微分方程初值问题的GDQR解法

本文用最近由Wu提出的一种数值方法-GDQR(GeneralizedDifferentialQuadra-tureRule)对工程和科学技术中常遇到的2—4阶微分方程初值问题进行了求解.部分结果与精确解或龙格-库塔方法所得结果作了对比,表明GDQR在解决常微分方程初值问题时简单方便有效.     

Analysis of 3D frame problems by DQEM using EDQ

The differential quadrature element method (DQEM) and extended differential quadrature (EDQ) have been proposed by the author. The development of a differential quadrature element analysis model of three-dimensional shear-undeformable frame problems adopting the EDQ is carried out. The element can be a nonprismatic beam. The EDQ technique is used to discretize the element-based governing differential equations, the transition conditions at joints and the boundary conditions on domain boundaries. An overall algebraic system can be obtained by assembling all of the discretized equations. A numerically rigorous solution can be obtained by solving the overall algebraic system. Mathematical formulations for the EDQ-based DQEM frame analysis are carried out. By using this DQEM model, accurate results of frame problems can efficiently be obtained. Numerical results demonstrate this DQEM model.     

Buckling and post-buckling analysis of extensible beam-columns by using the differential quadrature method

Buckling and post-buckling analysis of extensible beam-columns is performed numerically in this paper. It was experienced earlier that in some cases the numerical integration would not produce the convergent post-buckling solutions, especially under high loads. Therefore, a new differential quadrature (DQ) based iterative numerical integration method is proposed to solve post-buckling differential equations of extensible beam-columns. Six cases, including five classical Euler buckling cases, are analyzed. Critical loads and convergent post-buckling solutions under different applied loads are obtained. The results are compared with the existing multiple scales solutions. It is found that under high applied loads, the small rotation assumption in obtaining multiple scales solutions is no longer valid. The proposed iterative DQ based numerical integration method can yield reliable and accurate post-buckling solutions even at high applied loads for the cases investigated.     

Application of Bernoulli matrix method for solving two-dimensional hyperbolic telegraph equations with Dirichlet boundary conditions

The present article is devoted to develop a new approach and methodology to find the approximate solution of second order two-dimensional telegraph equations with the Dirichlet boundary conditions. We first transform the telegraph equations into equivalent partial integro-differential equations (PIDEs) which contain both initial and boundary conditions and therefore can be solved numerically in a more appropriate manner. Operational matrices of integration and differentiation of Bernoulli polynomials together with the completeness of these polynomials are used to reduce the PIDEs into the associated algebraic generalized Sylvester equations which can be solved by an efficient Krylov subspace iterative (i.e., BICGSTAB) method. The efficiency of the proposed method has been confirmed with several test examples and it is clear that the results are acceptable and found to be in good agreement with exact solutions. We have compared the numerical results of the proposed method with radial basis function method and differential quadrature method. Also, the method is simple, efficient and produces very accurate numerical results in considerably small number of basis functions and hence reduces computational effort. Moreover, the technique is easy to apply for multidimensional problems.     

Two-to-one internal resonances in nonlinear two degree of freedom system with parametric and external excitations

A system of two degrees of freedom with quadratic and cubic nonlinearities and subjected to parametric and external excitations is studied and solved. The method of multiple scale perturbation technique (MSPT) is used to analyze the response of this system. Four ordinary differential equations are derived to describe the modulation of the amplitudes and phases of the two modes of vibrations for the principal parametric resonances. The steady-state solutions and their stability are determined and representative numerical results are included. The theoretical resonance cases of this system have been obtained from the first approximation differential equations and some of them are confirmed by applying well-known numerical technique. The stability of the obtained numerical solution is studied using phase–plane method. The effects of different parameters on the vibration of this system are investigated.     

Differential quadrature domain decomposition method for a class of parabolic equations

In this paper the differential quadrature method (DQM) and the domain decomposition method (DDM) are combined to form the differential quadrature domain decomposition method (DQDDM), in which the boundary reduction technique (BRM) is adopted. The DQDDM is applied to a class of parabolic equations, which have discontinuity in the coefficients of the equation, or weak discontinuity in the initial value condition. Two numerical examples belonging to this class are computed. It is found that the application of this method to the above mentioned problems is seen to lead to accurate results with relatively small computational effort.     

Out-of-plane deflection of nonprismatic curved beam structures solved by DQEM

The differential quadrature element method (DQEM) is used to solve the out-of-plane deflections of nonprismatic curved beam structures. The extended differential quadrature is used to discretize the governing differential equations defined on all elements, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions. Numerical results obtained by DQEM are presented. They demonstrate the developed numerical solution procedure.     

