Šiame darbe nagrinėta išsigimstanti dalinių išvestinių diferencialinių lygčių sistema su laipsniniu eilės išsigimimu. Rasti sistemos sprendiniai yra analizinės funkcijos išsigimimo daugdarų taškų aplinkose. Sprendiniai išskleisti laipsninėmis eilutėmis, kintamojo pagal kurį išsigimsta sistemos eilė laipsniais. / In this work, there was analyzed the system of partial differential equations with gradual degeneration of row. There were found the solutions of analytical function in manifold point environments. The solutions are expanded by gradual lines, by variable where system number of degrees degenerates.

This thesis discusses the mesoscopic physics of two examples for soft condensed matter: (i) aeolian sand transport and the ensuing structure formation and (ii) solutions of stiff biopolymers. It emphasizes the impact of a heterogeneous mesoscale structure on the macroscopic phenomenology. For the aeolian sand transport, we start with a coarse-grained description of the collisions between mobile grains and the sand bed. Combining the collision geometry with basic physical principles, like momentum and energy conservation, we derive the full splash statistics as a function of impact velocity and impactor--bed grain-size ratio. This serves as a starting point for attacking the complicated transport statistics of wind-blown sand. Two approaches are proposed: first, a two-species approximation that distinguishes between high-energy rebounding grains and low-energy bed ejecta, second, a statistical description that resolves the full distribution of grain trajectories. While the former provides an ideal framework to accurately predict macroscopic averages, the latter resolves the heterogeneous mesoscale structure of the transport layer. Both approaches are shown to be in excellent agreement with various laboratory and field data. We moreover establish a new perspective on the transport's saturation transients that illustrates the crucial influence of the intermittent turbulent wind fluctuations in the field, thereby resolving a long-standing debate. Eventually, we address the formation of megaripples, an aeolian bedform that is made from strongly polydisperse sand. Combining our proposed theory with long-term field measurements, we clarify the importance of wind-driven sand sorting and, again, intermittent turbulent fluctuations. Our approach suggests to describe the megaripples as down-scaled dunes, as indeed support by various field data for their morphology and migration. In the second part of the thesis, we consider the mesoscale structure and its influence on the viscoelastic response of entangled biopolymer solutions. Their mechanics is determined by the topological entanglements of the filamentous polymers that cannot pass through each other. The tube model for semiflexible polymers represents this effect on a mean-field level, where test filament is confined to a tube-like cage formed by surrounding polymers. We employ it to investigate the solution's mesoscale packing structure and its change under finite shear deformations. Comparing our predictions with systematic computer simulations and experiments, we find the tube deformations to relax quickly after the deformation, while tube alignment and hairpin conformations are found to be long lived. In a second step, we propose a new perspective on the entangled solution's dynamics. Accounting for the nonaffine response at the level of the test filament, and assuming sticky or frictional polymer contacts, we argue that soft bending deformations of the filaments can couple to stiff axial stretching modes. This allows us to explain various experimentally observed similarities of the entangled solutions to crosslinked networks, like the increasing elastic modulus with increasing length and bending rigidity of the filaments or the softening--stiffening transition as a function of polymer concentration, polymer length, deformation rate, and various solvent properties.:I Aeolian sand transport and megaripple formation 1 Introduction 2 The splash process 3 Aeolian sand transport 4 Structure formation 5 Outlook Bibliography II Solutions of stiff biopolymers 6 Introduction 7 Entangled solutions 8 Viscoelastic response Appendix Bibliography Acknowledgments