An analytical approach to the unified solution of kinetic equations in the rarefied gas dynamics. II. Heat transfer problems

The ADO method, an analytical version of the discrete-ordinates method, is used here to solve a heat-transfer problem in a rarefied gas confined in a channel, as well as to solve a half-space problem in order to evaluate the temperature jump at the wall. This work is an extension of a previous work, devoted to flow problems, where the complete development of the solution, which is analytical in terms of the spatial variable, is presented in a way, such that, a wide class of kinetic models are considered, in an unified approach. A series of numerical results are showed and different simulations are used in order to establish a general comparative analysis based on this consistent set of results provided by the same methodology. In particular, numerical results for heat-flow profile, temperature and density perturbations are obtained for channels (walls), defined by different materials, on which different temperatures are imposed.     

An analytical approach to the unified solution of kinetic equations in rarefied gas dynamics. III. Evaporation and condensation problems

An analytical version of the discrete-ordinates method, the ADO method, is used here to solve two problems in the rarefied gas dynamics field, that describe evaporation/condensation between two parallel interfaces and the case of a semi-infinite medium. The modeling of the problems is based on a general expression which may represent four different kinetic models, derived from the linearized Boltzmann equation. This work is an extension of two other previous works, devoted to rarefied gas flow and heat transfer problems, where the complete development of the ADO solution, which is analytical in terms of the spatial variable, is presented in a way, such that, the four kinetic models are considered, in an unified approach. A series of numerical results are showed in order to establish a general comparative analysis between this consistent set of results provided by the same methodology, based on kinetic models, and results obtained from the linearized Boltzmann equation. In particular, the temperature and density jumps are evaluated.     

An analytical approach to the unified solution of kinetic equations in rarefied gas dynamics. III. Evaporation and condensation problems

An analytical version of the discrete-ordinates method, the ADO method, is used here to solve two problems in the rarefied gas dynamics field, that describe evaporation/condensation between two parallel interfaces and the case of a semi-infinite medium. The modeling of the problems is based on a general expression which may represent four different kinetic models, derived from the linearized Boltzmann equation. This work is an extension of two other previous works, devoted to rarefied gas flow and heat transfer problems, where the complete development of the ADO solution, which is analytical in terms of the spatial variable, is presented in a way, such that, the four kinetic models are considered, in an unified approach. A series of numerical results are showed in order to establish a general comparative analysis between this consistent set of results provided by the same methodology, based on kinetic models, and results obtained from the linearized Boltzmann equation. In particular, the temperature and density jumps are evaluated.     

An analytical approach to the unified solution of kinetic equations in rarefied gas dynamics. I. Flow problems

The ADO method, an analytical version of the discrete-ordinates method, is used to solve several classical problems in the rarefied gas dynamics field. The complete development of the solution, which is analytical in terms of the spatial variable, is presented in a way, such that, a wide class of kinetic models are considered, in an unified approach. A series of numerical results are showed and different simulations are used in order to establish a general comparative analysis based on this consistent set of results provided by the same methodology.     

An analytical approach to the unified solution of kinetic equations in rarefied gas dynamics. I. Flow problems

The ADO method, an analytical version of the discrete-ordinates method, is used to solve several classical problems in the rarefied gas dynamics field. The complete development of the solution, which is analytical in terms of the spatial variable, is presented in a way, such that, a wide class of kinetic models are considered, in an unified approach. A series of numerical results are showed and different simulations are used in order to establish a general comparative analysis based on this consistent set of results provided by the same methodology. Received: July 10, 2007; revised: October 29/December 4, 2007     

对流扩散方程的QC3无网格法

对流扩散方程作为偏微分运动方程的分支,在流体力学、气体动力学等领域有着重要应用.为解决对流扩散方程难以通过解析法得到解析解的难题,采用二阶一致3点积分(Quadratically Consistent 3-Point Integration,简称QC3)提高无网格法的计算效率,通过对积分点上形函数导数的修正,改善无网格...     

Error analysis of the numerical solution of split differential equations

The operator splitting method is a widely used approach for solving partial differential equations describing physical processes. Its application usually requires the use of certain numerical methods in order to solve the different split sub-problems. The error analysis of such a numerical approach is a complex task. In the present paper we show that an interaction error appears in the numerical solution when an operator splitting procedure is applied together with a lower-order numerical method. The effect of the interaction error is investigated by an analytical study and by numerical experiments made for a test problem.     