SDP-based approximation of stabilising solutions for periodic matrix Riccati differential equations

Numerically finding stabilising feedback control laws for linear systems of periodic differential equations is a nontrivial task with no known reliable solutions. The most successful method requires solving matrix differential Riccati equations with periodic coefficients. All previously proposed techniques for solving such equations involve numerical integration of unstable differential equations and consequently fail whenever the period is too large or the coefficients vary too much. Here, a new method for numerical computation of stabilising solutions for matrix differential Riccati equations with periodic coefficients is proposed. Our approach does not involve numerical solution of any differential equations. The approximation for a stabilising solution is found in the form of a trigonometric polynomial, matrix coefficients of which are found solving a specially constructed finite-dimensional semidefinite programming (SDP) problem. This problem is obtained using maximality property of the stabilising solution of the Riccati equation for the associated Riccati inequality and sampling technique. Our previously published numerical comparisons with other methods shows that for a class of problems only this technique provides a working solution. Asymptotic convergence of the computed approximations to the stabilising solution is proved below under the assumption that certain combinations of the key parameters are sufficiently large. Although the rate of convergence is not analysed, it appeared to be exponential in our numerical studies.     

A Numerical Method to Approximate Multi-Asset Option Pricing Under Exponential Lévy Model

In this paper, a modification of the original global radial basis functions-based differential quadrature (RBF-DQ) method is set forth and analyzed. The improved RBF-DQ method is applicable to the numerical approximation of solutions of a wide range of partial differential equations with mixed derivative terms. However, it appears to be considerably faster than the original method. In support of this contention, the multi-asset option pricing problems under exponential Lévy framework have been solved numerically by using the proposed method and compared with results obtained via the original RBF-DQ method. For accuracy achieved versus work expended, the improved method performs better.     

A new radial basis functions method for pricing American options under Merton's jump-diffusion model

A new radial basis functions (RBFs) algorithm for pricing financial options under Merton's jump-diffusion model is described. The method is based on a differential quadrature approach, that allows the implementation of the boundary conditions in an efficient way. The semi-discrete equations obtained after approximation of the spatial derivatives, using RBFs based on differential quadrature are solved, using an exponential time integration scheme and we provide several numerical tests which show the superiority of this method over the popular Crank–Nicolson method. Various numerical results for the pricing of European, American and barrier options are given to illustrate the efficiency and accuracy of this new algorithm. We also show that the option Greeks such as the Delta and Gamma sensitivity measures are efficiently computed to high accuracy.     

Collocation method for solving systems of Fredholm and Volterra integral equations

In this paper, a numerical method based on based quintic B-spline has been developed to solve systems of the linear and nonlinear Fredholm and Volterra integral equations. The solutions are collocated by quintic B-splines and then the integral equations are approximated by the four-points Gauss-Turán quadrature formula with respect to the weight function Legendre. The quintic spline leads to optimal approximation and O(h⁶) global error estimates obtained for numerical solution. The error analysis of proposed numerical method is studied theoretically. The results are compared with the results obtained by other methods which show that our method is accurate.     

A numerical algorithm for computation modelling of 3D nonlinear wave equations based on exponential modified cubic B-spline differential quadrature method

In this paper, the authors proposed a method based on exponential modified cubic B-spline differential quadrature method (Expo-MCB-DQM) for the numerical simulation of three dimensional (3D) nonlinear wave equations subject to appropriate initial and boundary conditions. This work extends the idea of Tamsir et al. [An algorithm based on exponential modified cubic B-spline differential quadrature method for nonlinear Burgers’ equation, Appl. Math. Comput. 290 (2016), pp. 111–124] for 3D nonlinear wave type problems. Expo-MCB-DQM transforms the 3D nonlinear wave equation into a system of ordinary differential equations (ODEs). To solve the resulting system of ODEs, an optimal five stage and fourth-order strong stability preserving Runge–Kutta (SSP-RK54) scheme is used. Stability analysis of the proposed method is also discussed and found that the method is conditionally stable. Four test problems are considered in order to demonstrate the accuracy and efficiency of the algorithm.     

Stability and accuracy of differential quadrature method in solving dynamic problems

In this paper, the differential quadrature method is used to solve dynamic problems governed by second-order ordinary differential equations in time. The Legendre, Radau, Chebyshev, Chebyshev–Gauss–Lobatto and uniformly spaced sampling grid points are considered. Besides, two approaches using the conventional and modified differential quadrature rules to impose the initial conditions are also investigated. The stability and accuracy properties are studied by evaluating the spectral radii and truncation errors of the resultant numerical amplification matrices. It is found that higher-order accurate solutions can be obtained at the end of a time step if the Gauss and Radau sampling grid points are used. However, the conventional approach to impose the initial conditions in general only gives conditionally stable time step integration algorithms. Unconditionally stable algorithms can be obtained if the modified differential quadrature rule is used. Unfortunately, the commonly used Chebyshev–Gauss–Lobatto sampling grid points would not generate unconditionally stable algorithms.     

