A semiexplicit method for numerical solution of functional differential algebraic equations

We study systems with delay effect that contain additional algebraic relations. We propose semiexplicit numerical methods of the Rosenbrock type. We prove the solvability of equations of a numerical model and estimate the order of the global error. The chosen parameters provide the third order of the error.     

A Taylor method for numerical solution of generalized pantograph equations with linear functional argument

This paper is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument. In this paper, we introduce a numerical method based on the Taylor polynomials for the approximate solution of the pantograph equation with retarded case or advanced case. The method is illustrated by studying the initial value problems. The results obtained are compared by the known results.     

A parametrization method for the numerical solution of singular differential equations

The paper explains the numerical parametrization method (PM), originally created for optimal control problems, for classical calculus of variation problems that arise in connection with singular implicit (IDEs) and differential-algebraic equations (DAEs) in frame of their regularization. The PM for IDEs is based on representation of the required solution as a spline with moving knots and on minimization of the discrepancy functional with respect to the spline parameters. Such splines are named variational splines. For DAEs only finite entering functions can be represented by splines, and the functional under minimization is the discrepancy of the algebraic subsystem. The first and the second derivatives of the functionals are calculated in two ways – for DAEs with the help of adjoint variables, and for IDE directly. The PM does not use the notion of differentiation index, and it is applicable to any singular equation having a solution.     

A Bessel collocation method for numerical solution of generalized pantograph equations

This article is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument. In this article, we introduce a collocation method based on the Bessel polynomials for the approximate solution of the pantograph equations. The method is illustrated by studying the initial value problems. The results obtained are compared by the known results. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A mesh-free numerical method for solution of the family of Kuramoto-Sivashinsky equations

In this paper, radial basis function (RBFs) based mesh-free method is implemented to find numerical solution of the Kuramoto-Sivashinsky equations. This approach has an edge over traditional methods such as finite-difference and finite element methods because it does not require a mesh to discretize the problem domain, and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method. The accuracy of the method is assessed in terms of the error norms L₂,L_∞, number of nodes in the domain of influence, free parameter, dependent parameter RBFs and time step length. Numerical experiments demonstrate accuracy and robustness of the method for solving a class of nonlinear partial differential equations.     

A mountain pass method for the numerical solution of semilinear wave equations

Summary It is well-known that periodic solutions of semilinear wave equations can be obtained as critical points of related functionals. In the situation that we studied, there is usually an obvious solution obtained as a solution of linear problem. We formulate a dual variational problem in such a way that the obvious solution is a local minimum. We then find additional non-obvious solutions via a numerical mountain pass algorithm, based on the theorems of Ambrosetti, Rabinowitz and Ekeland. Numerical results are presented.Research supported in part by grant DMS-9208636 from the National Science FoundationResearch supported in part by grant DMS-9102632 from the National Science Foundation     

A block by block method for the numerical solution of Volterra integral equations

4. Conclusions TheO(h ⁶) block-by-block method developed here gives accurate results without the requirement of a start-up procedure and with good computational efficiency. The computing cost, based upon the number of timesk is evaluated, is proportional ton ²/2+3n wheren is the number of mesh points. The examples are linear, but were solved by iterating the system of equations (4) fifteen times.     

A finite element method for the numerical solution of convection-dominated anisotropic diffusion equations

Summary. The proposed method is based on an additive decomposition of the differential operator and the subsequent fitted discretization of the resulting components. For standard situations, the derived stability and error estimates in the energy norm qualitatively coincide with well-known estimates. In the case of small diffusion, a uniform error estimate with reduced order is obtained. Received August 7, 1997 / Revised version received July 15, 1998 / Published online December 6, 1999     

A numerical solution of nonlinear Volterra-Fredholm integral equations

In this paper, a numerical procedure for solving a class ofnonlinear Volterra-Fredholm integral equations is presented. Themethod is based upon the globally defined sinc basis functions.Properties of the sinc procedure are utilized to reduce thecomputation of the nonlinear integral equations to some algebraicequations. Illustrative examples are included to demonstrate thevalidity and applicability of the method.     

