Numerical method of lines for parabolic functional differential equations

Initial boundary value problems for nonlinear parabolic functional differential equations are transformed by discretization in space variables into systems of ordinary functional differential equations. A comparison theorem for differential difference inequalities is proved. Sufficient conditions for the convergence of the numerical method of lines are given. An explicit Euler method is proposed for the numerical solution of systems thus obtained. This leads to difference scheme for the original problem. A complete convergence analysis for the method is given.     

Analytic and Experimental Studies of the Errors in Numerical Methods for the Valuation of Options

The value of a European option satisfies the Black-Scholes equation with appropriately specified final and boundary conditions.We transform the problem to an initial boundary value problem in dimensionless form.There are two parameters in the coefficients of the resulting linear parabolic partial differential equation.For a range of values of these parameters,the solution of the problem has a boundary or an initial layer.The initial function has a discontinuity in the first-order derivative,which leads to the appearance of an interior layer.We construct analytically the asymptotic solution of the equation in a finite domain.Based on the asymptotic solution we can determine the size of the artificial boundary such that the required solution in a finite domain in x and at the final time is not affected by the boundary.Also,we study computationally the behaviour in the maximum norm of the errors in numerical solutions in cases such that one of the parameters varies from finite (or pretty large) to small values,while the other parameter is fixed and takes either finite (or pretty large) or small values. Crank-Nicolson explicit and implicit schemes using centered or upwind approximations to the derivative are studied.We present numerical computations,which determine experimentally the parameter-uniform rates of convergence.We note that this rate is rather weak,due probably to mixed sources of error such as initial and boundary layers and the discontinuity in the derivative of the solution.     

Semi-analytical method for solving nonlinear heat diffusion problems in spherical medium

A semi-analytical methodology, based on the finite integral transform technique, is proposed to solve the heat diffusion problem in a spherical medium subject to nonlinear boundary conditions due to radiation exchange at the interface according to the fourth power law. The method proceeds by treating the nonlinearity term in the boundary condition as a source in the differential equation and keeping other conditions unchanged. The results obtained from this semi-analytical solutions are compared with those obtained from a numerical solution developed using an explicit finite difference method, which showed very good agreement.     

Boundary triples and quasi boundary triples for elliptic operators

The abstract concepts of boundary triples and their recent generalizations are useful tools to parametrize the self-adjoint and maximal dissipative/maximal accumulative extensions of formally symmetric elliptic differential expressions with the help of explicit boundary conditions. In the present note the parametrizations induced by the “natural” quasi boundary triple with the Dirichlet and Neumann trace as boundary maps are compared with the parametrizations induced by a “classical” ordinary boundary triple, where a regularized Neumann trace is used for one of the boundary maps. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

二维非线性Schr\"{o}dinger方程显式格式的最大模误差分析

本文讨论了数值求解二维非线性Schr\"{o}dinger方程周期边值问题的Du Fort-Frankel格式和蛙跳格式. 以解函数的一个广义时间导数作为独立变量, 将非线性方程初边值问题改写成一个混合方程组形式, 应用我们最新提出的离散能量技巧讨论这两个三层显式格式的收敛性. 分析表明, 在必要的网格条件下, 差分解在最大模意义下二阶收敛. 数值算例验证了理论分析结果.     

A Side-by-Side Single Sex Age-Structured Human Population Dynamic Model: Exact Solution and Model Validation

A side-by-side single sex age-structured population dynamic model is presented in this paper. The model consists of two coupled von Foerster-McKendrick-type quasi-linear partial differential equations, two initial conditions, and two boundary conditions. The state variables of the model are male and female population densities. The solutions of these partial differential equations provide explicit time and age dependence of the variables. The initial conditions define the male and female population densities at the initial time, while the boundary conditions compute the male and female births at zero-age by using fertility rates. The assumptions of the nontime-dependence of the death and fertility rates and a specific factorization of the migratory balances allow us to obtain exact solutions for male and female population densities. In addition, the hypotheses about the mathematical structure of the input variables are formulated, and the exact solution of the model is obtained. Next, the model is applied to the case study of Spain for the time period 1996–2004. Model validation demonstrates that this approach is a powerful prediction tool. Code and data are available upon request.     

