A Fourth-order Tridiagonal Finite Difference Method for General Two-point Boundary Value Problems with Non-linear Boundary Conditions

We present a fourth-order finite difference method for the generalsecond-order nonlinear differential equation y^" = f(x, y, y‘)subject to non-linear two-point boundary conditions g₁(y(a), — y^’()) = 0, g₂(y(b), y'(b)) = 0. When both the differential equation and the boundary conditionsare linear, the method leads to a tridiagonal linear system.We show that the finite difference method is O(h⁴)-convergent.Numerical examples are given to illustrate the method and itsfourth-order convergence. The present paper extends the methodgiven in Chawla (1978) to the case of non-linear boundary conditions.     

A Fourth-order Tridiagonal Finite Difference Method for General Non-linear Two-point Boundary Value Problems with Mixed Boundary Conditions

We present a new fourth-order finite difference method for thegeneral second-order non-linear differential equation y^N = f(x,y, y') subject to mixed two-point boundary conditions. An interestingfeature of our method is that each discretization of the differentialequation at an interior grid point is based on just three evaluationsof f. We establish, under appropriate conditions, O(h⁴)-convergenceof the finite difference scheme. In the case of linear differentialequations, our finite difference scheme leads to tridiagonallinear systems. Numerical examples are considered to demonstratecomputationally the fourth order of the method.     

A Factorization-Related Method for Elliptic Partial Differential Equations with Iterative Improvement

We consider a method for solving elliptic boundary-value problems.The method arises from a finite-difference discretization whichhas one form in the interior region, but is modified near theboundary. This permits the problem to be solved in terms ofsparse upper and lower triangular matrices. The result of thisdirect method is then improved by an iterative technique, whichis further enhanced by a multigrid-type process. For the type of problems we consider here, the total combinedmethod requires only O(N²) time and O(N²) space to compute thesolution of a system of N x N mesh points to good accuracy.The method is applied to a case where normal discretizationleads to a matrix that is not positive definite.     

Error estimates for a finite-difference approximation of a mean field model of superconducting vortices in one-dimension

Error estimates for a semi-implicit finite-difference approximationof a mean field model of superconducting vortices are obtained.The L(L¹) error between the approximate and the exact superconductingvortex density of the model is of order h^1/3.     

A Bound for Solutions of a Fourth-order Dynamical System

We consider the homogeneous linearized equation of yawing motionof a spinning projectile ⁿ+A)s)' + B(s) = 0, where A and B are complex functions of the real independentvariable s. It is shown that where K is a positive constant and v(s) is a real function ofs.     

Convergence of Finite-Difference Schemes for Elliptic Equations with Variable Coefficients

We study the convergence of finite-difference schemes for second-orderelliptic equations with variable coefficients. We prove thatthe convergence rate in the discrete W₂¹ norm is of the orderh^{s –1} if the solution of the boundary value problem belongsto the Sobolev space W₂^s (1 < s 3).     

Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square

Sorin Micu ^{This paper studies the numerical approximation of the boundarycontrol for the wave equation in a square domain. It is knownthat the discrete and semi-discrete models obtained by discretizingthe wave equation with the usual finite-difference or finite-elementmethods do not provide convergent sequences of approximationsto the boundary control of the continuous wave equation as themesh size goes to zero. Here, we introduce and analyse a newsemi-discrete model based on the space discretization of thewave equation using a mixed finite-element method with two differentbasis functions for the position and velocity. The main theoreticalresult is a uniform observability inequality which allows usto construct a sequence of approximations converging to theminimal L2-norm control of the continuous wave equation. Wealso introduce a fully discrete system, obtained from our semi-discretescheme, for which we conjecture that it provides a convergentsequence of discrete approximations as both h and t, the timediscretization parameter, go to zero. We illustrate this factwith several numerical experiments.}     

Convergence of a Finite-Difference Scheme for Second-Order Hyperbolic Equations with Variable Coefficients

We study the convergence of a finite-difference scheme for thefirst initial-boundary-value problem for linear second-orderhyperbolic equations with variable coefficients. Using the bilinearversion of the Bramble-Hilbert lemma we prove that the convergencerate in the discrete energy norm is of the order h^–2 ifthe exact solution belongs to the Sobolev space W₂(Q) with 2<<4.     

