This paper uses Biot's theory for poroelastic materials to develop a means for the clinical investigation of bone rigidity using acoustic fields. The problem is studied by considering an incident wave impinging on a bone sample and producing a scattered wave. In particular, this is modeled by supposing a bone sample is placed in a water tank with a transducer on one side and acoustic receiver on the other. The boundary-field equations are used to formulate this model as a nonlocal boundary problem in the Laplace transformed domain. Existence, uniqueness and stability of the weak solution to the nonlocal boundary problem are established in appropriate Sobolev spaces in terms of the Laplace transformed variable. The stability bounds are transformed into time-domain estimates.

An explicit formula for the polyharmonic Green function of any order for the unit disc of the complex plane based on the work of E. Almansi is widely known through I.N. Vekua’s 1967 book. A simple, however, not easy way to prove the formula is by verification of the defining properties. P.R. Garabedian has suggested a constructive way for this function in his 1964 pde book but only has applied it for the biharmonic case. His construction is not smoothly working for higher order as is explained here in the triharmonic case. The general case will demand very concentrated calculations. The procedure should similarly work also for the half plane.

Motivated by the desire to model the entry of 1,25D into a cell by receptor mediated endocytosis, we have formulated the problem as the dynamics of a bilayer membrane. We have discussed setting the problem as a variational problem using the Helfrich modeling of the bilayer in terms of the free energy. Using a Lagrangian formulation we arrive at the Euler–Lagrange equations for the system. The model we have used depends on the amount of reagent in the neighborhood of the upper membrane. The problem thereby reduces to a moving boundary problem, which is dependent on a diffusion equation for a region changing with time. In order to solve this problem we seek the correct Neumann function for this altered. This is accomplished by deriving a Hadamard variational formula for the diffusion equation. We also offer an iterative procedure for solving this non-linear problem.

In this paper, we investigate ultrasound waves passing through a skeletal muscle segment of a specimen. The model is solved using an extension of a method due to Ilya Vekua. An ansatz is made that a solution to a partial differential equation can be found in a form resembling the ansatz of Vekua. It is shown that this is indeed possible. This method is used continuously throughout the paper to find simple representations of the acoustic equations in the different muscle and bone regions.

aCentre de Mathématiques Appliquées, École Polytechnique, CNRS, Institut Polytechnique de Paris, Palaiseau, France; bBâtiment IPRA, Université de Pau et des Pays de l’Adour, Pau, France; cDepartment of Mathematics, University of Delaware, Newark, DE, USA; dInstitut für Angewandte Mathematik, Universität Heidelberg, Heidelberg, Germany; eThe Arctic University of Norway, campus Narvik, Narvik, Norway; fWorcester Polytechnic Institute, Worcester, MA, USA

We outline the mathematical model of the ultrasonic response of wet cortical bone and its time‐harmonic formulation. We employ an energetic approach based on the Reuss bound of the free energy of a porous material consisting of a piezo‐electric solid and a conducting fluid part. Magnetic effects are taken into consideration. Corresponding boundary value problems are stated, and associated theorems are established. A conclusion is included concerning future developments of this formulation.

Eukaryotic cells are complex systems which carry out a variety of different tasks. The current contribution gives insight into the modeling of some of their vital components and represents an overview of results achieved within the international D‐A‐CH project on computational modeling of transport processes in a cell. The first part of the contribution studies viscoelastic effects of cross‐linked actin network embedded in cytosol. The basic‐model is used to simulate the actin behavior at a microscopic level. It considers the influence of the physical length, the end‐to‐end distance and the stretch modulus in order to provide a relationship between the stretch of a single polymer chain and the applied tension force. The effective behavior of the cell cytoplasm is simulated by using the multiscale finite element method. Here, a standard large strain viscous approach is applied for the cytosol, while the generalized Maxwell model simulates viscous effects occurring in filaments due to deviatoric changes. The examples dealing with combinations of tension‐holding tests give insight into the effective behavior of the cytoplasm.

We outline the mathematical model of the ultrasonic response of cancellous bone and its time harmonic formulation. In contrast to the Biot model, the fluid is not inviscid. Our fluid is viscous, but does not interact with the solid components.

The present contribution focuses on the receptor driven endocytosis typical of viral entry into a cell. The process is characterized by a local increase in receptor density necessary to establish contact between the cell and the virus. While the receptors of the virus are fixed on its surface, the receptors of the cell are able to move over its membrane, which leads to a local change in their concentration. In the model developed, the receptor motion is described by the diffusion equation along with two boundary conditions. The boundary conditions represent the balance of fluxes at the front of the contact area, where the velocity is assumed to be proportional to the gradient of the chemical potential, and the energy balance behind and before the front, causing the fronts movement. The model provides a basis to incorporate different phenomena such as for example cooperativity. This property strongly influences the effective binder density necessary to establish contact. The moving boundary problem describing the process is numerically solved by using the finite difference method and applied to study the change of receptor density over the membrane as well as the motion of the adhesion front. Two features are investigated in particular: The process initiation is analyzed in order to obtain information on possibilities for inhibiting the viral entry, and the non‐dimensional analysis is performed to minimize the number of necessary process parameters and to check their priority.

However, differential equations which arise in “real” world problems, such as from physics or engineering, rarely have analytic solutions which are easy (or even possible) to find because the world is complex and chaotic, not nice and linear. Even in Calculus, you see differential equations such as y′ = e−t 2 which do not have solutions which are elementary functions. (Exercise: Look up the definition of elementary function in your Calculus textbook).