A CQM-based BEM for transient heat conduction problems in homogeneous materials and FGMs

A method for numerical solution of time-domain boundary integral formulations of transient problems governed by the heat equation is presented. The heat conduction problem is analyzed considering homogeneous and non-homogeneous media. In the case of the non-homogeneous media, the conductor material is assumed to be a functionally graded material, i.e., the material properties vary spatially according to known smooth functions. For some specific spatial variations of the material properties, the fundamental solution and the boundary integral equation of the problem are obtained thanks to a change of variables that transforms the original problem to the standard heat conduction problem for homogeneous materials. For the treatment of time-dependent terms, the convolution quadrature method is adopted to approximate numerically the integral equation of the time-domain boundary element method. In the case that the responses are required at a large number of interior points, the convolution performed to calculate them is very time consuming. It is shown that the discrete convolution of the proposed formulation can be computed by means of the fast Fourier transform technique, which considerably reduces the computational complexity. Results for some transient heat conduction examples are presented to validate the numerical techniques studied.     

Decay mode solution of nonlinear boundary–initial value problems for the cylindrical (spherical) Boussinesq–Burgers equations

The major target of this paper is to construct new nonlinear boundary–initial value problems for Boussinesq–Burgers Equations, and derive the solutions of these nonlinear boundary–initial value problems by the simplified homogeneous balance method. The nonlinear transformation and its inversion between the Boussinesq–Burgers Equations and the linear heat conduction equation are firstly derived; then a new nonlinear boundary–initial value problem for the Boussinesq–Burgers equations with variable damping on the half infinite straight line is put forward for the first time, and the solution of this nonlinear boundary–initial value problem is obtained, especially, the decay mode solution of nonlinear boundary–initial value problem for the cylindrical (spherical) Boussinesq–Burgers equations is obtained.     

A full multigrid method for nonlinear eigenvalue problems

We introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of the nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and nonlinear eigenvalue problems on the coarsest finite element space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as the linear problem solver. We prove that the computational work of this new scheme is truly optimal, the same as solving the linear corresponding boundary value problem. In this case, this type of iteration scheme certainly improves the overfull efficiency of solving nonlinear eigenvalue problems. Some numerical experiments are presented to validate the efficiency of the new method.     

A full multigrid method for semilinear elliptic equation

A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient numerical methods can also serve as the linear solver for solving boundary value problems. The optimality of the computational work is also proved. Compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only needs the Lipschitz continuation in some sense of the nonlinear term.     

The method of differential quadrature for transient nonlinear diffusion

Numerical solutions to transient nonlinear diffusion problems are obtained by the method of differential quadrature. The accuracy of the solutions is inferred by comparison with the analytical solution for the linear case. The particular problems associated with general boundary conditions are handled by the use of integral methods. Examples from heat diffusion include composite media and radiation-enhanced conduction.     

移动边界半线性热传导方程始边值问题分析近似解

本文应用Fourier方法求得移动边界非齐次线性热传导方程始边值问题解及半线性方程问题分析近似解     

Nonlinear heat equation for nonhomogeneous anisotropic materials: A dual‐reciprocity boundary element solution

A dual‐reciprocity boundary element method is presented for the numerical solution of initial‐boundary value problems governed by a nonlinear partial differential equation for heat conduction in nonhomogeneous anisotropic materials. To assess the validity and accuracy of the method, some specific problems are solved. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

正交各向异性功能梯度层合板稳态热传导问题的精确解

对轴对称正交各向异性功能梯度层合圆板稳态热传导问题进行精确分析.假设材料热传导率沿板厚方向按指数函数形式梯度分布,从正交各向异性功能梯度圆板稳态热传导的基本方程出发,利用分离变量法,获得了在上、下表面作用任意热分布情况下的精确解.通过数值算例的分析,指出材料性质的梯度变化、板厚边界条件等分析了对温度场分布的影响.所获得的精确结果,可以作为评价其它近似方法的标准解答.     