Matrix displacement mappings in the numerical solution of functional and nonlinear differential equations with the tau method

The use of matrix displacement mappings reduces most matrix operations required in the construction of an approximate solution of a functional or differential equation by means of Ortiz' formulation of the Tau method to index shifts. The coefficient vector of the approximate solution is defined implicitly by a very sparse system of linear algebraic equations. The contributions of the differential or functional operator, and of the supplementary conditions of the problem (initial, boundary, or multipoint conditions) are treated with a single and versatile procedure of remarkable simplicity, which can be easily implemented in a computer. We give two nontrivial examples on the application of this approach: the first is a nonlinear boundary value problem with a continuous locus of singular points and multiple solutions, where stiffness is present, the second is a functional differential equation arising in analytic number theory. In both cases we obtain results of nigh accuracy.     

Variable-step variable-order algorithm for the numerical solution of neutral functional differential equations

The variable-step variable-order algorithm based on predictor-corrector methods for neutral functional differential equations is described. The algorithm is implemented in interpolation mode, i.e. we use fixed-step formulation of Adams-Bashford Adams-Moulton formulas while all back-values are computed by interpolation. The local discretization error is estimated by Milne's device. Although the assumptions under which this estimate is asymptotically correct are rather strong and not always expected to be satisfied in practice, the numerical experiments up to date indicate that this algorithm can be quite accurate and efficient.     

Preconditioned Barzilai-Borwein method for the numerical solution of partial differential equations

The preconditioned Barzilai-Borwein method is derived and applied to the numerical solution of large, sparse, symmetric and positive definite linear systems that arise in the discretization of partial differential equations. A set of well-known preconditioning techniques are combined with this new method to take advantage of the special features of the Barzilai-Borwein method. Numerical results on some elliptic test problems are presented. These results indicate that the preconditioned Barzilai-Borwein method is competitive and sometimes preferable to the preconditioned conjugate gradient method.This author was partially supported by the Parallel and Distributed Computing Center at UCV.This author was partially supported by BID-CONICIT, project M-51940.     

Explicit multistep method for the numerical solution of stiff differential equations

An explicit multistep method of variable order for integrating stiff systems with high accuracy and low computational costs is examined. To stabilize the computational scheme, componentwise estimates are used for the eigenvalues of the Jacobian matrix having the greatest moduli. These estimates are obtained at preliminary stages of the integration step. Examples are given to demonstrate that, for certain stiff problems, the method proposed is as efficient as the best implicit methods.     

Local Linearization method for the numerical solution of stochastic differential equations

The Local Linearization (LL) approach for the numerical solution of stochastic differential equations (SDEs) is extended to general scalar SDEs, as well as to non-autonomous multidimensional SDEs with additive noise. In case of autonomous SDEs, the derivation of the method introduced gives theoretical support to one of the previously proposed variants of the LL approach. Some numerical examples are given to demonstrate the practical performance of the method.     

On Nordsieck's method for the numerical solution of ordinary differential equations

The connection between the class of methods suggested by Nordsieck and the class of linear multi-step methods is examined. It is shown that the starting procedure suggested by Nordsieck is specially suited to the Adams method.     

The numerical method of successive interpolations for Fredholm functional integral equations

A new numerical method for Fredholm functional integral equations is proposed. The method combines the fixed point technique with numerical integration and cubic spline interpolation. The convergence and the numerical stability of the method are proved and tested on some numerical examples.     

A note on order of convergence of numerical method for neutral stochastic functional differential equations

In this paper, we study the order of convergence of the Euler-Maruyama (EM) method for neutral stochastic functional differential equations (NSFDEs). Under the global Lipschitz condition, we show that the pth moment convergence of the EM numerical solutions for NSFDEs has order p/2 − 1/l for any p ? 2 and any integer l > 1. Moreover, we show the rate of the mean-square convergence of EM method under the local Lipschitz condition is 1 − ε/2 for any ε ∈  (0, 1), provided the local Lipschitz constants of the coefficients, valid on balls of radius j, are supposed not to grow faster than log j. This is significantly different from the case of stochastic differential equations where the order is 1/2.     

Moment matching method for numerical solution of MTL equations in interconnect analysis

Multiconductor transmission line (MTL) analysis is a popular technique for evaluating high-speed electrical interconnects. Typically, MTLs are modeled in the Laplace domain and similarity transformations are used to decouple the MTL equations. For high-speed systems, however, direct solution of the MTL equations at a large number of frequencies is computationally very expensive. Recent studies have employed moment matching techniques to approximate the solution for the MTL equations and improve the computational efficiency. In this study, a generalization of the method of characteristics is further studied for solving the MTL equations for lossy transmission lines. An efficient recursive solution for generating the moments of eigenvalues and eigenvectors is presented. Numerical results of this moment matching technique agree with the direct solution methods up to 10GHz.