Computation of moment functions for parabolic PDEs with random parameters

This paper deals with different approaches for the computation of stochastic characteristics (such as mean, variance, correlation function, …) of the solution to parabolic PDEs with random parameters, in particular with a random Neumann boundary conditions and a random initial condition. Thereby Finite-Element discretisation of the spatial variables is used for the construction of pathwise solutions. The random influences are assumed to be ε -correlated (cp. [4]) that means the correlation functions vanish if the difference of the arguments exceeds a given correlation length ε . So an asymptotic expansion of higher order with respect to the correlation length for the correlation function of the approximative solution is given. A second possibility is the so called explicit calculation. Examples which compare the different methods can be found in [2]. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a price formation free boundary model by Lasry and Lions: The Neumann problem

We discuss local and global existence and uniqueness for the price formation free boundary model with homogeneous Neumann boundary conditions introduced by Lasry and Lions in 2007. The results are based on a transformation of the problem to the heat equation with nonstandard boundary conditions. The free boundary becomes the zero level set of the solution of the heat equation. The transformation allows us to construct an explicit solution and discuss the behavior of the free boundary. Global existence can be verified under certain conditions on the free boundary and examples of non-existence are given.     

Application of finite difference method of lines on the heat equation

In this article, we apply the method of lines (MOL) for solving the heat equation. The use of MOL yields a system of first–order differential equations with initial value. The solution of this system could be obtained in the form of exponential matrix function. Two approaches could be applied on this problem. The first approach is approximation of the exponential matrix by Taylor expansion, Padé and limit approximations. Using this approach leads to create various explicit and implicit finite difference methods with different stability region and order of accuracy up to six for space and superlinear convergence for time variables. Also, the second approach is a direct method which computes the exponential matrix by applying its eigenvalues and eigenvectors analytically. The direct approach has been applied on one, two and three‐dimensional heat equations with Dirichlet, Neumann, Robin and periodic boundary conditions.     

An explicit staggered-grid method for numerical simulation of large-scale natural gas pipeline networks

We present an explicit second order staggered finite difference (FD) discretization scheme for forward simulation of natural gas transport in pipeline networks. By construction, this discretization approach guarantees that the conservation of mass condition is satisfied exactly. The mathematical model is formulated in terms of density, pressure, and mass flux variables, and as a result permits the use of a general equation of state to define the relation between the gas density and pressure for a given temperature. In a single pipe, the model represents the dynamics of the density by propagation of a non-linear wave according to a variable wave speed. We derive compatibility conditions for linking domain boundary values to enable efficient, explicit simulation of gas flows propagating through a network with pressure changes created by gas compressors. We compare our staggered grid method with an explicit operator splitting method and a lumped element scheme, and perform numerical experiments to validate the convergence order of the new discretization approach. In addition, we perform several computations to investigate the influence of non-ideal equation of state models and temperature effects on pipeline simulations with boundary conditions on various time and space scales.     

Multipoint boundary value problems of Neumann type for functional differential equations

We obtain the expression of the explicit solution to a class of multipoint boundary value problems of Neumann type for linear ordinary differential equations and apply these results to study sufficient conditions for the existence of solution to linear functional differential equations with multipoint boundary conditions, considering the particular cases of equations with delay and integro-differential equations.     

Time-dependent boundary conditions for the 2-D linear anisotropic-viscoelastic wave equation

Wave propagation simulation requires a correct implementation of boundary conditions to avoid numerical instabilities. A boundary treatment based on characteristics, which includes as special cases more simple rheologies involving isotropy and elastic behavior, is applied to the anisotropic-viscoelastic wave equation. The method introduces the boundary conditions by specifying the values of the incoming variables, which depend on the solution outside the model volume. The formulation ends up with a wave equation for the boundaries that implicitly includes the boundary conditions. The examples illustrate common problems in geophysical modeling, including free surface and nonreflecting conditions. © 1994 John Wiley & Sons, Inc.     

Finite difference reaction-diffusion systems with coupled boundary conditions and time delays

This paper is concerned with finite difference solutions of a coupled system of reaction-diffusion equations with nonlinear boundary conditions and time delays. The system is coupled through the reaction functions as well as the boundary conditions, and the time delays may appear in both the reaction functions and the boundary functions. The reaction-diffusion system is discretized by the finite difference method, and the investigation is devoted to the finite difference equations for both the time-dependent problem and its corresponding steady-state problem. This investigation includes the existence and uniqueness of a finite difference solution for nonquasimonotone functions, monotone convergence of the time-dependent solution to a maximal or a minimal steady-state solution for quasimonotone functions, and local and global attractors of the time-dependent system, including the convergence of the time-dependent solution to a unique steady-state solution. Also discussed are some computational algorithms for numerical solutions of the steady-state problem when the reaction function and the boundary function are quasimonotone. All the results for the coupled reaction-diffusion equations are directly applicable to systems of parabolic-ordinary equations and to reaction-diffusion systems without time delays.     

