCP; (ii) if the generalized Jacobian is nonsingular at a solution point, then the whole sequence converges to the (unique) solution of the P₀ LCP superlinearly; (iii) for the P₀ LCP, if an accumulation point of the iteration sequence satisfies the strict complementary condition, then the whole sequence converges to this accumulation point superlinearly. Preliminary numerical experiments are reported to show the efficiency of the algorithm.     

The comparison of different mathematical methods to determine the BOD parameters,a new developed method and impacts of these parameters variations on the design of WWTPs

One of the most common tests for the determination of strength and organic content of wastewater is the biochemical oxygen demand (BOD). This test is widely applied to define organic water pollution and to control the performance of wastewater treatment plants. Generally, BOD is standardized by the measurement of oxygen consumption in 5 days (BOD₅). But, determination of the ultimate biochemical oxygen demand (BOD_u), which is taken 28 days and the reaction rate constant (k) are necessary to understand the organic strength of the wastewater. In this study, the different mathematical methods in order to determine the BOD parameters (BOD_u, k) and two different BOD test method (respirometer and dilution method) are investigated comparatively. Also, a new method based on cubic spline method to estimate ultimate BOD values is developed. Moreover, the impacts of BOD parameters on the design of an activated sludge and aerated lagoon systems are analyzed by using a written user-friend program, which is developed for designing WWTPs by the mean of C++ programming language.     

Computation of double integrals over a triangle

Let T denote the triangle with vertices (u_i, v_i) i = 1, 2, 3. The purpose of this paper is to present a family of cubature rules for approximating the double integral of f(u, v) over T. A FORTRAN implementation of the method is given.     

Mathematical and numerical aspects of one-dimensional laminar flame simulation

The aim of this paper is to solve several mathematical and numerical questions related to the simulation of stationary and nonstationary premixed flat flames. Most of the results are obtained in the general context of complex chemical and diffusion mechanisms. The main mathematical results concern: (i) thea priori positivity of the mass fractions, and (ii) the sensitivity of the flame speed to the computational domain. The numerical method proposed for solving the stationary problem is a new combination of the pseudo-nonstationary approach, the Newton iterations, and the adaptive gridding. The computation of H₂-O₂-N₂ flames with various initial concentrations (including the chemical extinction zone) shows the efficiency of this method.     

An evaluation of interface capturing methods in a VOF based model for multiphase flow of a non-Newtonian ceramic in tape casting

The aim of the present study is to evaluate the different interface capturing methods as well as to find the best approach for flow modeling of the ceramic slurry in the tape casting process. The conventional volume of fluid (VOF) method with three different interpolation methods for interface capturing, i.e. the Geometric Reconstruction Scheme (GRS), High Resolution Interface Capturing (HRIC) and Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM), are investigated for the advection of the VOF, both for Newtonian and non-Newtonian cases. The main purpose is to find the best method for the free surface capturing during the flow of a ceramic slurry described by a constitutive power law equation in the tape casting process. First the developed model is tested against well-documented and relevant solutions from literature involving free surface tracking and subsequently it is used to investigate the flow of a La_0.85Sr_0.15MnO₃ (LSM) ceramic slurry modeled with the Ostwald de Waele power law. Results of the modeling are compared with corresponding experimental data and good agreement is found.     

Elastoplastic stress analysis of thick laminated metal-matrix composite plates by the finite-element method

The elastoplastic stress state of a laminated stainless-steel-fiber-reinforced aluminum-matrix plates, with or without a hole, subjected to a pressure on their top is examined by using the finite-element method. The analysis is carried out for three layouts: (0/90/0/90)_s, (45/-45/45/-45)_s, and (30/60/30/60)_s. The Newton-Raphson method is used to solve the nonlinear problem. The distributions of equivalent stresses and the plastic zones of the plates without a hole and with a hole of various diameters are determined. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 4, pp. 531–544, July–August, 2006.     

Accelerated Projected Gradient Method for Linear Inverse Problems with Sparsity Constraints

Regularization of ill-posed linear inverse problems via ? ₁ penalization has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an ? ₁ penalized functional is via an iterative soft-thresholding algorithm. We propose an alternative implementation to ? ₁-constraints, using a gradient method, with projection on ? ₁-balls. The corresponding algorithm uses again iterative soft-thresholding, now with a variable thresholding parameter. We also propose accelerated versions of this iterative method, using ingredients of the (linear) steepest descent method. We prove convergence in norm for one of these projected gradient methods, without and with acceleration.     

Numerical solution of the nonlinear Klein-Gordon equation

A numerical method is developed to solve the nonlinear one-dimensional Klein-Gordon equation by using the cubic B-spline collocation method on the uniform mesh points. We solve the problem for both Dirichlet and Neumann boundary conditions. The convergence and stability of the method are proved. The method is applied on some test examples, and the numerical results have been compared with the exact solutions. The L₂, L_∞ and Root-Mean-Square errors (RMS) in the solutions show the efficiency of the method computationally.     

