Multivariate approximation by PDE splines

This work deals with an approximation method for multivariate functions from data constituted by a given data point set and a partial differential equation (PDE). The solution of our problem is called a PDE spline. We establish a variational characterization of the PDE spline and a convergence result of it to the function which the data are obtained. We estimate the order of the approximation error and finally, we present an example to illustrate the fitting method.     

A second-order splitting method for a finite difference scheme for the Sivashinsky equation

A finite difference method is analyzed for the approximation of the solution of the evolutionary fourth-order in space, Sivashinsky equation.We prove that the scheme has a unique solution and we study error estimation for the numerical scheme.     

An efficient approach for the numerical solution of the Monge-Ampère equation

In this paper, we present a new method to compute the numerical solution of the elliptic Monge-Ampère equation. This method is based on solving a parabolic Monge-Ampère equation for the steady state solution. We study the problem of global existence, uniqueness, and convergence of the solution of the fully nonlinear parabolic PDE to the unique solution of the elliptic Monge-Ampère equation. Some numerical experiments are presented to show the convergence and the regularity of the numerical solution.     

A Variational Analysis for the Moving Finite Element Method for Gradient Flows

By using the Onsager principle as an approximation tool, we give a novel derivation for the moving finite element method for gradient flow equations. We show that the discretized problem has the same energy dissipation structure as the continuous one. This enables us to do numerical analysis for the stationary solution of a nonlinear reaction diffusion equation using the approximation theory of free-knot piecewise polynomials. We show that under certain conditions the solution obtained by the moving finite element method converges to a local minimizer of the total energy when time goes to infinity. The global minimizer, once it is detected by the discrete scheme, approximates the continuous stationary solution in optimal order. Numerical examples for a linear diffusion equation and a nonlinear Allen-Cahn equation are given to verify the analytical results.     

一类非线性离散系统的指数二分性及其在数值分析和计算中的应用

本文中我们考虑一类二阶非线性常微分方程的边值问题的迎风差分格式.我们运用奇异摄动方法构造了该迎风差分方程解的渐近近似,并利用指数二分性理论证明了有一个低阶方程其解是该迎风方程式的在边界外的一个良好近似.我们还构造了校正项,使校正项与低阶方程的解之和是一个渐近近似.最后一些数值例子用于显示本文方法的应用.     

Stabilization for the Kirchhoff type equation from an axially moving heterogeneous string modeling with boundary feedback control

The first objective of this paper is to make the mathematical model for vibration suppression of an axially moving heterogeneous string. In order to describe the geometrical nonlinearity due to finite transverse deformation, the exact expression of the strain is used. The mathematical modeling is derived first by using Hamilton’s principle and variational lemma and the derived nonlinear PDE system is the Kirchhoff type equation with boundary feedback control. Next, we show the existence and uniqueness of strong solutions of the PDE system via techniques of functional analysis, mainly a theorem of compactness for the analysis of the approximation of the Faedo–Galerkin method and estimate a decay rate for the energy. The theoretical results are assured by numerical results of the solution’s shape and asymptotic behavior for the system.     

Well-posedness and finite element approximation of time dependent generalized bioconvective flow

We consider the finite element approximation of a time dependent generalized bioconvective flow. The underlying system of partial differential equations consists of incompressible Navier–Stokes type convection equations coupled with an equation describing the transport of micro-organisms. The viscosity of the fluid is assumed to be a function of the concentration of the micro-organisms. We show the existence and uniqueness of the weak solution of the system in two dimensions and construct numerical approximations based on the finite element method, for which we obtain error estimates. In addition, we conduct several numerical experiments to demonstrate the accuracy of the numerical method and perform simulations of the bioconvection pattern formations based on realistic model parameters to demonstrate the validity of the proposed numerical algorithm.     

Point approximation of a space-homogeneous transport equation

Summary We present an approximation method of a space-homogeneous transport equation which we prove is convergent. The method is very promising for numerical computation. Comparison of a numerical computation with an exact solution is given for the Master equation.     

