These notes contain an introduction of the theory of multiscale analysis and periodic homogenization of PDEs. Basic tools, such as multiscale expansion, 2-scale convergence, and Gamma-convergence are introduced and carefully explained. Application to diffusion equation or porous media are also considered, as well as numerical methods. Exercises are also provided to help the reader to become familiar with the theory.

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We study a delayed parabolic functional differential equation on a circle that is coupled with an initial value problem for the Schrodinger equation. Such equations arise as models of nonlinear optical systems with a time-delayed feedback loop, when diffusion of molecular excitation and diffraction are taken into account. The goal of this paper is to prove the existence of spatially inhomogeneous rotating-wave solutions bifurcating from homogeneous equilibria. We pass to a rotating coordinate system and seek an inhomogeneous solution to an ordinary functional differential equation. We find the solution in the form of a small parameter expansion and explicitly compute the first-order coefficients. We also provide examples of parameters that satisfy the constraints imposed throughout the analysis.

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The paper is devoted to the specific problem of continuum mechanics - numerical computation of the solution of two-dimensional Navier–Stokes equations for viscous incompressible fluids. The author plans to use the constructed numerical methods in hemodynamics to compute blood flow in elastic vessels. The support operators technique was chosen to construct the methods because it allows to construct conservative numerical methods, which can be relatively easy implemented on unstructured meshes. These properties are very important in hemodynamics. The whole family of such conservative methods was built. One of the methods was tested on the problem of fluid flow between two plane-parallel plates with different values of Reynolds number.

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The effect of operating conditions on the clearance of a countercurrent hollow fiber dialyzer has been investigated by utilizing the membrane transport model based on the volume averaging theory. The three-dimensional numerical method for describing the mass transport phenomena within a hollow fiber membrane dialyzer has been proposed to estimate performances under the several volume flow rates for blood and dialysate phases. Clearances obtained from the present numerical simulation are compared against available set of experimental data to elucidate the validity of the present three-dimensional numerical method. A series of calculations reveal the effect of the volume flow rate for blood and dialysate phases on urea clearance under the several total ultrafiltration rates. Moreover, the removal efficiency, which is the ratio of the mass flow rate of urea removed from the blood phase within a dialyzer to that at the blood phase inlet, is introduced in order to estimate an appropriate volume flow rate for blood and dialysate phases in the hemodialysis treatment. The present study clearly indicates that the present numerical method is quite useful for determining the best clinical protocol of the hemodialysis treatment and developing new dialysis systems such as home hemodialysis, nocturnal dialysis and even wearable artificial kidney.

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