Application of homotopy perturbation method to multidimensional partial differential equations

In this paper, an application of homotopy perturbation method (HPM) is applied to solve the kindly of multidimensional partial differential equation such as Helmholtz equation. Comparisons are made between the Adomians decomposition method and HPM. The results reveal that the HPM is very effective and simple and gives the exact solution.     

A third-order convergent iterative method for solving non-linear equations

In this paper, we present and analyse a new predictor-corrector iterative method for solving non-linear single variable equations. The convergence analysis establishes that the new method is cubically convergent. Numerical tests show that the method is comparable with the well-known existing methods and in many cases gives better results.     

边坡稳定性分析最小二乘法

利用经典土压力理论设定合理土条推力线位置,对土条底滑面采用摩尔-库伦破坏准则,根据静力平衡及力矩平衡条件建立线性超定方程组,应用MATLAB软件基于最小二乘法原理对此方程组求解,得到比较精确的安全系数。该法从设定合理土条推力线位置出发,避免了对条间力函数的不合理设定,采用MATLAB求解线性超定方程组得解,克服了求解非线性方程组不能迭代收敛得解的缺点,经算例验证其在精度方面比较可靠,对于评价边坡的稳定性具有参考意义。     

Solitons and periodic solutions of coupled nonlinear evolution equations by using the sine–cosine method

In this paper, we establish exact solutions for coupled nonlinear evolution equations. The sine–cosine method is used to construct exact periodic and soliton solutions of coupled nonlinear evolution equations. Many new families of exact travelling wave solutions of the (2+1)-dimensional Konopelchenko–Dubrovsky equations and the coupled nonlinear Klein–Gordon and Nizhnik–Novikov–Veselov equations are successfully obtained. The obtained solutions include compactons, solitons, solitary patterns and periodic solutions. These solutions may be important and of significance for the explanation of some practical physical problems.     

一类Krylov子空间方法在求解Sylvester方程的应用

提出了一种求解Sylvester方程Ax＋XB=EFT的块Krylov子空间方法。当矩阵A和B非常大，并且右侧的的秩很小时，给出如何求解精确低秩近似解。理论结果和数值实例证明了方法的有效性。     

A new analytical method for solving systems of Volterra integral equations

In this paper, a new modified homotopy perturbation method (NHPM) is introduced for solving systems of Volterra integral equations of the second kind. Theorems of existence and uniqueness of the solutions to these equations are presented. Comparison of the results of applying the NHPM with those of the homotopy perturbation method and Adomian's decomposition method leads to significant consequences. Several examples, including the system of linear and nonlinear Volterra integral equations, are given to demonstrate the efficiency of the new method.     

Comments on “Robust circularly orthogonal moment based on Chebyshev rational function” [Digit. Signal Process. 62 (2017) 249–258]

Comments on the recent work of Guo et al. [1] are presented, in order to correct the erroneous equations. The corrected equations of the Chebychev rational moments, for gray-level images, are presented. In addition, one numerical experiment is performed to ensure the validity of the corrected equations.     

Simflowny: A general-purpose platform for the management of physical models and simulation problems

Simflowny is a software platform which aims to formalize the main elements of a simulation flow. It allows users to manage (i) formal representations of physical models based on Initial Value Problems (hyperbolic, parabolic and mixed-type partial differential equations), (ii) simulation problems based on such models, and (iii) discretization schemes to translate the problem to a finite mesh. Additionally, Simflowny generates automatically code for general-purpose simulation frameworks. This paper first presents an introductory example of such problems. Then, formal representations are explained. Afterwards, it summarizes the platform’s architecture. Finally, validation results are provided.     

Modeling and analysis of an over-constrained flexure-based compliant mechanism

Bridge-type micro-displacement amplifier with flexure hinges is a classic displacement amplification mechanism. Existing theoretic models cannot predict its amplification ratio and input stiffness accurately and make it very difficult to confirm the amplifier’s performance and error compensation by means of these models, which is very significant in ultra-precision positioning. This paper focuses on the development of design equations that can accurately calculate the ideal displacement amplification ratio and input stiffness of the amplifier based on the thought of statically indeterminate structure. Force Method, Maxwell–Mohr Method, principle of superposition and deformation compatibility are used together to establish uncanonical linear homogeneous equations. The analytical results are verified by FEA simulations. The influence of the geometric parameters on the amplifier performance is investigated. It is noted that amplifier performance is more sensitive to the longitudinal distance of flexure hinges. Besides, two same-sized amplifiers with the opposite output directions can be clearly differentiated by these equations.     

Tidal turbine array optimisation using the adjoint approach

Oceanic tides have the potential to yield a vast amount of renewable energy. Tidal stream generators are one of the key technologies for extracting and harnessing this potential. In order to extract an economically useful amount of power, hundreds of tidal turbines must typically be deployed in an array. This naturally leads to the question of how these turbines should be configured to extract the maximum possible power: the positioning and the individual tuning of the turbines could significantly influence the extracted power, and hence is of major economic interest. However, manual optimisation is difficult due to legal site constraints, nonlinear interactions of the turbine wakes, and the cubic dependence of the power on the flow speed. The novel contribution of this paper is the formulation of this problem as an optimisation problem constrained by a physical model, which is then solved using an efficient gradient-based optimisation algorithm. In each optimisation iteration, a two-dimensional finite element shallow water model predicts the flow and the performance of the current array configuration. The gradient of the power extracted with respect to the turbine positions and their tuning parameters is then computed in a fraction of the time taken for a flow solution by solving the associated adjoint equations. These equations propagate causality backwards through the computation, from the power extracted back to the turbine positions and the tuning parameters. This yields the gradient at a cost almost independent of the number of turbines, which is crucial for any practical application. The utility of the approach is demonstrated by optimising turbine arrays in four idealised scenarios and a more realistic case with up to 256 turbines in the Inner Sound of the Pentland Firth, Scotland.