In this paper, we consider a problem of recovering a space-dependent source term for the Rayleigh–Stokes equation, where the additional data is the observation at a final moment $t = T$, which is ill-posed in the sense of Hadamard. Firstly, the uniqueness, ill-posedness and the conditional stability of inverse source problem is given. Next, we develop a filter regularization method to overcome the ill-posedness of the problem. Under reasonable a priori bound assumption about the source function, a Hölder-type error estimate of the regularized solution is proved for a priori regularization parameter choice rule. Furthermore, a logarithmic-type error estimate between the exact solution and the regularized solution is established based on a posteriori regularization parameter choice rule.

In this paper, we consider the Cauchy problem for the Laplace operator. We construct approximate solutions by using the iterative method proposed by Bastay, Kozlov and Turesson. In the iterative method, we solve the corresponding boundary value problems repeatedly. Then, we show that we construct them more stably when we choose the smaller domain where we consider the boundary value problems. Furthermore, since the iterative method is applicable to the case where we know only approximations to the exact data with error, we also deal with this case.

Given a simple graph $G$ with $m$ edges, we are looking for a bijection $f$ from $E(G)$ to the integer set $\{ k+1,k+2,\ldots,k+m \}$ such that the vertex sum of each vertex $v$, $\phi(v)$, defined as the sum of $f(e)$ over all edges $e$ incident to $v$ is unique. If such a bijection $f$ exists, we say $G$ is $k$-shifted antimagic. This is a generalization of the antimagic graphs proposed by Hartsfield and Ringel [7]. In this paper, we proved that every tree of diameter four or five, except for two previous known examples, is $k$-shifted antimagic for every integer $k$.