Equivalent Initial Value Problem Research Articles

Initial value technique for singularly perturbed two point boundary value problems using an exponentially fitted finite difference scheme

In this paper, we present an approximate method (Initial value technique) for the numerical solution of quasilinear singularly perturbed two point boundary value problems in ordinary differential equations having a boundary layer at one end (left or right) point. It is motivated by the asymptotic behavior of singular perturbation problems. The original problem is reduced to an asymptotically equivalent first order initial value problem by approximating the zeroth order term by outer solution obtained by asymptotic expansion, and then this initial value problem is solved by an exponentially fitted finite difference scheme. Some numerical examples are given to illustrate the given method. It is observed that the presented method approximates the exact solution very well for crude mesh size h .

Numerical solution of hybrid fuzzy differential equation IVPs by a characterization theorem

In this paper, we study hybrid fuzzy differential equation initial value problems (IVPs). We consider the problem of finding their numerical solutions by using a recent characterization theorem of Bede for fuzzy differential equations. We prove a corollary to Bede’s characterization theorem and give a characterization theorem for hybrid fuzzy differential equation IVPs. Then we prove that any suitable numerical method for ODEs can be applied piecewise to numerically solve hybrid fuzzy differential equation IVPs. Numerical examples are provided which connect the new results with previous findings.

A family of two-stage two-step methods for the numerical integration of the Schrödinger equation and related IVPs with oscillating solution

We develop a family of six methods for the numerical integration of the Schrodinger equation and related initial value problems with oscillating solution. Three of the methods are constructed so that they are P-stable, using the methodology of Wang (Comp Phys Comm 171(3):162–174, 2005). Also two of these three methods are trigonometrically fitted with trigonometric orders one and two. The other three methods are constructed so that they are trigonometrically fitted with orders one, two and three. We show that there is an equivalence between the three pairs of methods, as if the property of P-stability can be substituted by an extra trigonometric order, that is the P-stable method is equivalent to the method with trigonometric order one, the P-stable method with trigonometric order one is equivalent to the method with order two, and the P-stable method with order two is equivalent to the method with order three. There is a condition that we choose the same frequency for the P-stability test problem y′′ = −θ 2 y and the functions that the method has to integrate exactly, in order to be trigonometrically fitted: {cos(ω x), sin(ω x), x cos(ω x), x sin(ω x), x 2 cos(ω x), x 2 sin(ω x)}. A stability analysis and a local truncation error analysis are performed on the methods and also the v–s diagrams are produced, where v = ω h and s = θ h. Finally the methods are applied to IVPs with oscillating solutions, such as the one-dimensional time independent Schrodinger equation and the nonlinear problem.

A MONOTONE-ITERATIVE METHOD FOR FINDING PERIODIC SOLUTION OF AN IMPULSIVE DIFFERENTIAL EQUATIONS AND APPLICATION

In this paper, using the method of upper and lower solutions and its associated monotone iterations, we establish a new monotone-iterative scheme for finding periodic solutions of an impulsive differential equations. This method leads to the existence of maximal and minimal periodic solutions which can be computed from a linear iteration process in the same fashion as for impulsive differential equations initial value problem. This method is constructive and can be used to develop a computational algorithm for numerical solution of the periodic impulsive system. Our existence result improves a result established in [1]. The result is applied to a model of mutualism of Lotka–Volterra type which involves interactions among a mutualist-competitor, a competitor and a mutualist, the existence of positive periodic solutions of the model is obtained.

The Blasius Function: Computations Before Computers, the Value of Tricks, Undergraduate Projects, and Open Research Problems

The Blasius flow is the idealized flow of a viscous fluid past an infinitesimally thick, semi-infinite flat plate. The Blasius function is the solution to $2 f_{xxx} + f f_{xx} = 0$ on $x \in [0, \infty]$ subject to $f(0)=f_{x}(0)=0, f_{x}(\infty) = 1$. We use this famous problem to illustrate several themes. First, although the flow solves a nonlinear partial differential equation (PDE), Toepfer successfully computed highly accurate numerical solutions in 1912. His secret was to combine a Runge-Kutta method for integrating an ordinary differential equation (ODE) initial value problem with some symmetry principles and similarity reductions, which collapse the PDE system to the ODE shown above. This shows that PDE numerical studies were possible even in the precomputer age. The truth, both a hundred years ago and now, is that mathematical theorems and insights are an arithmurgist's best friend, and they can vastly reduce the computational burden. Second, we show that special tricks, applicable only to a given problem, can be as useful as the broad, general methods that are the fabric of most applied mathematics courses: the importance of “particularity.” In spite of these triumphs, many properties of the Blasius function $f(x)$ are unknown. We give a list of interesting projects for undergraduates and another list of challenging issues for the research mathematician.

On the perturbation of the observability equation in linear control systems

The aim of this paper is to investigate stability and sensitivity of the observability variable in linear control systems, (LCS) for short. We first present two results of Holder continuity in the abstract framework of the ordinary differential equation initial-value problem x?(t) = f(t,x(t)),x(t 0) = x 0. Afterwards, we apply our results to automatic systems, providing henceforth the sharpest bounds for the parametric input-output relation in LCS.

