In this paper the Generalized Pulse-Spectrum Technique (GPST) is extended to solve an inverse problem for the Burgers equation. We prove that the GPST is equivalent in some sense to the Newton-Kantorovich iteration method. A feasible numerical implementation is presented in the paper and some examples are executed. The numerical results show that this procedure works quite well.

This paper presents a random numerical method which combines the random vortex method and the random choice method. A random choice method is used for the modeling reaction-diffusion system. The splitting of source terms in two dimensions for the random choice method is tested. A hybrid random vortex method is used for solving the Navier-Stokes equation which governs the fluid motion. With the assumption of incompressibility, the fluid motion can be uncoupled from the chemistry. The method is applied to a flow passing a circular cylinder which is kept cold or heated. In both cases the method demonstrates an ability to resolve turbulent effects on flame front propagation.

In this paper linear programming method for minimax approximation is used to obtain an approximation to the analytical solution of a Dirichlet problem using the logarithmic potential function as an approximating function. This approach has the advantage of producing a better approximation than that using other solution of the potential equation as an approximating or basis function for a problem in $n=2$ dimensions.  

In the paper, we first deduce an optimization problem from an inverse problem for a general operator equation and prove that the optimization problem possesses a unique, stable solution that converges to the solution of the original inverse problem, if it exists, as a regularization factor goes to zero. Secondly, we apply the above results to an inverse problem determining the spatially varying coefficients of a second order hyperbolic equation and obtain a necessary condition, which can be used to get an approximate solution to the inverse problem.  

In this paper, we generalize the paracontracting matrices to pseudocontracting matrices. The convergence of (parallel) iteration $$x_i=P_{ji,x_{i-1}}x_{i-1}$$ and $$x_{i+r_i}=α_{j_i}x_{i+r_i-1}+(1-α_{j_i})P_{j_i,x_i}x_i$$ where $P_{j_i,x_i}, j=1,\cdots,n$, are paracontracting and/or pseudocontracting matrices is analyzed. These iterations can also be applied to solve consistent systems with incomplete data.  

A numerical method for singularly perturbed semilinear boundary value problems is given. The method uses the fourth order Hermite scheme on a special discretization mesh. Its stability and convergence are investigated in the discrete $L^1$ norm.  

To solve a class of operator equations numerically, some general streamline diffusion methods with satisfactory convergence properties are presented in this paper. It is proved that the approximation accuracy is only half a power of $h$, the mesh size, from being optimal when these methods are applied to mixed problems and convection-diffusion problems.  

This paper develops an optimal-order multigrid method for the TRUNC plate element.  

The general interpolation formulas for discrete functional spaces of discrete functions with one index are presented in the note. These are the relationship among the discrete norms in the forms of the summation of powers, the maximum modulo, the Hölder and Lipschitz coefficients for the discrete functions.