Mobile and immobile gaseous transport: Embedded analytical solutions to finite volume methods

The motivation is driven by deposition processes based on chemical vapor problems. The underlying model problem is based on coupled transport–reaction equations with mobile and immobile areas. We deal with systems of ordinary and partial differential equations. Such equation systems are delicate to solve and we introduce a novel solver method, that takes into account ways to solve analytically parts of the transport and reaction equations. The main idea is to embed the analytical and semianalytical solutions, which can then be explicitly given to standard numerical schemes of higher order. The numerical scheme is based on flux‐based characteristic methods, which is a finite volume method. Such a method is an attractive alternative to the standard numerical schemes, which fully discretize the full equations. We instead reduce the computational time while embedding fast computable analytical parts. Here, we can accelerate the solver process, with a priori explicitly given solutions. We will focus on the derivation of the analytical solutions for general and special solutions of the characteristic methods that are embedded into a finite volume method. In the numerical examples, we illustrate the higher‐order method for different benchmark problems. Finally, the method is verified with realistic results. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

Optimization technique in solving laminar boundary layer problems with temperature dependent viscosity

In this paper, a novel numerical method is proposed to solve specific third order ODE on semi-infinite interval. These kinds of problems often occur in laminar boundary layer with temperature dependent viscosity. Runge-Kutta method incorporating with optimization techniques is used to solve the problem. First, the semi-infinite interval is transformed into a finite interval. Second, by converting the boundary value problem, with some initial and distributed unknowns, into an optimization problem, solving the original problem is limited to solving a multiobjective optimization problem. Third, we use shooting-Newton’s method for solving this optimization problem. It is shown that the Falkner-Skan problem with constant surface temperature, that arise during the solution for the laminar forced convection heat transfer from wedges to flow, can be solved accurately and simultaneously by this strategy. Numerical results for different values of wedge angle and Prandtl number are presented, which are in good agreement with some of the successful provided solutions in the literature.     

The temperature-jump problem based on the CES model of the linearized Boltzmann equation

An analytical discrete-ordinates method is used to solve the temperature-jump problem as defined by a synthetic-kernel model of the linearized Boltzmann equation. In particular, the temperature and density perturbations and the temperature-jump coefficient defined by the CES model equation are obtained (essentially) analytically in terms of a modern version of the discrete-ordinates method. The developed algorithms are implemented for general values of the accommodation coefficient to yield numerical results that compare well with solutions derived from more computationally intensive techniques.     

The temperature-jump problem based on the CES model of the linearized Boltzmann equation

An analytical discrete-ordinates method is used to solve the temperature-jump problem as defined by a synthetic-kernel model of the linearized Boltzmann equation. In particular, the temperature and density perturbations and the temperature-jump coefficient defined by the CES model equation are obtained (essentially) analytically in terms of a modern version of the discrete-ordinates method. The developed algorithms are implemented for general values of the accommodation coefficient to yield numerical results that compare well with solutions derived from more computationally intensive techniques.     

Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method

This paper investigates the numerical solutions of singular second order three-point boundary value problems using reproducing kernel Hilbert space method. It is a relatively new analytical technique. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel Hilbert space method cannot be used directly to solve a singular second order three-point boundary value problem, so we convert it into an equivalent integro-differential equation, which can be solved using reproducing kernel Hilbert space method. Four numerical examples are given to demonstrate the efficiency of the present method. The numerical results demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems.     

A low-order meshless model for multidimensional heat conduction problems

In this paper, a perturbation method is used to solve a two-dimensional unsteady heat conduction problem. Low-order transfer functions are defined. Step responses are obtained and compared to the complete numerical solutions given by a meshless method. The analytical results are found to be in good agreement with numerical solutions which reveals the effectiveness and convenience of the used method.     

Higher order optimization and adaptive numerical solution for optimal control of monodomain equations in cardiac electrophysiology

In this work adaptive and high resolution numerical discretization techniques are demonstrated for solving optimal control of the monodomain equations in cardiac electrophysiology. A monodomain model, which is a well established model for describing the wave propagation of the action potential in the cardiac tissue, will be employed for the numerical experiments. The optimal control problem is considered as a PDE constrained optimization problem. We present an optimal control formulation for the monodomain equations with an extra-cellular current as the control variable which must be determined in such a way that excitations of the transmembrane voltage are damped in an optimal manner.The focus of this work is on the development and implementation of an efficient numerical technique to solve an optimal control problem related to a reaction-diffusions system arising in cardiac electrophysiology. Specifically a Newton-type method for the monodomain model is developed. The numerical treatment is enhanced by using a second order time stepping method and adaptive grid refinement techniques. The numerical results clearly show that super-linear convergence is achieved in practice.     

