Monte Carlo method via a numerical algorithm to solve a parabolic problem

This paper is intended to provide a numerical algorithm consisted of the combined use of the finite difference method and Monte Carlo method to solve a one-dimensional parabolic partial differential equation. The numerical algorithm is based on the discretize governing equations by finite difference method. Due to the application of the finite difference method, a large sparse system of linear algebraic equations is obtained. An approach of Monte Carlo method is employed to solve the linear system. Numerical tests are performed in order to show the efficiency and accuracy of the present work.     

Finite-difference methods for solving the reaction-diffusion equations of a simple isothermal chemical system

Numerical methods are proposed for the numerical solution of a system of reaction-diffusion equations, which model chemical wave propagation. The reaction terms in this system of partial differential equations contain nonlinear expressions. Nevertheless, it is seen that the numerical solution is obtained by solving a linear algebraic system at each time step, as opposed to solving a nonlinear algebraic system, which is often required when integrating nonlinear partial differential equations. The development of each numerical method is made in the light of experience gained in solving the system of ordinary differential equations, which model the well-stirred analogue of the chemical system. The first-order numerical methods proposed for the solution of this initialvalue problem are characterized to be implicit. However, in each case it is seen that the numerical solution is obtained explicitly. In a series of numerical experiments, in which the ordinary differential equations are solved first of all, it is seen that the proposed methods have superior stability properties to those of the well-known, first-order, Euler method to which they are compared. Incorporating the proposed methods into the numerical solution of the partial differential equations is seen to lead to two economical and reliable methods, one sequential and one parallel, for solving the travelling-wave problem. © 1994 John Wiley & Sons, Inc.     

The dynamic analysis of the functionally graded piezoelectric (FGP) shell panel based on three-dimensional elasticity theory

In this paper, three-dimensional elasticity solution is extended to investigate a FGPM finite length, simply supported shell panel under dynamic pressure excitation. The host panel is assumed to be of some functionally graded piezoelectric material (FGPM). The ordinary differential equations (o.d.e.) are derived from the highly coupled partial differential equations (p.d.e.) using series expansions of mechanical and electrical displacements. The resulting system of ordinary differential equations is solved by means of Galerkin finite element method. At last, numerical examples are presented for a FGPM shell panel. To verify the validity of code and formulation, the results of a FGM panel and a FGM plate are compared with the published results.     

A hydrodynamic model of human cochlea

A two-compartment model of the human cochlea is proposed. When stretched out, the bony spiral tube looks like two chambers separated by a membrane. Both chambers are filled with viscous fluid called perilymph; they communicate with one another via a canal. Sound vibrations enter the cochlea through the oval window and cause periodic change of pressure in the perilymph, which, in turn, causes the membrane to vibrate. The motion of the fluid is described by hydrodynamic equations, which are supplemented with the membrane vibration equation. The equations are linearized in the amplitude of the vibrations, and their solution is sought in the form of Fourier harmonics with a given frequency. To determine the harmonics, a system of linear boundary value problems for ordinary differential equations with variable coefficients is obtained. The numerical solution of this system using finite difference method fails because it involves a large parameter and the problem is close to a singular one. We propose a novel numerical method without saturation that enables us to obtain solutions in a wide range of frequencies up to an arbitrary and controllable accuracy. The computations confirm the Bekesy theory stating that high-frequency sounds cause the membrane to bend near the apex of the cochlea, and low-frequency sounds cause it to bend near the base of the cochlea.     

非均匀薄板弯曲的精确元法

本文在阶梯折算法的基础上,提出构造有限元的新方法——精确元法.它不用一般变分原理,可适用于任意变系数正定和非正定偏微分方程.利用该方法,得到薄板弯曲一个非协调三角形单元,它具有6个自由度.文中给出证明,位移和内力均收敛于精确解,并有很好的精度.文末给出算例.算例表明利用本文的方法,内力和位移均可获得满意的结果.     

