We study a nonstandard formulation of the Neumann initial value problem \begin{equation} \begin{array}{rl} u_t(x,t) = \Delta \phi(u(x,t)), & x \in \Omega \subseteq \mathbb{R}^k, \ t \mathbb{R} \label{abstract}\\ u(x,0) u_0(x), \Omega. \end{array} \end{equation} with boundary conditions. The function $\phi C^1(\mathbb{R})$ is assumed to be decreasing either in bounded interval $(u^-,u^+)$, or an...

We treat an ill-posed system of linear equations by transforming it into a linear system of stiff ordinary differential equations (SODEs), adding a differential term on the left-hand side. In order to overcome the difficulty of numerical instability when integrating the SODEs, Liu [20] has combined nonstandard finite difference method and group-preserving scheme, namely the nonstandard group-pr...

We consider the classic problem of minimizing a quadratic cost functional for well-posed linear systems over the class of inputs that are square integrable and that produce a square integrable output. As is well-known, the minimum cost can be expressed in terms of a bounded nonnegative selfadjoint operator X that in the finite-dimensional case satisfies a Riccati equation. Unfortunately, the in...

considering the nonlinear hyperelastic or viscoelastic nature of soft tissues has an important effect on modeling results. in medicalapplications, accounting nonlinearity begets an ill posed problem, due to absence of external force. myocardium can be consideredas a hyperelastic material, and variational approaches are proposed to estimate stiffness matrix, which take into account the linearand...