Solution of magnetohydrodynamic flow in a rectangular duct by differential quadrature method

The polynomial based differential quadrature and the Fourier expansion based differential quadrature method are applied to solve magnetohydrodynamic (MHD) flow equations in a rectangular duct in the presence of a transverse external oblique magnetic field. Numerical solution for velocity and induced magnetic field is obtained for the steady-state, fully developed, incompressible flow of a conducting fluid inside of the duct. Equal and unequal grid point discretizations are both used in the domain and it is found that the polynomial based differential quadrature method with a reasonable number of unequally spaced grid points gives accurate numerical solution of the MHD flow problem. Some graphs are presented showing the behaviours of the velocity and the induced magnetic field for several values of Hartmann number, number of grid points and the direction of the applied magnetic field.     

用余弦微分求积法数值求解KdV—Burgers方程

采用余弦微分求积法（CDQM）对（1＋1）维非线性KdV—Burgers方程进行了数值求解．结果表明,所得数值解与方程的精确解相比具有明显的高精度且稳定性高,相对于其他常用方法,且公式简单,使用方便;计算量小,时间复杂性好．     

Error analysis of solutions of second-order boundary value problems obtained with residual correction method

Based on the maximum principle of differential equations and with the aid of asymptotic iteration technique, this paper tries to establish monotonic relation of second-order obstacle boundary value problems with their approximate solutions to eventually obtain the upper and lower approximate solutions of the exact solution. To obtain numerical solutions, the cubic spline approximation method is applied to discretize equations, and then according to the ‘residual correction method’ proposed in this paper, residual correction values are added into discretized grid points to translate once complex inequalities’ constraint mathematical programming problems into simple equational iteration problems. The numerical results also show that such method has the characteristic of correcting residual values to symmetrical values for such problems, as a result, the mean approximate solutions obtained even with a considerably small quantity of grid points still quite approximate the exact solution. Furthermore, the error range of approximate solutions can be identified very easily by using the obtained upper and lower approximate solutions, even if the exact solution is unknown.     

New analytical method for solving Burgers' and nonlinear heat transfer equations and comparison with HAM

In this study, we present a numerical comparison between the differential transform method (DTM) and the homotopy analysis method (HAM) for solving Burgers' and nonlinear heat transfer problems. The first differential equation is the Burgers' equation serves as a useful model for many interesting problems in applied mathematics. The second one is the modeling equation of a straight fin with a temperature dependent thermal conductivity. In order to show the effectiveness of the DTM, the results obtained from the DTM is compared with available solutions obtained using the HAM [M.M. Rashidi, G. Domairry, S. Dinarvand, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 708-717; G. Domairry, M. Fazeli, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 489-499] and whit exact solutions. The method can easily be applied to many linear and nonlinear problems. It illustrates the validity and the great potential of the differential transform method in solving nonlinear partial differential equations. The obtained results reveal that the technique introduced here is very effective and convenient for solving nonlinear partial differential equations and nonlinear ordinary differential equations that we are found to be in good agreement with the exact solutions.     

An approximate method of shallow shells with variable thickness

An approximate method is developed to study the static bending of shallow shells with variable thickness. The solutions are obtained by transforming the partial differential equations into the integral equations and applying the numerical integrations. Some numerical examples are shown together with other solutions, and as an application of this method, the results of shallow shell with variable thickness are shown.     

A non-uniform Haar wavelet method for numerically solving two-dimensional convection-dominated equations and two-dimensional near singular elliptic equations

A non-uniform Haar wavelet based collocation method has been developed in this paper for two-dimensional convection dominated equations and two-dimensional near singular elliptic partial differential equations, in which traditional Haar wavelet method produces oscillatory solutions or low accurate solutions. The main idea behind the proposed method is to transform the computation of numerical solution of considered partial differential equations to computation of solution of a linear system of equations. This process is done by discretizing space variables with non-uniform Haar wavelets. To confirm efficiency of the proposed method seven benchmark problems are solved and the obtained results are compared with exact solutions and with local meshless methods, finite element method, finite difference method and polynomial collocation method. Numerical experiments show that the proposed method gives convincing results even in less number of collocation nodes.