弹性力学平面问题的精确元法

本文在阶梯折算法[1]和精确解析法[2]的基础上,提出构造有限元的新方法——精确元法.该方法不用一般变分原理,可适用于任意变系数正定和非正定偏微分方程.利用该方法得到弹性力学平面问题的一个非协调任意四边形单元.它具有八个自由度.由于没有采用雅可比变换,该单元可以蜕化为三角形单元,在工程中使用起来较为方便.文中给出收敛性证明.文末给出算例,位移和应力均给出较好的结果,在单元的节点上有较好的数值精度.     

A highly accurate Jacobi collocation algorithm for systems of high‐order linear differential–difference equations with mixed initial conditions

In this paper, a shifted Jacobi–Gauss collocation spectral algorithm is developed for solving numerically systems of high‐order linear retarded and advanced differential–difference equations with variable coefficients subject to mixed initial conditions. The spatial collocation approximation is based upon the use of shifted Jacobi–Gauss interpolation nodes as collocation nodes. The system of differential–difference equations is reduced to a system of algebraic equations in the unknown expansion coefficients of the sought‐for spectral approximations. The convergence is discussed graphically. The proposed method has an exponential convergence rate. The validity and effectiveness of the method are demonstrated by solving several numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier. Copyright © 2015 John Wiley & Sons, Ltd.     

Dynamic motion of whirling rods with Coriolis effect

Longitudinal vibrations coupled with transverse vibrations of whirling rods are investigated. It is known that longitudinal and transverse vibrations are governed by second and fourth order differential equations, respectively. Due to the Coriolis effect, a system of equations that governs the longitudinal and transverse displacements will be constructed by coupling these two equations together. Solutions of the equations assume small oscillations of vibration being superimposed on the steady state of the whirling rod. Exact and approximate solutions are obtained from the proposed governing equations, where the approximate solutions on displacements and natural frequencies are acquired by neglecting the Coriolis effect. A proposed numerical scheme known as complete function collocation method is implemented to solve the governing equations coupled with longitudinal and transverse displacements. The approximate results on both longitudinal and transverse natural frequencies show that natural frequencies are decreasing while the angular velocity of the rod is increasing. Exact and numerical results on both longitudinal and transverse natural frequencies show that there are no predictable trends whether natural frequencies are increasing or decreasing while the angular velocity of the rod is increasing.     

Practical finite‐analytic method for solving differential equations by compact numerical schemes

Methodology for development of compact numerical schemes by the practical finite‐analytic method (PFAM) is presented for spatial and/or temporal solution of differential equations. The advantage and accuracy of this approach over the conventional numerical methods are demonstrated. In contrast to the tedious discretization schemes resulting from the original finite‐analytic solution methods, such as based on the separation of variables and Laplace transformation, the practical finite‐analytical method is proven to yield simple and convenient discretization schemes. This is accomplished by a special universal determinant construction procedure using the general multi‐variate power series solutions obtained directly from differential equations. This method allows for direct incorporation of the boundary conditions into the numerical discretization scheme in a consistent manner without requiring the use of artificial fixing methods and fictitious points, and yields effective numerical schemes which are operationally similar to the finite‐difference schemes. Consequently, the methods developed for numerical solution of the algebraic equations resulting from the finite‐difference schemes can be readily facilitated. Several applications are presented demonstrating the effect of the computational molecule, grid spacing, and boundary condition treatment on the numerical accuracy. The quality of the numerical solutions generated by the PFAM is shown to approach to the exact analytical solution at optimum grid spacing. It is concluded that the PFAM offers great potential for development of robust numerical schemes. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Numerical treatment for solving fractional Riccati differential equation

This paper presents an accurate numerical method for solving fractional Riccati differential equation (FRDE). The proposed method so called fractional Chebyshev finite difference method (FCheb-FDM). In this technique, we approximate FRDE with a finite dimensional problem. The method is based on the combination of the useful properties of Chebyshev polynomials approximation and finite difference method. The Caputo fractional derivative is replaced by a difference quotient and the integral by a finite sum. By this method the given problem is reduced to a problem for solving a system of algebraic equations, and by solving this system, we obtain the solution of FRDE. Special attention is given to study the convergence analysis and estimate an error upper bound of the obtained approximate formula. Illustrative examples are included to demonstrate the validity and applicability of the proposed technique.     