Multistep approximation of linear sectorial evolution equations

This paper concerns the linear multistep approximation of alinear sectorial evolution equation u_t = Au on a complex Banachspace X. Given a strictly A()-stable q-step method of orderp whose stability region includes a sectorial region containingthe spectrum of the operator A, the corresponding evolutionsemigroup for the method is Cⁿ(hA), n 0, defined on X^q, whereC(z) L (C^q) denotes the one-step map associated with the method.It is shown that for appropriately chosen V, Y: C C^q, basedon the principal right and left eigenvectors of C(z), Cⁿ(hA)approximates the semigroup V(hA)e^nhAY^H(hA) with optimal orderp.     

A qualocation method for a unidimensional single phase semilinear Stefan problem

Based on straightening the free boundary, a qualocation methodis proposed and analysed for a single phase unidimensional Stefanproblem. This method may be considered as a discrete versionof the H¹-Galerkin method in which the discretization is achievedby approximating the integrals by a composite Gauss quadraturerule. Optimal error estimates are derived in L(W^j,), j = 0,1,and L (H^j), j = 0,1,2, norms for a semidiscrete scheme withoutany quasi-uniformity assumption on the finite element mesh.     

A fourth-order cubic spline method for linear second-order two-point boundary-value problems

A cubic spline method for linear second-order two-point boundary-valueproblems is analysed. The method is a Petrov-Galerkin methodusing a cubic spline trial space, a piecewise-linear test space,and a simple quadrature rule for the integration, and may beconsidered a discrete version of the H¹-Galerkin method. Themethod is fully discrete, allows an arbitrary mesh, yields alinear system with bandwidth five, and under suitable conditionsis shown to have an 0(h^4– rate of convergence in the W_p¹norm for i = 0, 1, 2, 1p. The H¹-Galerkin method and orthogonalspline collocation with Hermite cubics are also discussed.     

Khayyam with Cabri: experiences of pre-service mathematics teachers with Khayyam's solution of cubic equations in dynamic geometry environment

The study reported in this article deals with the observed actionsof Turkish pre-service mathematics teachers in dynamic geometryenvironment (DGE) as they were learning Khayyam's method forsolving cubic equations formed as x³ + ax = b. Having learnedthe method, modelled it in DGE and verified the correctnessof the solution, students generated their own methods for solvingdifferent types of cubic equations such as x³ + ax² = b andx³ + a = bx in the light of Khayyam's method. With the presentedteaching experiment, students realized that Khayyam's mathematicsis different from theirs. We consider that this gave them anopportunity to have an insight about the cultural and socialaspects of mathematics. In addition, the teaching experimentshowed that dynamic geometry software is an excellent tool fordoing mathematics because of their dynamic nature and accurateconstructions. And, it can be easily concluded that the historyof mathematics is useful resource for enriching mathematicslearning environment.     

Optimum Runge-Kutta-Fehlberg Methods for Second-order Differential Equations

^* Presently at Deparment of Mathematics, Indian Institute of Technology, Madras, India. The optimum Runge-Kutta method of a particular order is theone whose truncation error is minimum. In this paper, we havederived optimum Runge-Kutta mehtods of 0(h^m+4), 0(h^m+5) and0(h^m+6) for m = 0(1)8, which can be directly used for solvingthe second order differential equation yⁿ = f(x, y, y'). Thesemethods are based on a transformation similar to that of Fehlbergand require two, three and four evaluations of f(x, y, y') respectively,for each step. The numercial solutions of one example obtainedwith these methods are given. It has been assumed that f(x,y, y')is sufficiently differentiable in the entire region ofintegration.     

