Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

The differential quadrature method is a numerical solution technique for initial and/or boundary problems. It was developed by the late Richard Bellman and his associates in the early 70s and, since then, the technique has been successfully employed in a variety of problems in engineering and physical sciences. The method has been projected by its proponents as a potential alternative to the conventional numerical solution techniques such as the finite difference and finite element methods. This paper presents a state-of-the-art review of the differential quadrature method, which should be of general interest to the computational mechanics community.

Differential Quadrature Method in Computational Mechanics: A Review

Barycentric interpolation is a variant of Lagrange polynomial interpolation that is fast and stable. It deserves to be known as the standard method of polynomial interpolation.

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Barycentric Lagrange Interpolation

A global method of generalised differential quadrature is applied to solve the two-dimensional incompressible Navier-Stokes equations in the vorticity-stream-function formulation. Numerical results for the flow past a circular cylinder were obtained using just a few grid points. A good agreement is found with the experimental data.

Application of generalized differential quadrature to solve two-dimensional incompressible navier-stokes equations

New insights in solving distributed system equations by the quadrature method—I. Analysis

Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations

can be determinedusing interpolating polynomials. Although integral quadratures with a variety of interpolatingpolynomials are fully developed, the differential quadratures are still in an early stage ofdevelopment. Differentialquadrature hasobviousapplicationsinthe numericalsolutionof partialdifferential equations. ordinary differential equations withboundary conditions and identificationproblems. In this paper, we deal mostly with the former case and refer the reader to [1-3Jfor thediscussions of other applications.Ingeneral, the interpolationformulas are expressed intwoways,either intermsof differencesof the function or in terms of values of the function. The derivatives of a function can also beexpressed in the same mannerin both cases. The firstcase, i.e. the approximationof derivativeswith differences, provides the basis of many methods of solving differential equations (thedifference methods). This approximation can also be used for the estimation of errors innumerical differentiation. However, the application of the second case (i.e. approximatingthederivatives in terms of the values of the function) for the purpose of numerical differentiationwas hastily dropped, mainly due to the hazards of the roundoff errors in calculation, especiallywhen dealing with low accuracy empirical data. A better approach used for numericaldifferentiation was to first "smooth" the data and then to differentiate[4J. In the differentialquadrature method we use the second approximationfor solutionof partialdifferentialequations,and effectively smooth the "data" by employing spline interpolation for determination ofcoeffi cient

Differential quadrature and splines

In this paper, we outline a new method for numerical solution of the nonlinear partial differential equation of the Hodgkin-Huxley model. Preliminary results of numerical experimentation are presented.

Solution of the partial differential equation of the Hodgkin-Huxley model using differential quadrature

A model of the electrical activity of the heart relating the ventricular dipoles to cardiogram measurements naturally leads to the inverse problem of estimating the heart parameters from the observation of skin potentials. The quasilinearization method for handling this multipoint boundary value problem necessitates the use of an initial guess for some of the parameters and the use of a large digital computer for the solution of a large system of linear equations. An alternate technique involving differential quadrature which obviates the use of any initial guess, if feasible. In this technique, a small on-line computer will produce the results on a real-time basis for as many as 20 myocardial segments. The details of this procedure and some numerical experiments form the contents of this article.

The inverse problem of estimating heart parameters from cardiograms

With the recent advance in parallel processing computers, it is important to develop a library of computational procedures which can be applied one after the other to obdurate problems in an adaptive fashion. The methods of differential and integral quadrature are discussed and are shown to be such powerful computational procedures. To illustrate the approach, these methods, combined with other numerical procedures such as spline, are applied to system identification and radiative transfer.

Bayesteh Kashef

Papers

Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations

Differential quadrature and splines

Solution of the partial differential equation of the Hodgkin-Huxley model using differential quadrature

The inverse problem of estimating heart parameters from cardiograms

Solving hard problems by easy methods: differential and integral quadrature