John Rozier Cannon

Structural identification of an unknown source term in a heat equation

The identification of an unknown state-dependent source term in a reaction-diffusion equation is considered. Integral identities are derived which relate changes in the source term to corresponding changes in the measured output. The identities are used to show that the measured boundary output determines the source term uniquely in an appropriate function class and to show that a source term that minimizes an output least squares functional based on this measured output must also solve the inverse problem. The set of outputs generated by polygonal source functions is shown to be dense in the set of all admissible outputs. Results from some numerical experiments are discussed.

An Inverse Problem for a Nonlinear Diffusion Equation

In this paper we consider the determination of an unknown diffusion coefficient in a nonlinear diffusion equation from overspecified data measured at the boundary. This inverse problem is reformulated as an “auxiliary inverse problem,” where we seek a member of a class of admissible coefficients which minimizes a given error functional. It is shown that this auxiliary problem has at least one solution in a specified admissible class. Finally, the auxiliary problem is approximated by an associated identification problem and some numerical results are presented.

Numerical procedures for the determination of an unknown coefficient in semi-linear parabolic differential equations

We consider a finite difference approximation to an inverse problem of determining an unknown source parameter p(t) which is a coefficient of the solution u in a linear parabolic equation subject to the specification of the solution u at an internal point along with the usual initial boundary conditions. The backward Euler scheme is studied and its convergence is proved via an application of the discrete maximum principle for a transformed problem. Error estimates For u and p involve numerical differentiation of the approximation to the transformed problem. Some experimental numerical results using the newly proposed numerical procedure are discussed.

Sinc-Galerkin method for solving linear sixth-order boundary-value problems

mmThere are few techniques available to numerically solve sixth-order boundary-value problems with two-point boundary conditions. In this paper we show that the Sinc-Galerkin method is a very effective tool in numerically solving such problems. The method is then tested on examples with homogeneous and nonhomogeneous boundary conditions and a comparison with the modified decomposition method is made. It is shown that the Sinc-Galerkin method yields better results.

Determining Unknown Coefficients in a Nonlinear Heat Conduction Problem

This paper considers the problem of determining the positive unknown functions

A modified nonlinear Galerkin method for the viscoelastic fluid motion equations

a( u ),b( u )

An Inverse Problem for an Unknown Source Term in a Wave Equation

and the unknown function

A Numerical Method for a Nonlocal Elliptic Boundary Value Problem

u = u( {x,t} )

On solutions of some non-linear differential equations arising in third grade fluid flows

in the nonlinear diffusion equation

Determination of the coefficient of ux in a linear parabolic equation

a( u )u_t = ( {b( u )u_x } )_x ,x > 0,t > 0