Supraconvergence of a finite difference scheme for solutions in Hs(0, L)

^** Email: silvia{at}mat.uc.pt^*** Email: ferreira{at}mat.uc.pt^**** Email: grigo{at}math.tu-berlin.de In this paper we study the convergence of a centred finite differencescheme on a non-uniform mesh for a 1D elliptic problem subjectto general boundary conditions. On a non-uniform mesh, the schemeis, in general, only first-order consistent. Nevertheless, weprove for s (1/2, 2] order O(h^s)-convergence of solution andgradient if the exact solution is in the Sobolev space H^1+s(0,L), i.e. the so-called supraconvergence of the method. It isshown that the scheme is equivalent to a fully discrete linearfinite-element method and the obtained convergence order isthen a superconvergence result for the gradient. Numerical examplesillustrate the performance of the method and support the convergenceresult.     

Mixed block elimination for linear systems with wider borders

The paper is about the stable solution of possibly ill-conditionedbordered linear systems. Given stable solvers for matrix A andfor A^T, we prove that the Govaerts Mixed Block Elimination (BEM)method constitutes a stable solver for the matrix consistingof A or A^T with a border of width 1, and hence by recursionfor a border of any width. We express the algorithm in an efficient,iterative, form. We analyse its operation count, and verifythe theory by extensive numerical experiments. ^*Senior Research Associate of the Belgian National Fund of ScientificResearch NFWO.     

An Efficient Enthalpy-type Method for the Stefan Problem

The paper presents a new finite-difference method for solvingthe one-dimensional two-phase Stefan problem. Under assumptionson the data which guarantee the temperature u and the movingboundary s to belong to and , respectively, we obtain L₂-errorestimates of order O(h + h^–?) provided the time step is chosen such that Numerical aspects are discussed.     

Propagation of Smallness for Harmonic and Analytic Functions in Arbitrary Domains

In this paper the authors develop a new approach to the problemof ‘propagation of smallness’ for harmonic functionsin arbitrary domains, in Rⁿ (n 2). The main result of thispaper is a certain logarithmic-convexity relation for the L²-normsof harmonic functions. As a consequence, new kinds of uniquenessresults for harmonic functions are obtained. The method worksalso for analytic functions in C, with L^p-norms (p > 0).1991 Mathematics Subject Classification 31B05.     

Numerical Methods for y' =f(x, y) via Rational Approximations for the Cosine

To investigate stability and phase lag, a numerical method isapplied to the test equation y' = –²y. Frequently, thecharacteristic equation of the resulting recurrence relationhas the form ²– 2R_nm(v²) + 1 = 0, where v = h, with hthe steplength, and R_nm(v²) is a rational approximation forcos v. In this paper, properties of such approximations areused to provide a general framework for the study of stabilityintervals and orders of dispersion of a variety of one- andtwo-step methods. Upper bounds on the intervals of periodicityof explicit methods with maximum order of dispersion are established.It is shown that the order of dispersion of a P-stable method,for given n and m, cannot exceed 2m; a consequence is that,of the Pad? approximants for cos v, only the [0/2m] approximantshave modulus less than unity for all v² >0. A complete characterizationof P-stable methods of fourth order corresponding to the rationalapproximation R₂₂(v²) is followed by several results for methodswhich have finite intervals of periodicity; in particular, weidentify methods which have order of dispersion 6 or 8 withlarge intervals of periodicity. There is also a detailed discussionof P-stable methods of sixth order corresponding to the rationalapproximation R₃₃(v²).     

A qualocation method for parabolic partial differential equations

In this paper a qualocation method is analysed for parabolicpartial differential equations in one space dimension. Thismethod may be described as a discrete H¹-Galerkin method inwhich the discretization is achieved by approximating the integralsby a composite Gauss quadrature rule. An O (h^4-i) rate of convergencein the W^i.p norm for i = 0, 1 and 1 p is derived for a semidiscretescheme without any quasi-uniformity assumption on the finiteelement mesh. Further, an optimal error estimate in the H² normis also proved. Finally, the linearized backward Euler methodand extrapolated Crank-Nicolson scheme are examined and analysed.     

Numerical Investigations into the Stokes Phenomenon. II

The Stokes phenomenon associated with the differential equationsW " = WZ (z²–a²) and W" = w(z² –a²)(x²–b²)isconsidered. As an application to the method introduced in paper I, somenumerical and analytical results concerning the Stokes constantsof these equations are presented.