The QNET method for two-moment analysis of open queueing networks

Consider an open network of single-server stations, each with a first-in-first-out discipline. The network may be populated by various customer types, each with its own routing and service requirements. Routing may be either deterministic or stochastic, and the interarrival and service time distributions may be arbitrary. In this paper a general method for steady-state performance analysis is described and illustrated. This analytical method, called QNET, uses both first and second moment information, and it is motivated by heavy traffic theory. However, our numerical examples show that QNET compares favorably with W. Whitt's Queueing Network Analyzer (QNA) and with other approximation schemes, even under conditions of light or moderate loading. In the QNET method one first replaces the original queueing network by what we call an approximating Brownian system model, and then one computes the stationary distribution of the Brownian model. The second step amounts to solving a certain highly structured partial differential equation problem; a promising general approach to the numerical solution of that PDE problem is described by Harrison and Dai [8] in a companion paper. Thus far the numerical solution technique has been implemented only for two-station networks, and it is clear that the computational burden will grow rapidly as the number of stations increases. Thus we also describe and investigate a cruder approach to two-moment network analysis, called ΠNET, which is based on a product form approximation, or decomposition approximation, to the stationary distribution of the Brownian system model. In very broad terms, ΠNET is comparable to QNA in its level of sophistication, whereas QNET captures more subtle system interactions. In our numerical examples the performance of ΠNET and QNA is similar; the performance of QNET is generally better, sometimes much better.     

Existence and convergence of Galerkin approximation for second order hyperbolic equations with memory term

We study a second order hyperbolic initial‐boundary value partial differential equation (PDE) with memory that results in an integro‐differential equation with a convolution kernel. The kernel is assumed to be either smooth or no worse than weakly singular, that arise for example, in linear and fractional order viscoelasticity. Existence and uniqueness of the spatial local and global Galerkin approximation of the problem is proved by means of Picard's iteration. Then, spatial finite element approximation of the problem is formulated, and optimal order a priori estimates are proved by the energy method. The required regularity of the solution, for the optimal order of convergence, is the same as minimum regularity of the solution for second order hyperbolic PDEs. Spatial rate of convergence of the finite element approximation is illustrated by a numerical example. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 548–563, 2016     

APPROXIMATION, STABILITY AND FAST EVALUATION OF EXACT ARTIFICIAL BOUNDARY CONDITION FOR THE ONE-DIMENSIONAL HEAT EQUATION

In this paper we consider the numerical solution of the one-dimensional heat equation on unbounded domains. First an exact semi-discrete artificial boundary condition is derived by discretizing the time variable with the Crank-Nicolson method. The semi-discretized heat equation equipped with this boundary condition is then proved to be unconditionally stable, and its solution is shown to have second-order accuracy. In order to reduce the computational cost, we develop a new fast evaluation method for the convolution operation involved in the exact semi-discrete artificial boundary condition. A great advantage of this method is that the unconditional stability held by the semi-discretized heat equation is preserved. An error estimate is also given to show the dependence of numerical errors on the time step and the approximation accuracy of the convolution kernel. Finally, a simple numerical example is presented to validate the theoretical results.     

Galerkin approximations with embedded boundary conditions for retarded delay differential equations

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs.     

Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

In this paper, we consider a two‐dimensional multi‐term time‐fractional Oldroyd‐B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time‐fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L_∞ ‐norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two‐dimensional multi‐term time‐fractional Oldroyd‐B model.     

Numerical simulation of blowup in nonlocal reaction–diffusion equations using a moving mesh method

In this paper we implement the moving mesh PDE method for simulating the blowup in reaction–diffusion equations with temporal and spacial nonlinear nonlocal terms. By a time-dependent transformation, the physical equation is written into a Lagrangian form with respect to the computational variables. The time-dependent transformation function satisfies a parabolic partial differential equation — usually called moving mesh PDE (MMPDE). The transformed physical equation and MMPDE are solved alternately by central finite difference method combined with a backward time-stepping scheme. The integration time steps are chosen to be adaptive to the blowup solution by employing a simple and efficient approach. The monitor function in MMPDEs plays a key role in the performance of the moving mesh PDE method. The dominance of equidistribution is utilized to select the monitor functions and a formal analysis is performed to check the principle. A variety of numerical examples show that the blowup profiles can be expressed correctly in the computational coordinates and the blowup rates are determined by the tests.     

Numerical method for solving linear Fredholm fuzzy integral equations of the second kind

In this paper we use parametric form of fuzzy number and convert a linear fuzzy Fredholm integral equation to two linear system of integral equation of the second kind in crisp case. We can use one of the numerical method such as Nystrom and find the approximation solution of the system and hence obtain an approximation for fuzzy solution of the linear fuzzy Fredholm integral equations of the second kind. The proposed method is illustrated by solving some numerical examples.     