Convergence of generalized AOR iterative method for linear systems with strictly diagonally dominant matrices

In this paper, some improvements on Darvishi and Hessari [On convergence of the generalized AOR method for linear systems with diagonally dominant coefficient matrices, Appl. Math. Comput. 176 (2006) 128–133] are presented for bounds of the spectral radius of _lω,r$l_{\omega ,r}$, which is the iterative matrix of the generalized AOR (GAOR) method. Subsequently, some new sufficient conditions for convergence of GAOR method will be given, which improve some results of Darvishi and Hessari [On convergence of the generalized AOR method for linear systems with diagonally dominant coefficient matrices, Appl. Math. Comput. 176 (2006) 128–133].     

Existenzsätze mit der linienmethode für parabolische probleme und periodische lösungen von ut=f(t,x,u,ux,uxx)

The (longitudinal) method of lines transforms a parabolic equation into a first order system of ordinary differential equations by discretization of the spatial variable. It is shown how to obtain existence theorems for nonlinear parabolic equations from those for ordinary differential equations under general growth conditions and weak regularity assumptions. The method is demonstrated in proving a new existence theorem for periodic solutions to u_t=f(t,x,u,u_x,u_xx) with boundary conditions of Dirichlet type.     

A hybrid radial basis functions collocation technique to numerically solve fractional advection–diffusion models

In this work, we propose a hybrid radial basis functions (RBFs) collocation technique for the numerical solution of fractional advection–diffusion models. In the formulation of hybrid RBFs (HRBFs), there exist shape parameter (c* ) and weight parameter (ϵ) that control numerical accuracy and stability. For these parameters, an adaptive algorithm is developed and validated. The proposed HRBFs method is tested for numerical solutions of some fractional Black–Sholes and diffusion models. Numerical simulations performed for several benchmark problems verified the proposed method accuracy and efficiency. The quantitative analysis is made in terms of L_∞, L₂, L_rms , and L_rel error norms as well as number of nodes N over space domain and time-step δt. Numerical convergence in space and time is also studied for the proposed method. The unconditional stability of the proposed HRBFs scheme is obtained using the von Neumann methodology. It is observed that the HRBFs method circumvented the ill-conditioning problem greatly, a major issue in the Kansa method.     

Acoustic estimation of suspended sediment concentration

In this paper, the acoustic estimation of suspended sediment concentration is discussed and two estimation methods of suspended sediment concentration are presented. The first method is curve fitting method, in which, according to the acoustic backscattering theory we assume that the fitting factor K₁ (r) between the concentration M(r) obtained by acoustic observation and the concentration M₀ ( r) obtained by sampling water is a high order power function of distancer. Using least-square algorithm, we can determine the coefficients of the high order power function by minimizing the difference betweenM( r) and M₀ ( r) in the whole water profile. To the absorption coefficient of sound due to the suspension in water we do not give constraint in the first method. The second method is recursive fitting method, in which we take M₀ ( r) as the conditions of initialization and decision and give rational constraints to some parameters. The recursive process is stable. We analyzed the two methods with a lot of experimental data. The analytical results show that the estimate error of the first method is less than that of the second method and the latter can not only estimate the concentration of suspended sediment but also give the absorption coefficient of sound. Good results have been obtained with the two methods.     

在再生核空间中求解非线性奇异两点边值问题

该文建立了一个迭代方法求解一类奇异两点边值问题(x_αu')'=f (x, u, u'), 其中x∈ (0,1),α< 2. 解的表达式是在再生核空间W₂[0,1]中以级数的形式给出的. 近似解一致收敛到准确解. 并且, 误差是单调下降的. 最后通过一些数值算例论述了所提方法的正确性与有效性.     

A mesh-free numerical method for solution of the family of Kuramoto-Sivashinsky equations

In this paper, radial basis function (RBFs) based mesh-free method is implemented to find numerical solution of the Kuramoto-Sivashinsky equations. This approach has an edge over traditional methods such as finite-difference and finite element methods because it does not require a mesh to discretize the problem domain, and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method. The accuracy of the method is assessed in terms of the error norms L₂,L_∞, number of nodes in the domain of influence, free parameter, dependent parameter RBFs and time step length. Numerical experiments demonstrate accuracy and robustness of the method for solving a class of nonlinear partial differential equations.     

Chaos game representation of the Dst index and prediction of geomagnetic storm events

This paper proposes a two-dimensional chaos game representation (CGR) for the D_st index. The CGR provides an effective method to characterize the multifractality of the D_st time series. The probability measure of this representation is then modeled as a recurrent iterated function system in fractal theory, which leads to an algorithm for prediction of a storm event. We present an analysis and modeling of the D_st time series over the period 1963–2003. The numerical results obtained indicate that the method is useful in predicting storm events one day ahead.     

Simulation of Piezoelectric Induced Lamb Waves in Plates

In the paper, piezoelectric induced Lamb wave propagation on homogeneous plates is studied by applying different approaches. Assuming the presence of piezoelectric actuators and sensors the transient wave analysis is made using the classical formulation of the finite element method (FEM) and the novel approach of spectral finite element method (SE). Pure mode excitation and mode conversion are numerically simulated for the fundamental modes S₀ and A₀. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerical technique for solution of the MRLW equation using quartic B-splines

Numerical scheme based on quartic B-spline collocation method is designed for the numerical solution of modified regularized long wave (MRLW) equation. Unconditional stability is proved using Von-Neumann approach. Performance of the method is checked through numerical examples. Using error norms L₂ and L_∞ and conservative properties of mass, momentum and energy, accuracy and efficiency of the new method is established through comparison with the existing techniques.