A modified Markov model for the estimation of computer software performance

The model proposed by Trivedo and Shooman [8] is extended and modified by assuming that (1) the error occurrence rate when the machine is running is proportional to the number of errors in the system; (2) the error correction rate has two components, either an error is corrected with correction rate μ₀ or an error is corrected but a new error is created with ineffective correction rate μ₁. The solution of the differential equations corresponding to the model is obtained in closed form.     

Algebraic solutions of the Lamé equation, revisited

A minor error in the necessary conditions for the algebraic form of the Lamé equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out (see Baldassarri, J. Differential Equations 41 (1) (1981) 44). It is shown that if the group is the octahedral group S₄, then the degree parameter of the equation may differ by ±1/6 from an integer; this possibility was missed. The omission affects a recent result on the monodromy of the Weierstrass form of the Lamé equation (see Churchill, J. Symbolic Comput. 28 (4-5) (1999) 521). The Weierstrass form, which is a differential equation on an elliptic curve, may have, after all, an octahedral projective monodromy group.     

A family of implicit Chebyshev methods for the numerical integration of second-order differential equations

A family of implicit methods based on intra-step Chebyshev interpolation is developed for the solution of initial-value problems whose differential equations are of the special second-order form y″ = f(y(x); x). The general procedure allows stepsizes which are considerably larger than commonly used in conventional methods. Computation overhead is comparable to that required by high-order single or multistep procedures. In addition, the iterative nature of the method substantially reduces local errors while maintaining a low rate of global error growth.     

On error operators related to the arbitrary functions principle

The error on a real quantity Y due to the graduation of the measuring instrument may be asymptotically represented, when the graduation is regular and fines down, by a Dirichlet form on R whose square field operator does not depend on the probability law of Y as soon as this law possesses a continuous density. This feature is related to the “arbitrary functions principle” (Poincaré, Hopf). We give extensions of this property to Rd and to the Wiener space for some approximations of the Brownian motion. This gives new approximations of the Ornstein-Uhlenbeck gradient. These results apply to the discretization of some stochastic differential equations encountered in mechanics.     

Higher order fully discrete scheme combined withH 1-Galerkin mixed finite element method for semilinear reaction-diffusion equations

We first apply a first order splitting to a semilinear reaction-diffusion equation and then discretize the resulting system by anH ¹-Galerkin mixed finite element method in space. This semidiscrete method yields a system of differential algebraic equations (DAEs) ofindex one. Apriori error estimates for semidiscrete scheme are derived for both differential as well as algebraic components. For fully discretization, an implicit Runge-Kutta (IRK) methods is applied to the temporal direction and the error estimates are discussed for both components. Finally, we conclude the paper with a numerical example.     

Generating differential equations satisfied by products of functions

Based on the coefficients of two homogeneous linear differential equations, a method is proposed to construct a third homogeneous linear differential equations which is satisfied by all products of the form uv, where u and v satisfy, respectively, the first and the second given differential equation. The method was used recently in the computation of rapidly oscillatory integrals with kernels which are products of Bessel functions and their variants.     

Approximation, integration and differentiation of time functions using a set of orthogonal hybrid functions (HF) and their application to solution of first order differential equations

Differential equations of different types and orders are of utmost importance for mathematical modeling of control system problems. State variable method uses the concept of expressing n number of first order differential equations in vector matrix form to model and analyze/synthesize control systems.The present work proposes a new set of orthogonal hybrid functions (HF) which evolved from synthesis of sample-and-hold functions (SHF) and triangular functions (TF). This HF set is used to approximate a time function in a piecewise linear manner with the mean integral square error (MISE) much less than block pulse function based approximation which always provides staircase solutions.The operational matrices for integration and differentiation in HF domain are also derived and employed for solving non-homogeneous and homogeneous differential equations of the first order as well as state equations. The results are compared with exact solutions, the 4th order Runge-Kutta method and its further improved versions proposed by Simos [6]. The presented HF domain theory is well supported by a few illustrations.     

Invariant solutions as internal singularities of nonlinear differential equations and their use for qualitative analysis of implicit and numerical solutions

Lie group analysis of nonlinear differential equations reveals existence of singularities provided by invariant solutions and invisible from the form of the equation in question. We call them internal singularities in contrast with external singularities manifested by the form of the equation. It is illustrated by way of examples that internal singularities are useful for analyzing a behaviour of solutions of nonlinear differential equations near external singularities.     

A combination between the reduced basis method and the ANOVA expansion: On the computation of sensitivity indices

We consider a method to efficiently evaluate in a real-time context an output based on the numerical solution of a partial differential equation depending on a large number of parameters. We state a result allowing to improve the computational performance of a three-step RB–ANOVA–RB method. This is a combination of the reduced basis (RB) method and the analysis of variations (ANOVA) expansion, aiming at compressing the parameter space without affecting the accuracy of the output. The idea of this method is to compute a first (coarse) RB approximation of the output of interest involving all the parameter components, but with a large tolerance on the a posteriori error estimate; then, we evaluate the ANOVA expansion of the output and freeze the least important parameter components; finally, considering a restricted model involving just the retained parameter components, we compute a second (fine) RB approximation with a smaller tolerance on the a posteriori error estimate. The fine RB approximation entails lower computational costs than the coarse one, because of the reduction of parameter dimensionality. Our result provides a criterion to avoid the computation of those terms in the ANOVA expansion that are related to the interaction between parameters in the bilinear form, thus making the RB–ANOVA–RB procedure computationally more feasible.     