An initial boundary value problem for a pseudoparabolic equation with a nonlinear boundary condition

An initial boundary value problem for a quasilinear equation of pseudoparabolic type with a nonlinear boundary condition of the Neumann–Dirichlet type is investigated in this work. From a physical point of view, the initial boundary value problem considered here is a mathematical model of quasistationary processes in semiconductors and magnets, which takes into account a wide variety of physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions in problems where the boundary conditions are linear with respect to the desired function and its derivatives. Among these methods, the Galerkin method leads to the simplest calculations. On the basis of a priori estimates, we prove a local existence theorem and uniqueness for a weak generalized solution of the initial boundary value problem for the quasilinear pseudoparabolic equation. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, or, as they are called in another way, blow-up regimes. Nonlinear evolutionary problems admitting unbounded solutions are globally unsolvable. In the article, sufficient conditions for the blow-up of a solution in a finite time in a limited area with a nonlinear Neumann–Dirichlet boundary condition are obtained.     

Novel numerical techniques based on Fokas transforms,for the solution of initial boundary value problems

The unified transform method of A. S. Fokas has led to important new developments, regarding the analysis and solution of various types of linear and nonlinear PDE problems. In this work we use these developments and obtain the solution of time-dependent problems in a straightforward manner and with such high accuracy that cannot be reached within reasonable time by use of the existing numerical methods. More specifically, an integral representation of the solution is obtained by use of the A. S. Fokas approach, which provides the value of the solution at any point, without requiring the solution of linear systems or any other calculation at intermediate time levels and without raising any stability problems. For instance, the solution of the initial boundary value problem with the non-homogeneous heat equation is obtained with accuracy 10⁻¹⁵, while the well-established Crank–Nicholson scheme requires 2048 time steps in order to reach a 10⁻⁸ accuracy.     

Approximating solutions to linear and nonlinear differential equations by the method of maximum entropy

A technique to approximate the solution to linear and nonlinear boundary value problems is developed and numerical examples are presented. The technique is based on the method of maximum entropy with moments of the differential equation used as constraints. The method is very general and has the advantage that additional information can be fed into the solution, such as the function's domain or the positivity or negativity of the solution. The technique should find applications in approximating solutions to equations which may or may not contain noise and as an alternative to finite difference and Fourier series solutions and may have applications to large scale simulations.     

Solving a nonlinear system of second order boundary value problems

In this paper, a method is presented to obtain the analytical and approximate solutions of linear and nonlinear systems of second order boundary value problems. The analytical solution is represented in the form of series in the reproducing kernel space. In the mean time, the approximate solution un(x) is obtained by the n-term intercept of the analytical solution and is proved to converge to the analytical solution. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.     

Numerical method for solving the nonlinear four-point boundary value problems

In this paper, a new reproducing kernel space is constructed skillfully in order to solve a class of nonlinear four-point boundary value problems. The exact solution of the linear problem can be expressed in the form of series and the approximate solution of the nonlinear problem is given by the iterative formula. Compared with known investigations, the advantages of our method are that the representation of exact solution is obtained in a new reproducing kernel Hilbert space and accuracy of numerical computation is higher. Meanwhile we present the convergent theorem, complexity analysis and error estimation. The performance of the new method is illustrated with several numerical examples.     

An effective approach for numerical solution of linear and nonlinear singular boundary value problems

In this study, an effective approach is presented to obtain a numerical solution of linear and nonlinear singular boundary value problems. The proposed method is constructed by combining reproducing kernel and Legendre polynomials. Legendre basis functions are used to get the kernel function, and then the approximate solution is obtained as a finite series sum. Comparison of numerical results is made with the results obtained by other methods available in the literature. Furthermore, efficiency and accuracy of the method are demonstrated in tabulated results and plotted graphs. The numerical outcomes demonstrate that our method is very effective, applicable, and convenient.     

Numerical treatment of singularly perturbed two point boundary value problems using initial-value method

In this paper, we describe an initial-value method for linear and nonlinear singularly perturbed boundary value problems in the interval [p,q]. For linear problems, the required approximate solution is obtained by solving the reduced problem and one initial-value problems directly deduced from the given problem. For nonlinear problems the original second-order nonlinear problem is linearized by using quasilinearization method. Then this linear problem is solved as previous method. The present method has been implemented on several linear and non-linear examples which approximate the exact solution. We also present the approximate and exact solutions graphically.     

