Bayesteh Kashef

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

Abstract The numerical solution of nonlinear partial differential equations plays a prominent role in numerical weather forecasting, optimal control theory, radiative transfer, and many other areas of physics, engineering, and biology. In many cases all that is desired is a moderately accurate solution at a few points which can be calculated rapidly. In this paper we wish to present a simple direct technique which can be applied in a large number of cases to circumvent the difficulties of programming complex algorithms for the computer, as well as excessive use of storage and computer time. We illustrate this technique with the solution of some partial differential equations arising in various simplified models of fluid flow and turbulence.

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

Abstract 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.

The inverse problem of estimating heart parameters from cardiograms

Abstract 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.

Applications of dynamic programming and scan-rescan processes to nuclear medicine and tumor detection

Abstract Scan-rescan processes are assuming a more and more important role in applications of nuclear medicine to cancer detection as well as in detection of other medical abnormalities, especially when real time processing is desirable. In this paper it is shown that dynamic programming can both improve accuracy and decrease the time required for examination. Very interesting analytic questions arise in this fashion. The numerical resolution of these questions and the methods developed will have extensive application in other areas where pattern recognition and search are involved. In particular, a hierarchal application of scan-rescan processes, suitably generalized, could have extensive use in the national Medicare program.

Splines via dynamic programming

Abstract Approximating a function with prescribed values at given points on a real interval by a cubic spline is based on the minimum curvature property of the approximation. This essential feature can be used as the criterion to determine the cubic polynomial approximation in each interval in a sequential manner by methods of dynamic programming. A stable system of recurrence relations for the coefficients of the spline in successive intervals is obtained by the methods of dynamic programming and they are shown to be identical with the usual relations of the spline approximation. Extension of this method to other types of splines is also considered.

SOLVING HARD PROBLEMS BY EASY METHODS: DIFFERENTIAL AND INTEGRAL QUADRATURE

Abstract 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.

Dynamic programming and bicubic spline interpolation

Abstract Dynamic programming techniques were used to obtain the spline approximation for a function with prescribed values on the knot points along a line. Extending this procedure to two dimensions, the bicubic spline approximation defined over a two-dimensional region is obtained in this paper employing the methods of dynamic programming. A regular rectangular region as well as a region with irregular boundaries can be handled by this method, avoiding the difficulties of large storage and high dimensionality.

On a class of hereditary processes in biomechanics

Abstract The description of biomechanical heredity processes leads to the study of non-linear Volterra integral equations such as that given by Eq. 3. It is shown that under convenient assumptions regarding the nature of the memory of the system approximate equations of evolution of the process can be given in terms of a system of difference-differential equations subject to initial conditions. Alternative formulations are discussed, mainly emphasizing numerical aspects. The identification problem is briefly discussed, and some numerical examples, obtained using synthetic data, are presented to exhibit the application and the principal ideas of the method.

Mean square spline approximation

Abstract The approximation to a specified function on the real line by fitting a cubic in a piecewise fashion is achieved by minimizing the deviations in the mean square sense. The coefficients of the cubic are determined sequentially employing the method of dynamic programming. Employing this method a known function is approximated and the results of the computation are tabulated.

A useful approximation to ⁻2

Using differential approximation, we obtain a remarkably accurate representa- tion of e-9 as a sum of three exponentials.