Modal analysis of magneto-electro-elastic plates using the state-vector approach

The state-vector approach is proposed to analyze the free vibration of magneto-electro-elastic laminate plates. The extended displacements and stresses can be divided into the so-called in-plane and out-of-plane variables. Once the state equation for the out-of-plane variables is obtained, a complex boundary value problem is converted into an equivalent simple initial value problem. Through the state equation, the propagator matrix between the top and bottom interfaces of every layer can be easily derived. The global propagator matrix can also be assembled using the continuity conditions. It is obvious that the order of global propagator matrix is not related to the number of layers. Consequently, this approach possesses certain virtues including simple formulation, less expensive computation, etc. To test the formulation, the developed solution is then applied to a simply supported multilayered plate constructed of piezoelectric and/or piezomagnetic materials. The natural frequencies and corresponding mode shapes are computed and compared with existing results. Furthermore, the fundamental modes along with a couple of other higher modes, which have never been reported in previous literature, are presented. Therefore, this completed set of frequencies and mode shapes can be used as benchmarks for future research in this field. It is also believed that the approach could be useful in the analysis and design of smart structures constructed from piezoelectric/piezomagnetic composites.

Compact finite difference method for American option pricing

A compact finite difference method is designed to obtain quick and accurate solutions to partial differential equation problems. The problem of pricing an American option can be cast as a partial differential equation. Using the compact finite difference method this problem can be recast as an ordinary differential equation initial value problem. The complicating factor for American options is the existence of an optimal exercise boundary which is jointly determined with the value of the option. In this article we develop three ways of combining compact finite difference methods for American option price on a single asset with methods for dealing with this optimal exercise boundary. Compact finite difference method one uses the implicit condition that solutions of the transformed partial differential equation be nonnegative to detect the optimal exercise value. This method is very fast and accurate even when the spatial step size h is large ( h ⩾ 0.1 ) . Compact difference method two must solve an algebraic nonlinear equation obtained by Pantazopoulos (1998) at every time step. This method can obtain second order accuracy for space x and requires a moderate amount of time comparable with that required by the Crank Nicolson projected successive over relaxation method. Compact finite difference method three refines the free boundary value by a method developed by Barone-Adesi and Lugano [The saga of the American put, 2003], and this method can obtain high accuracy for space x. The last two of these three methods are convergent, moreover all the three methods work for both short term and long term options. Through comparison with existing popular methods by numerical experiments, our work shows that compact finite difference methods provide an exciting new tool for American option pricing.

Arbitrary order Krylov deferred correction methods for differential algebraic equations

In this paper, a new framework for the construction of accurate and efficient numerical methods for differential algebraic equation (DAE) initial value problems is presented. The methods are based on applying spectral deferred correction techniques as preconditioners to a Picard integral collocation formulation for the solution. The resulting preconditioned nonlinear system is solved using Newton–Krylov schemes such as the Newton-GMRES method. Least squares based orthogonal polynomial approximations are computed using Gaussian type quadratures, and spectral integration is used to avoid the numerically unstable differentiation operator. The resulting Krylov deferred correction (KDC) methods are of arbitrary order of accuracy and very stable. Preliminary results show that these new methods are very competitive with existing DAE solvers, particularly when high precision is desired.

Global existence of solutions for a free boundary problem modeling the growth of necrotic tumors

In this paper we study a free boundary problem of a reaction diffusion equation modeling the growth of necrotic tumors. We first reduce this problem into an equivalent initial boundary value problem for a nonlinear parabolic equation on a fixed domain. This parabolic equation is strongly singular in the sense that not only it contains a discontinuous nonlinear function of the unknown function, but also all its coefficients are discontinuous nonlinear functionals of the unknown function. We use approximation method and the Schauder fixed point theorem combined with $L^p$ estimates to prove the existence of a global solution.

Equivalent histories, minimal state and initial value problem in viscoelasticity

The modelling of materials with memory is characterized by the process and the initial state. Because of the equivalence between histories, the state is identified with the response generated by the trivial process on the given initial strain. The process is the strain since the initial time. The occurrence of histories up to past infinity is then avoided. Within this setting, the inversion of the constitutive equation and a variational formulation are examined. Moreover a virtual-work type formulation provides a simpler weak form of the initial-value problem. Uniqueness of the solution to the initial-value problem is then established. Copyright © 2004 John Wiley & Sons, Ltd.

Asymptotic Initial-Value Method for Singularly-Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations

A computational method is presented to solve a class of nonturning-point singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a small parameter multiplying the highest derivative, subject to Dirichlet-type boundary conditions. In this method, first we construct a zeroth order asymptotic expansion for the solution of the given boundary-value problem. Then, this problem is integrated to get an equivalent initial-value problem for first-order ordinary differential equations. This initial-value problem is solved by either a classical method or a fitted operator method after approximating some of the terms in the differential equations by using the zeroth order asymptotic expansion. This method is effective and easy to implement. An error estimate is derived for the numerical solution. Examples are given to illustrate the method.