Nonlinear analysis of buckling,free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment

Employing Euler–Bernoulli beam theory and the physical neutral surface concept, the nonlinear governing equation for the functionally graded material beam with two clamped ends and surface-bonded piezoelectric actuators is derived by the Hamilton’s principle. The thermo-piezoelectric buckling, nonlinear free vibration and dynamic stability for the piezoelectric functionally graded beams, subjected to one-dimensional steady heat conduction in the thickness direction, are studied. The critical buckling loads for the beam are obtained by the existing methods in the analysis of thermo-piezoelectric buckling. The Galerkin’s procedure and elliptic function are adopted to obtain the analytical solution of the nonlinear free vibration, and the incremental harmonic balance method is applied to obtain the principle unstable regions of the piezoelectric functionally graded beam. In the numerical examples, the good agreements between the present results and existing solutions verify the validity and accuracy of the present analysis and solving method. Simultaneously, validation of the results achieved by rule of mixture against those obtained via the Mori–Tanaka scheme is carried out, and excellent agreements are reported. The effects of the thermal load, electric load, and thermal properties of the constituent materials on the thermo-piezoelectric buckling, nonlinear free vibration, and dynamic stability of the piezoelectric functionally graded beam are discussed, and some meaningful conclusions have been drawn.     

Hybrid differential transform and finite difference method to solve the nonlinear heat conduction problem

In this paper, the differential transform is employed to discuss the behaviors of nonlinear heat conduction problem. A hybrid method of differential transform and finite difference approach is proposed to solve the transient responses of a nonlinear heat conduction problem. Different parameters of the equation and boundary conditions are considered to verify the feasibility of the proposed method to such problems. Simulation results are illustrated and discussed in comparison with the linear case. The results show that the hybrid method can achieve good results for such problems.     

Analytic approximate solutions of parameterized unperturbed and singularly perturbed boundary value problems

A novel approach is presented in this paper for approximate solution of parameterized unperturbed and singularly perturbed two-point boundary value problems. The problem is first separated into a simultaneous system regarding the unknown function and the parameter, and then a methodology based on the powerful homotopy analysis technique is proposed for the approximate analytic series solutions, whose convergence is guaranteed by optimally chosen convergence control parameters via square residual error. A convergence theorem is also provided. Several nonlinear problems are treated to validate the applicability, efficiency and accuracy of the method. Vicinity of the boundary layer is shown to be adequately treated and satisfactorily resolved by the method. Advantages of the method over the recently proposed conventional finite-difference or Runga–Kutta methods are also discussed.     

Multigrid method for coupled semilinear elliptic equation

This paper introduces a kind of multigrid finite element method for the coupled semilinear elliptic equations. Instead of the common way of directly solving the coupled semilinear elliptic problems on some fine spaces, the presented method transforms the solution of the coupled semilinear elliptic problem into a series of solutions of the corresponding decoupled linear boundary value problems on the sequence of multilevel finite element spaces and some coupled semilinear elliptic problems on a very low dimensional space. The decoupled linearized boundary value problems can be solved by some multigrid iterations efficiently. The optimal error estimate and optimal computational work are proved theoretically and demonstrated numerically. Moreover, the requirement of bounded second‐order derivatives of the nonlinear term in the existing multigrid method is reduced to a Lipschitz continuous condition in the proposed method.     

A finite difference scheme for nonlinear ultra-parabolic equations

In this paper, our aim is to study a numerical method for an ultraparabolic equation with nonlinear source function. Mathematically, the bibliography on initial–boundary value problems for ultraparabolic equations is not extensive although the problems have many applications related to option pricing, multi-parameter Brownian motion, population dynamics and so forth. In this work, we present the approximate solution by virtue of finite difference scheme and Fourier series. For the nonlinear case, we use an iterative scheme by linear approximation to get the approximate solution and obtain error estimates. A numerical example is given to justify the theoretical analysis.