Bifurcation analysis of a non-cooperative differential game with one weak player

We study a bifurcation problem for a system of two differential equations in implicit form. For each value of the parameter θ, the solution yields a pair of Nash equilibrium strategies in feedback form, for a non-cooperative differential game. When θ=0, the second player has no power to influence the dynamics of the system, and his optimal strategy is myopic. The game thus reduces to an optimal control problem for the first player. By studying the bifurcation in the solutions to the corresponding system of Hamilton-Jacobi equations, one can establish existence and multiplicity of solutions to the differential game, as θ becomes strictly positive.     

Zero loci of differential modular forms

The theory of p-adic modular forms initiated by Serre, Dwork, and Katz (p-Adic Properties of Modular Schemes and Modular Forms, Lecture Notes in Mathematics, Vol. 350, Springer, Berlin, 1973) “lives” on the complement (in the p-adic completion of the appropriate modular curve) of the zero locus of the Eisenstein form E_p−1. On the other hand, most of the interesting phenomena in the theory of differential modular forms (J. Reine Angew. Math. (520) (2000) 95) take place on the complement of the zero locus of a fundamental differential modular form called f_jet. We establish that the zero locus of the reduction modulo p for p not congruent to one modulo 12 of the Eisenstein form E_p−1 is not contained in the zero locus of the reduction modulo p of the differential modular form f_jet implying that the theory of differential modular forms is applicable in certain situations not covered by the theory of p-adic modular forms.     

Efficient computable error bounds for discontinuous Galerkin approximations of elliptic problems

We present guaranteed and computable both sided error bounds for the discontinuous Galerkin (DG) approximations of elliptic problems. These estimates are derived in the full DG-norm on purely functional grounds by the analysis of the respective differential problem, and thus, are applicable to any qualified DG approximation. Based on the triangle inequality, the underlying approach has the following steps for a given DG approximation: (1) computing a conforming approximation in the energy space using the Oswald interpolation operator, and (2) application of the existing functional a posteriori error estimates to the conforming approximation. Various numerical examples with varying difficulty in computing the error bounds, from simple problems of polynomial-type analytic solution to problems with analytic solution having sharp peaks, or problems with jumps in the coefficients of the partial differential equation operator, are presented which confirm the efficiency and the robustness of the estimates.     

Parallel domain decomposition procedures of improved D-D type for parabolic problems

Two parallel domain decomposition procedures for solving initial-boundary value problems of parabolic partial differential equations are proposed. One is the extended D-D type algorithm, which extends the explicit/implicit conservative Galerkin domain decomposition procedures, given in [5], from a rectangle domain and its decomposition that consisted of a stripe of sub-rectangles into a general domain and its general decomposition with a net-like structure. An almost optimal error estimate, without the factor H^−1/2 given in Dawson-Dupont’s error estimate, is proved. Another is the parallel domain decomposition algorithm of improved D-D type, in which an additional term is introduced to produce an approximation of an optimal error accuracy in L²-norm.     

Error control of rational approximations to the exponential function

Several methods for the numerical solution of stiff ordinary differential equations require approximation of an exponential of a matrix. In the present paper we present a technique for estimating the error incurred in replacing a matrix exponential by a rational approximation. This estimation is done by introducing another approximation, of superior order, whose aposteriori evaluation is cheap. Properties of the new approximation pertaining to both its stability and the behavior of the error for matrices with negative eigenvalues are analyzed.     

Variable multistep methods for delay differential equations

In this paper, variable stepsize multistep methods for delay differential equations of the type y(t) = f(t, y(t), y(t − τ)) are proposed. Error bounds for the global discretization error of variable stepsize multistep methods for delay differential equations are explicitly computed. It is proved that a variable multistep method which is a perturbation of strongly stable fixed step size method is convergent.     

A collocation method for systems of nonlinear ordinary differential equations

A collocation method based on piecewise polynomials is applied to boundary value problems for mth order systems of nonlinear ordinary differential equations. Optimal a priori estimates are obtained for the error of approximation in the maximum norm and superconvergence is verified at particular points.     

On the relationship between the proportional hazards and accelerated failure time models in survival analysis

The slope parameters in the proportional hazards and accelerated failure time models are shown to be proportional if and only if the error variable in the accelerated failure time model has a generalized extreme value distribution. A differential equation relating the two score functions φ(u) and Φ(u) is established. The error distribution is characterized when φ (u) and Φ(u) are linearly related; subject to this restriction, censored data linear rank procedures are studied.     

The Kačanov method for some nonlinear problems

The Kačanov method is an iteration method for solving some nonlinear partial differential equation problems. In each iteration, a linear problem is solved. In this paper, we discuss the use of the Kačanov method in the context of two model problems. We show the convergence of the Kačanov iteration sequences, and derive a posteriori error estimates for the Kačanov iterates. Numerical examples are given showing the convergence of the method and the effectiveness of the a posteriori error estimates.     

Criterion for the basis property of the eigenfunction system of a multiple differentiation operator with an involution

We consider the spectral problem for a model second-order differential operator with an involution. The operator is given by the differential expression Lu = ?u??(?x) and boundary conditions of general form. We obtain a criterion for the basis property of the systems of eigenfunctions of this operator in terms of the coefficients in the boundary conditions.