A finite difference scheme for two-dimensional semiconductor device of heat conduction on composite triangular grids

The momentary state of a semiconductor device of heat conduction is described by a system of four nonlinear partial differential equations. One elliptic equation is for the electrostatic, two parabolic equations are for the electron concentration and the hole concentration, and one heat exchange equation is for the temperature. According to the necessary of practical numerical simulations and based on the balance equation, finite difference schemes for two-dimensional transient behavior of a semiconductor device of heat conduction on composite triangular grids are constructed. Studying their stability and convergence properties, the error estimate in the energy norm is obtained. Finally, a numerical example is given.     

A multigrid compact finite difference method for solving the one‐dimensional nonlinear sine‐Gordon equation

The aim of this paper is to propose a multigrid method to obtain the numerical solution of the one‐dimensional nonlinear sine‐Gordon equation. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a compact finite difference scheme of fourth‐order for discretizing the spatial derivative and the standard second‐order central finite difference method for the time derivative. The proposed method uses the Richardson extrapolation method in time variable. The obtained system has been solved by V‐cycle multigrid (VMG) method, where the VMG method is used for solving the large sparse linear systems. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional sine‐Gordon equation. Copyright © 2014 John Wiley & Sons, Ltd.     

Influence of several factors in the generalized finite difference method

In this paper it is possible to appreciate the great efficiency of the generalized finite difference method (GFD), that is to say with an irregular arrangements of nodes, to solve second-order partial differential equations which represent the behaviour of many physical processes. The method solves any type of second-order differential equation, in any type of domain and boundary condition (Dirichlet, Neumann and mixed), and immediately obtains the values of derivatives of the nodes through the application of the formulae in differences obtained. This paper analyses the influences of key parameters of the method, such as the number of nodes of the star, the arrangement of the same, the weight function and the stability parameter in time-dependent problems. This analysis includes solutions obtained for different types of problems, represented by different differential equations, including time-dependent equations and under different boundary conditions.     

Mixed two‐grid finite difference methods for solving one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations

The aim of this paper is to propose mixed two‐grid finite difference methods to obtain the numerical solution of the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. The finite difference equations at all interior grid points form a large‐sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a family of finite difference methods for discretizing the spatial and time derivatives. The obtained system has been solved by two‐grid method, where the two‐grid method is used for solving the large‐sparse linear systems. Also, in the proposed method, the spectral radius with local Fourier analysis is calculated for different values of h and Δt. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Method of lines solutions of the parabolic inverse problem with an overspecification at a point

The present work is motivated by the desire to obtain numerical solution to a quasilinear parabolic inverse problem. The solution is presented by means of the method of lines. Method of lines is an alternative computational approach which involves making an approximation to the space derivatives and reducing the problem to a system of ordinary differential equations in the variable time, then a proper initial value problem solver can be used to solve this ordinary differential equations system. Some numerical examples and also comparison with finite difference methods will be investigated to confirm the efficiency of this procedure.     

A fast and efficient two-grid method for solving d-dimensional poisson equations

The aim of this paper is to introduce a fast and efficient new two-grid method to solve the d-dimensional (d=1,2,3) Poisson elliptic equations. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The finite difference equations are based on applying a finite difference scheme of two- and four-orders (compact finite difference method) for discretizing the spatial derivative. The obtained linear systems of Poisson elliptic equations have been solved by a new two-grid (NTG) method and we also note that the NTG method which is used for solving the large sparse linear systems is faster and more effective than that of the standard two-grid method. We utilize the local Fourier analysis to show that the spectral radius of the new two-grid method for 1D and 2D models is less than that of the standard two-grid method. As well as, we expand the corresponding algorithm to the new multi-grid method. The numerical examples show the efficiency of the new algorithms for solving the d-dimensional Poisson equations.     