High order compact solution of the one‐space‐dimensional linear hyperbolic equation

In this article, we introduce a high‐order accurate method for solving one‐space dimensional linear hyperbolic equation. We apply a compact finite difference approximation of fourth order for discretizing spatial derivative of linear hyperbolic equation and collocation method for the time component. The main property of this method additional to its high‐order accuracy due to the fourth order discretization of spatial derivative, is its unconditionally stability. In this technique the solution is approximated by a polynomial at each grid point that its coefficients are determined by solving a linear system of equations. Numerical results show that the compact finite difference approximation of fourth order and collocation method produce a very efficient method for solving the one‐space‐dimensional linear hyperbolic equation. We compare the numerical results of this paper with numerical results of (Mohanty, 3 .© 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

A numerical method to compute the dissolution of second phases in ternary alloys

Dissolution of stoichiometric multi-component particles in ternary alloys is an important process occurring during the heat treatment of as-cast aluminium alloys prior to hot extrusion. A mathematical model is proposed to describe such a process. In this model an equation is given to determine the position of the particle interface in time, using two diffusion equations which are coupled by nonlinear boundary conditions at the interface. Some results concerning existence, uniqueness, and monotonicity are given. Furthermore, for an unbounded domain an analytical approximation is derived. The main part of this work is the development of a numerical solution method. Finite differences are used on a grid which changes in time. The discretization of the boundary conditions is important to obtain an accurate solution. The resulting nonlinear algebraic system is solved by the Newton-Raphson method. Numerical experiments illustrate the accuracy of the numerical method. The numerical solution is compared with the analytical approximation.     

Solving complex PDE systems for pricing American options with regime‐switching by efficient exponential time differencing schemes

In this article, we study the numerical solutions of a class of complex partial differential equation (PDE) systems with free boundary conditions. This problem arises naturally in pricing American options with regime‐switching, which adds significant complexity in the PDE systems due to regime coupling. Developing efficient numerical schemes will have important applications in computational finance. We propose a new method to solve the PDE systems by using a penalty method approach and an exponential time differencing scheme. First, the penalty method approach is applied to convert the free boundary value PDE system to a system of PDEs over a fixed rectangular region for the time and spatial variables. Then, a new exponential time differncing Crank–Nicolson (ETD‐CN) method is used to solve the resulting PDE system. This ETD‐CN scheme is shown to be second order convergent. We establish an upper bound condition for the time step size and prove that this ETD‐CN scheme satisfies a discrete version of the positivity constraint for American option values. The ETD‐CN scheme is compared numerically with a linearly implicit penalty method scheme and with a tree method. Numerical results are reported to illustrate the convergence of the new scheme. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Time accurate fast wavelet‐Taylor Galerkin method for partial differential equations

We introduce the concept of fast wavelet‐Taylor Galerkin methods for the numerical solution of partial differential equations. In wavelet‐Taylor Galerkin method discretization in time is performed before the wavelet based spatial approximation by introducing accurate generalizations of the standard Euler, θ and leap‐frog time‐stepping scheme with the help of Taylor series expansions in the time step. We will present two different time‐accurate wavelet schemes to solve the PDEs. First, numerical schemes taking advantage of the wavelet bases capabilities to compress the operators and sparse representation of functions which are smooth, except for in localized regions, up to any given accuracy are presented. Here numerical experiments deal with advection equation with the spiky solution in one dimension, two dimensions, and nonlinear equation with a shock in solution in two dimensions. Second, our schemes deal with more regular class of problems where wavelets are not efficient procedure for data compression but we can use the good approximation properties of wavelet. Here time‐accurate schemes lead to consistent mass matrix in an explicit time stepping, which can be solved by approximate factorization techniques. Numerical experiment deals with more regular class of problems like heat equation as well as coupled linear system in two dimensions. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Numerical Solutions for Option Pricing Models Including Transaction Costs and Stochastic Volatility

The option pricing problem when the asset is driven by a stochastic volatility process and in the presence of transaction costs leads to solving a nonlinear partial differential equation. The nonlinear term in the PDE reflects the presence of transaction costs. Under a particular market completion assumption we derive the nonlinear PDE whose solution may be used to find the price of options. In this paper under suitable conditions, we give an algorithmic scheme to obtain the solution of the problem by an iterative method and provide numerical solutions using the finite difference method.