Analytical description of the radiative-conductive heat transfer in a gray medium contained between two diffuse parallel plates

Explicit analytical solutions for the temperature and heat flux in a gray medium contained between two diffuse parallel plates are derived for both pure thermal radiation and coupled conduction-radiation heat transfer. This is achieved by combining the integral equations for the heat flux and temperature predicted by the radiative transfer equation with the corresponding predictions of the discrete ordinates method. The algebraic formulation of this well-known method is used to derive analytical results that agree with their corresponding numerical ones with an accuracy greater than 99.9%, for a large interval of optical thicknesses and conduction-to-radiation factors. The explicit and original solutions, for both pure radiation and radiative-conductive heat transfer, therefore solve the problem of one dimensional steady-state heat transfer in gray cavities.     

A variational iteration method for solving Troesch’s problem

Troesch’s problem is an inherently unstable two-point boundary value problem. A new and efficient algorithm based on the variational iteration method and variable transformation is proposed to solve Troesch’s problem. The underlying idea of the method is to convert the hyperbolic-type nonlinearity in the problem into polynomial-type nonlinearities by variable transformation, and the variational iteration method is then directly used to solve this transformed problem. Only the second-order iterative solution is required to provide a highly accurate analytical solution as compared with those obtained by other analytical and numerical methods.     

Use of Adomian decomposition method in the study of parallel plate flow of a third grade fluid

In this paper, Adomian’s decomposition method is used to solve non-linear differential equations which arise in fluid dynamics. We study basic flow problems of a third grade non-Newtonian fluid between two parallel plates separated by a finite distance. The technique of Adomian decomposition is successfully applied to study the problem of a non-Newtonian plane Couette flow, fully developed plane Poiseuille flow and plane Couette–Poiseuille flow. The results obtained show the reliability and efficiency of this analytical method. Numerical solutions are also obtained by solving non-linear ordinary differential equations using Chebyshev spectral method. We present a comparative study between the analytical solutions and numerical solutions. The analytical results are found to be in good agreement with numerical solutions which reveals the effectiveness and convenience of the Adomian decomposition method.     

Exact analytical solution of the Cauchy problem for a linear reaction–diffusion equation with time-dependent coefficients and space–time-dependent source term

The present work derives the exact analytical solution of the Cauchy problem for a linear reaction–diffusion equation with time-dependent coefficients and space–time-dependent source term. The work also emphasizes the role of reaction–diffusion models as important particular cases of much more general equations in the kinetic theory of active particles. The analytical expression derived shows the structure of the solution and the contributions of different terms of the model to it. The result obtained enables one to solve the Cauchy problem indicated by using the exact analytical representation rather than numerical methods, which are usually time-consuming, especially when the number of spatial dimensions is greater than 2.     

Investigation of a problem of an elastic half space subjected to stochastic temperature distribution at the boundary

The present work investigates the responses of stochastic type temperature distribution applied at the boundary of an elastic medium in the context of thermoelasticity without energy dissipation. We consider an one dimensional problem of half space and assume that the bounding surface of the half space is traction free and is subjected to two types of time dependent temperature distributions which are of stochastic types. In order to compare the results predicted by stochastic temperature distributions with the results of deterministic type temperature distribution, the stochastic type temperature distributions applied at the boundary are taken in such a way that they reduce to the cases of deterministic types as special cases. Integral transform technique along with stochastic calculus is used to solve the problem. The approximated solutions for physical fields like, stress, temperature, displacement etc. are derived for very small values of time where stochastic type boundary conditions are taken to be of white noise type. The problem is further illustrated with graphical representation of numerical solutions of the problem for a particular case. A detailed comparison of the results of stochastic temperature, displacement and stress distributions inside the half space with the corresponding results of deterministic distributions is presented and special features of the effects of stochastic type boundary conditions are highlighted.     

Homotopy perturbation-reproducing kernel method for nonlinear systems of second order boundary value problems

In this paper, based on homotopy perturbation method (HPM) and reproducing kernel method (RKM), a new method is presented for solving nonlinear systems of second order boundary value problems (BVPs). HPM is based on the use of traditional perturbation method and homotopy technique. The HPM can reduce a nonlinear problem to a sequence of linear problems and generate a rapid convergent series solution in most cases. RKM is also an analytical technique, which can solve powerfully linear BVPs. Homotopy perturbation-reproducing kernel method (HP-RKM) combines advantages of these two methods and therefore can be used to solve efficiently systems of nonlinear BVPs. Three numerical examples are presented to illustrate the strength of the method.