Three-layer finite difference method for solving linear differential algebraic systems of partial differential equations

A boundary value problem is examined for a linear differential algebraic system of partial differential equations with a special structure of the associate matrix pencil. The use of an appropriate transformation makes it possible to split such a system into a system of ordinary differential equations, a hyperbolic system, and a linear algebraic system. A three-layer finite difference method is applied to solve the resulting problem numerically. A theorem on the stability and the convergence of this method is proved, and some numerical results are presented.     

Numerical performance of parallel group explicit solvers for the solution of fourth order elliptic equations

Many applications in applied mathematics and engineering involve numerical solutions of partial differential equations (PDEs). Various discretisation procedures such as the finite difference method result in a problem of solving large, sparse systems of linear equations. In this paper, a group iterative numerical scheme based on the rotated (skewed) five-point finite difference discretisation is proposed for the solution of a fourth order elliptic PDE which represents physical situations in fluid mechanics and elasticity. The rotated approximation formulas lead to schemes with lower computational complexities compared to the centred approximation formulas since the iterative procedure need only involve nodes on half of the total grid points in the solution domain. We describe the development of the parallel group iterative scheme on a cluster of distributed memory parallel computer using Message-Passing Interface (MPI) programming environment. A comparative study with another group iterative scheme derived from the centred difference formula is also presented. A detailed performance analysis of the parallel implementations of both group methods will be reported and discussed.     

在动载荷作用下框架结构大变形分析的微分代数方法

采用弧坐标首先建立了在动载荷作用下,具有不连续性条件和初始位移的框架结构大变形分析的非线性数学模型．其次, 在空间区域内, 采用微分求积单元法（DQEM）来离散非线性数学模型, 并提出了在使用DQEM来求解结构大变形分析中,多个变量具有间断性条件的有效方法,得到了一组非线性DQEM的离散化方程,它是时间域内的一组具有奇异性的非线性微分-代数方程．同时也给出了求解非线性微分-代数方程组的一个解法A·D2作为应用,求解了受集中力和分布力作用的框架和组合框架的大变形静动力学问题,并与现有结果进行了比较．数值算例表明,处理多个变量具有间断性条件的方法和求解代数-微分系统的方法是一个有效的和一般的方法,它具有较少的节点、 较小的计算工作量、 较高的精度、良好的收敛性、 操作简单以及应用广泛等优点．     

Solution for a problem of linear plane elasticity with mixed boundary conditions on an ellipse by the method of boundary integrals

A numerical boundary integral scheme is proposed for the solution of the system of field equations of plane, linear elasticity in stresses for homogeneous, isotropic media in the domain bounded by an ellipse under mixed boundary conditions. The stresses are prescribed on one half of the ellipse, while the displacements are given on the other half. The method relies on previous analytical work within the Boundary Integral Method [1], [2].The considered problem with mixed boundary conditions is replaced by two subproblems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way and the problem at this stage is reduced to the solution of a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution inside the domain, and the unknown boundary values of stresses or displacements on proper parts of the boundary.On the basis of the obtained results, it is inferred that the tangential stress component on the fixed part of the boundary has a singularity at each of the two separation points, thought to be of logarithmic type. A tentative form for the singular solution is proposed to calculate the full solution in bulk directly from the given boundary conditions using the well-known Boundary Collocation Method. It is shown that this addition substantially decreases the error in satisfying the boundary conditions on some interval not containing the singular points.The obtained results are discussed and boundary curves for unknown functions are provided, as well as three-dimensional plots for quantities of practical interest. The efficiency of the used numerical schemes is discussed, in what concerns the number of boundary nodes needed to calculate the approximate solution.