A formulation of an inverse problem of a partial differential equation with multi-parameter to be determined is introduced. The numerical algorithm, pulse-spectrum technique, is extended to solve this type of inverse problem. An example for remote sensing of the thermal conductivity and specific heat of a nonhomogeneous material is demonstrated. Numerical simulations are carried out to test the feasibility and to study the general characteristics of this technique without real measurement data. It is found that the extended pulse-spectrum technique gives excellent results.

The estimation for solutions for the ill-posed Cauchy problems of the differential equation $\frac{du(t)}{dt}=A(t)u(t)+N(t)u(t),\forall t\in (0,1)$ is discussed, where $A(t)$ is a 2-nd order p.d.o. and $N(t)$ is a uniformly bounded $h-›H$ linear operator. Two estimates of $||u(t)||$ are obtained.

Let $l$ and $u$ be upper and lower semicontinuous extended functions on [a,b], respectively, with $l≤u$. Let $H$ be an n-dimensional Haar subspace and $K=\{p\in H:l\leq p\leq u\}$. This paper gives complete characterizations of K satisfying $$card \ K=0 \ or \ 1 \  or \ ∞$$ under certain assumptions, where card K denotes the cardinality of K.

In this paper, some general kinds of lacunary spline interpolations are discussed. Existence and uniqueness are proved. Order of convergence and saturation of approximation are also obtained. The results generalize some individual results in references.

An order interval test for existence and uniqueness of solutions to a nonlinear system is given. It combines the interval technique and the monotone iterative technique. It has the main merits of interval iterative methods but need not use interval arithmetic. An order interval Newton method is also given, which is globally convergent. It is a generalization of the results in [3,4,13.3].

The correction procedure has been discussed by L. Fox and V. Pereyra for accelerating the convergence of a certain approximate solution. Its theoretical basis is the existence of an asymptotic expansion for the error of discretization proved by Filippov and Rybinskii and Stetter: $u-u_h=h^2 v+O(h^4)$, where $u$ is the solution of the original differential equation, $u_h$ the solution of the approximate finite difference equation with parameter $h$ and $v$ the solution of a correction differential equation independent of $h$. Stetter et al. used the extrapolation procedure to eliminate the auxiliary function $v$ while Pereyra et al. used some special procedure to solve v approximately.  
In the present paper we will present a difference procedure for solving $v$ easily.  

This paper suggests a family of parallel iterations with parameter p for finding all roots of a polynomial simultaneously. The convergence of the methods is of order p+2. The methods may also be applied to interval iterations.