Asai Asaithambi

A finite-difference method for the Falkner-Skan equation

We present a finite-difference method for the solution of the Falkner-Skan equation. The method uses a coordinate transformation to map a semi-infinite physical domain to the unit interval [0, 1]. After using a suitable change of variables, the transformed third order boundary value problem for this domain is solved using a finite-difference scheme. The numerical solutions obtained by the present method are in agreement with those obtained by previous authors.

Numerical solution of the Falkner-Skan equation using piecewise linear functions

We present a finite-element method for the solution of the Falkner-Skan equation. The method uses a coordinate transformation to map the semi-infinite domain of the problem to the unit interval [0,1]. By means of a suitable change of variables, the transformed third-order boundary value problem is further rewritten as a system of differential equations consisting of a second-order equation and a first-order equation. The second-order equation is approximated using a Galerkin formulation with piecewise linear elements, and the first-order equation is approximated using a centered-difference approximation. In the process of the initial transformation, a finite computational boundary is introduced. This boundary is obtained as part of the computational solution by imposing an additional asymptotic boundary condition. The solutions obtained thus are in excellent agreement with those obtained by previous authors.

Numerical solution of the Burgers' equation by automatic differentiation

Abstract We compute the solution of the one-dimensional Burgers’ equation by marching the solution in time using a Taylor series expansion. Our approach does not require symbolic manipulation and does not involve the solution of a system of linear or non-linear algebraic equations. Instead, we use recursive formulas obtained from the differential equation to calculate exact values of the derivatives needed in the Taylor series. We illustrate the effectiveness of our method by solving four test problems with known exact solutions. The numerical solutions we obtain are in excellent agreement with the exact solutions, while being superior to other previously reported numerical solutions.

A shooting method for nonlinear heat transfer using automatic differentiation

The steady-state temperature distribution in a cylinder of unit radius is modeled by a nonlinear two-point boundary value problem with an endpoint singularity. We present a simple shooting method for the solution of this problem. It is well known that shooting methods solve initial-value problems repeatedly until the boundary conditions are satisfied. For the solution of the initial value problems, the method of this paper uses a technique known as automatic differentiation and obtains a Taylor series expansion for the solution. Automatic differentiation is the process of obtaining the exact values of derivatives needed in the Taylor series expansion using recursive formulas derived from the governing differential equation itself. Thus, the method does not face the need to deal with step-size issues or the need to carry out lengthy algebraic manipulations for obtaining higher-order derivatives. The method successfully reproduces the solutions obtained by previous researchers.

Construction of all non-reducible descriptors

We present a computational procedure based on a decision-tree model for the identification and construction of all non-reducible descriptors in a supervised pattern recognition problem in which pattern descriptions consist of Boolean features. We illustrate the use of the procedure through its application to the standard problem of recognizing Arabic numerals. We also present an analysis of the computational complexity of the procedure.

On computational complexity of non-reducible descriptors

We present a supervised pattern recognition model that uses Boolean formulas for non-reducible descriptors. This model leads to computational problem which is shown to be NP-complete. In the paper, we identify two open combinatorial problems in the construction of non-reducible descriptors that can be applied to a large set of applications.

Numerical solution of the one-dimensional time-independent Schrödinger's equation by recursive evaluation of derivatives

We develop a simple numerical method for solving the one-dimensional time-independent Schrodingers equation. Our method computes the desired solutions as Taylor series expansions of arbitrarily large orders. Instead of using approximations such as difference quotients for the derivatives needed in the Taylor series expansions, we use recursive formulas obtained using the governing differential equation itself to calculate exact derivatives. Since our approach does not use difference formulas or symbolic manipulation, it requires much less computational effort when compared to the techniques previously reported in the literature. We illustrate the effectiveness of our method by obtaining numerical solutions of the one-dimensional harmonic oscillator, the hydrogen atom, and the one-dimensional double-well anharmonic oscillator.

Numerical solution of Stefan problems using automatic differentiation

Abstract One-dimensional Stefan problems are described by a parabolic partial–differential equation, along with two boundary conditions on a moving boundary. The moving boundary needs to be determined as part of the solution. In this paper, we develop a simple numerical method for solving such problems using a technique known as automatic differentiation. The method obtains a Taylor series expansion for the solution whose coefficients are computed using recursive formulas derived from the differential equation itself. We illustrate the method using the Stefan problem concerning the heat transfer in an ice–water medium. The computational results obtained by the present method are in excellent agreement with the results reported previously by other researchers.

Multidimensional pattern recognition problems and combining classifiers

Abstract An algebraic approach for solving multidimensional supervised pattern recognition problems is considered. The original problem is first partitioned to a number of individual problems of lower dimensions. Then the optimal output classifier is constructed by combining classifiers for the individual problems.

Using automatic differentiation to solve concentration dependent diffusion problems

We present a Taylor series method for the solution of a class of nonlinear diffusion problems involving a concentration dependent diffusion coefficient. The computational initial-boundary value problem is to determine the concentration of the diffusing substance in a semi-infinite domain at any time, starting with a given initial concentration. The method of solution begins by first mapping the semi-infinite physical domain to a finite computational domain. Then the solution at each spatial grid point is advanced in time using a Taylor series expansion. The method employs a technique known as automatic differentiation, which is neither numerical nor symbolic. This technique is the evaluation of the coefficients in a Taylor series expansion using recursive formulas derived from the differential equation describing the initial value problem. The results obtained using the method of this paper is are in excellent agreement with approximate similarity solutions obtained previously.