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Нелокальные краевые задачи для параболических уравнений, в том числе уравнения теплопроводности, стали объектом исследований достаточно давно. Интерес к такого рода задачам вызван необходимостью дальнейшего развития теории краевых задач со смещением (задач Нахушева), а также в связи с их многочисленными приложениями. Настоящая статья посвящена исследованию вопроса однозначной разрешимости одного класса нелокальных краевых задач для уравнения теплопроводности. Рассмотрена задача отыскания регулярного решения уравнения теплопроводности с дробной производной Римана -Лиувилля в граничных условиях. Рассмотрена задача Коши для уравнения, эквивалентного исходному уравнению, при этом доказано, что рассматриваемая краевая задача редуцируется к первой краевой задаче для уравнения теплопроводности при условии, что задача Коши имеет единственное решение в классе функций, удовлетворяющих условиям А. Н. Тихонова. При этом решение представимо в виде интегрального уравнения, содержащим функцию Барретта в ядре. Также редукцией к системе дифференциальных уравнений с дробной производной Римана – Лиувилля решается вопрос единственности и существования решения поставленной задачи, когда в условии стоят значения решения на другом конце. Полученные в работе результаты послужат основой для дальнейшего исследования нелокальных краевых задач для дифференциальных уравнений параболического типа, лежащих в основе математического моделирования процессов в системах с фрактальной структурой, а также развития теории дифференциальных уравнений дробного порядка.

Strain rate sensitivity of stress-strain curves family generated by the Boltzmann–Volterra linear viscoelasticity constitutive equation (with an arbitrary relaxation modulus) under uni-axial loadings at constant strain rates is studied analytically as the function of strain and strain rate. The general expression for strain rate sensitivity index is derived and analyzed assuming relaxation modulus being arbitrary. Dependence of the strain rate sensitivity index on strain and strain rate and on relaxation modulus qualitative characteristics is examined, conditions for its monotonicity and for existence of extrema, the lower and the upper bounds and the limit values of the strain rate sensitivity as strain rate tends to zero or to infinity are studied. It is found out that (within the framework of the linear viscoelasticity) the strain rate sensitivity index which is, generally speaking, the function of two independent variables (namely strain and strain rate), depends on the single argument only that is the ratio of strain to strain rate. So defined function of one real variable is termed the strain rate sensitivity function and it may be regarded as a material function. The explicit integral expression (and the two-sided bound) for relaxation modulus in terms of strain rate sensitivity function is derived which enables one to restore relaxation modulus assuming a strain rate sensitivity function is given. The strain rate sensitivity function is represented as a linear function of ratio of tangent modulus to secant modulus of a stress-strain curve at any fixed constant strain rate and can be evaluated in such a way using experimental data. It is proved that the strain rate sensitivity value is confined in the interval from zero to unity (the upper bound of strain rate sensitivity index for pseudoplastic media) whatever strain and strain rate magnitudes. It is found out that the linear theory can reproduce increasing or decreasing or non-monotone dependences of strain rate sensitivity on strain rate (for any fixed strain) and it can provide existence of local maximum or minimum or several extrema as well without any complex restrictions on the relaxation modulus. General properties and peculiarities of the theoretic strain rate sensitivity function are illustrated by the examination of the classical regular and singular rheological models (consisting of two, three or four spring and dashpot elements) and fractional models. Namely, the Maxwell, Kelvin–Voigt, standard linear solid, Zener, anti-Zener, Burgers, anti-Burgers, Scott–Blair, fractional Kelvin–Voigt models and their parallel connections are considered. The carried out analysis let us to conclude that the linear viscoelasticity theory (supplied with common relaxation function which are non-exotic from any point of view) is able to produce high values of strain rate sensitivity index close to unity (the upper bound of strain rate sensitivity index for pseudoplastic media) and to provide existence of the strain rate sensitivity index maximum with respect to strain rate. Thus, it is able to simulate qualitatively existence of a flexure point on log-log graph of stress dependence on strain rate and its sigmoid shape which is one of the most distinctive features of superplastic deformation regime observed in numerous materials tests.

In this paper, a new class of the periodic boundary value problem for nonlinear implicit fractional differential equations involving Hilfer fractional derivative is considered in the weighted space of functions. We establish sufficient conditions for existence, uniqueness, Ulam-Hyers and Ulam-Hyers-Rassias stability of the given problem. The main results are based upon the technique of the Schaefer fixed point theorem, the Banach fixed point theorem, generalized Gronwall inequality, and with the help of some properties of Mittag-Leffler functions. An example is presented to illustrate our main results.

Some subjects of the well-formed initial value problems for ordinary differential equa- tions with Riemann-Liouville derivatives are discussed. As an example the simplest linear homogeneous differential equation with two fractional derivatives is cited. It's shown, that the requirement of the highest derivative summability influence the value of the lowest derivative order or the initial values in Cauchy type conditions. The specific class of functions, allowing the non-summability of the highest derivative, is intro- duced. The correctness of the modified Cauchy type problem and initial value problems with local and nonlocal conditions is substantiated.

In this work a solution is obtained for the boundary problem for two-dimensional thermal conductivity equation with derivatives of fractional order on time and space variables by grid method. Explicit and implicit difference schemes are developed. Stability criteria of these difference schemes are proven. It is shown that approximation order-by time equal but by space variables it equal two. A solution method is suggested using fractional steps. It is proved that the transition module, corresponding to two half-steps, approximates the transition module for given equation.

The problems of the mathematical modeling of the contaminated groundwater migration, which cannot be described in the context of the mass transfer theory classical approach, are considered. For the study of space-time conformity of nonlinear effects, determined by scaled invariance, the well-set nonlocal boundary problem for the model differential equation of nonlinear migration is proposed.