Parametric equations of motion for the transition operator and the Green’s operator

An approach based on parametric equations of motion for the transition operator and the Green's operator is presented. The formulation applies generally to discrete and continuous states of quantum-mechanical many-body systems. The method provides an alternative to solving the Schr\"odinger equation as a boundary value problem by replacing it with an equivalent initial value problem.

Sources of quantum waves

Due to the space and time dependence of the wave function in the time dependent Schroedinger equation, different boundary conditions are possible. The equation is usually solved as an ``initial value problem'', by fixing the value of the wave function in all space at a given instant. We compare this standard approach to "source boundary conditions'' that fix the wave at all times in a given region, in particular at a point in one dimension. In contrast to the well-known physical interpretation of the initial-value-problem approach, the interpretation of the source approach has remained unclear, since it introduces negative energy components, even for ``free motion'', and a time-dependent norm. This work provides physical meaning to the source method by finding the link with equivalent initial value problems.

A matrix D'Alembert formula for coupled wave initial value problems

If C is an invertible matrix in C r × r , the coupled wave equation initial value problem u tt = C 2 u xx , −∞ < x < ∞, t > 0, u( x, 0) = f( x), and u t ( x, 0) = g( x) for −∞ < x < ∞ is studied. A matrix D'Alembert formula for the closed form solution of the coupled wave equation is given. The approach is based on the Fourier transform, the holomorphic matrix functional calculus and some elements of complex variable functions. For the scalar case, the proposed D'Alembert formula coincides with the classical one.

Perturbed Scale-Invariant initial Value Problems in One-Dimensional Dynamic Elastoplasticity

he author considers an initial value problem for equations describing the longitudinal motion of an elastoplastic rod. Conditions on the stress $\sigma $ determine whether the deformation of the rod is plastic or elastic, bdth of which are described by wave equations with different wave speeds. Also, plastic deformation is quasi-linear while elastic deformation is assumed to be linear. The initial conditions are continuous, piecewise $C^1 $, and have a jump in the first derivative only at the origin. This is a generalization of the scale-invariant problem solved by D. Schaeffer and M. Shearer, in which plastic deformation is assumed to be linear and the initial conditions are piecewise linear. The analysis is divided into cases according to the structure of the corresponding scale-invariant problem; the most interesting case reduces to a free boundary problem for the plastic equations on a wedge withh two free boundaries.

A New Theory for Oscillating and Translating Slender Ships

A new theory is presented for the radiation problem of heave and pitch modes of a slender ship advancing at arbitrary forward speed. The theory has no restrictions on the order of forward speed and oscillation frequency, embracing both of unified theory developed by Newman and high-speed slender-body theory (HSSBT). The inner velocity potential consists of a particular solution, which is the same as that of HSSBT, and a homogeneous component, which is constructed by the superposition of the real-flow velocity potential and the reverse-flow reverse-time velocity potential.Since precise computations of HSSBT are prerequisite for the present theory, the calculation procedure adopted for HSSBT is described, converting the formulation into an equivalent initial-value problem in the time domain and applying the higher-order boundary-element method at each time step.Numerical results of added-mass and damping coefficients based on the present theory are shown for a modified Wigley model and compared with corresponding experiments and other computed results by the strip method, HSSBT, and 3-D Rankine-panel method.

The starting procedure in variable-stepsize variable-order PECE codes

An existing starting procedure used in codes for solving ordinary differential equation initial-value problems by variable-stepsize variable-order PECE formulas is studied. The actual error propagation and the estimate formed in the code are analyzed and compared. Based on the behavior of a bound on error propagation, we present a modified starting procedure which improves efficiency significantly for stringent error tolerances.

Error control for initial value problems with discontinuities and delays

When using software for ordinary differential equation (ODE) initial value problems, it is not unreasonable to expect the global error to decrease linearly with the user-supplied error tolerance. For standard ODEs, conditions on an algorithm that guarantee such “tolerance proportionality” asymptotically (as the error tolerance tends to zero) were derived by Stetter. Here we extend the analysis to cover a certain class of ODEs with low-order derivative discontinuities, and the class of ODEs with constant delays. We show that standard error control techniques will be successful if discontinuities are handled correctly and delay terms are calculated with sufficiently accurate interpolants. It is perhaps surprising that several delay ODE algorithms that have been proposed do not use sufficiently accurate interpolants to guarantee asymptotic proportionality. Our theoretical results are illustrated numerically.

Automated selection of mathematical software

Current approaches to recommending mathematical software are qualitative and categorical. These approaches are unsatisfactory when the problem to be solved has features that can “trade-off” in the recommendation process. A quantitative system is proposed that permits tradeoffs and can be built and modified incrementally. This quantitative approach extends other knowledge-engineering techniques in its knowledge representation and aggregation facilities. The system is demonstrated on the domain of ordinary differential equation initial value problems. The results are significantly superior to an existing qualitative (decision tree) system.