The behavior of plasma with an arbitrary degree of degeneracy of electron gas in the conductive layer

We obtain an analytic solution of the boundary problem for the behavior (fluctuations) of an electron plasma with an arbitrary degree of degeneracy of the electron gas in the conductive layer in an external electric field. We use the kinetic Vlasov–Boltzmann equation with the Bhatnagar–Gross–Krook collision integral and the Maxwell equation for the electric field. We use the mirror boundary conditions for the reflections of electrons from the layer boundary. The boundary problem reduces to a one-dimensional problem with a single velocity. For this, we use the method of consecutive approximations, linearization of the equations with respect to the absolute distribution of the Fermi–Dirac electrons, and the conservation law for the number of particles. Separation of variables then helps reduce the problem equations to a characteristic system of equations. In the space of generalized functions, we find the eigensolutions of the initial system, which correspond to the continuous spectrum (Van Kampen mode). Solving the dispersion equation, we then find the eigensolutions corresponding to the adjoint and discrete spectra (Drude and Debye modes). We then construct the general solution of the boundary problem by decomposing it into the eigensolutions. The coefficients of the decomposition are given by the boundary conditions. This allows obtaining the decompositions of the distribution function and the electric field in explicit form.     

On an upwind difference scheme for strongly degenerate parabolic equations modelling the settling of suspensions in centrifuges and non-cylindrical vessels

We prove the convergence of an explicit monotone finite difference scheme approximating an initial-boundary value problem for a spatially one-dimensional quasilinear strongly degenerate parabolic equation, which is supplied with two zero-flux boundary conditions. This problem arises in a model of sedimentation–consolidation processes in centrifuges and vessels with varying cross-sectional area. We formulate the definition of entropy solution of the model in the sense of Kružkov and prove the convergence of the scheme to the unique BV entropy solution of the problem. The scheme and the model are illustrated by numerical examples.     

Leitmann’s direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales

The fundamental problem of the calculus of variations on time scales concerns the minimization of a delta-integral over all trajectories satisfying given boundary conditions. This includes the discrete-time, the quantum, and the continuous/classical calculus of variations as particular cases. In this note we follow Leitmann’s direct method to give explicit solutions for some concrete optimal control problems on an arbitrary time scale.     

Outer solution for elastic torsion by the method of boundary layer residual states

By the method of boundary layer residual state (BLRS), it is possible to specify the unknown parameters in the general form of the outer asymptotic solution of the governing differential equations for linear boundary value problems (BVP) without any reference to the inner asymptotic solutions of the same problem and the matching procedure. The method accomplishes this task by rationally assigning a portion of the prescribed boundary data to the outer solution. Specifically, the method requires certain weighted averages of the outer solution to be equal to the same averages of the data over the (localized) boundary where the data is prescribed. These weighted averages are consequences of a reciprocity relation inherent in the BVP and the stipulation that the difference between the outer solution and the exact solution (called the residual solution) of the BVP be a boundary layer phenomenon.¶The weighted average requirements are only necessary conditions for the residual state to be a boundary layer. Unfortunately, there are generally countably infinite number of (2) states, many more than the available degrees of freedom in the outer solution to satisfy them. We must show that there is no over-determination or non-uniqueness of the outer asymptotic solution, the abundance of necessary conditions notwithstanding. The present note describes an approach to assuring a well-specified outer solution (up to the expected accuracy) by way of the problem of Saint-Venant torsion. The same approach also also applies to other linear BVP, deducing the appropriate outer solution whenever the determination of the relevant inner solutions is not practical.     

A note on the solution of a class of two-point boundary value problems involving a mixed boundary condition

In this note, we consider a class of two-point boundary value problems involving a pa- rameter in one of the boundary conditions. We shall show that if we know the solution corresponding to a particular value of the parameter, then the solution for any other value of the parameter can be obtained by a simple algebraic method.     

On the existence for viscoelastodynamic problems with unilateral boundary conditions

This note deals with a damped wave equation and the evolution of a Kelvin–Voigt viscoelastic material, both problems being subject to unilateral boundary conditions. The functional properties of all the traces are precisely identified through Fourier analysis, which implies the existence of a solution satisfying almost everywhere the unilateral boundary conditions. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

含裂纹平面问题Erdogan基本解的显式表达

基本解是边界元法、基本解法和无网格法等数值方法的重要理论基础.在断裂问题中,采用含裂纹的基本解可以避免将裂纹表面作为边界条件,从而大大简化问题的求解.在复变函数表示的含裂纹平面问题Erdogan基本解的基础上,对Erdogan基本解的使用条件进行了注解,修正了Erdogan基本解的一些错误,并推导出Erdogan基本解中位移函数解答的显式表达形式.编写了基于Erdogan基本解显式表达的样条虚边界元法（spline fictitious boundary element method, SFBEM）计算程序,计算了具有复合边界条件平面问题的位移、应力和应力强度因子.数值算例结果表明了该文提出的Erdogan基本解显式表达形式的正确性.