On the Existence of Eigenvalues of a Boundary Value Problem with Transmitting Condition of the Integral Form for a Parabolic-Hyperbolic Equation

In the paper, we investigate a local boundary value problem with transmitting condition of the integral form for mixed parabolic-hyperbolic equation with non-characteristic line of type changing. Theorem on strong solvability of the considered problem has been proved and integral representation of the solution is obtained in a functional space. Using Lidskii Theorem on coincidences of matrix and spectral traces of nuclear operator and Gaal’s formula for evaluating traces of nuclear operator, which is represented as a product of two Hilbert-Schmidt operators, we prove the existence of eigenvalues of the considered problemThe research of the first author is supported by the grant of the Committee of Sciences, Ministry of Education and Science of the Republic of Kazakhstan to the Institute of Information and Computational Technologies, project AP05131026. Second author is supported by the Agencia Estatal de Investigación (AEI) of Spain under grant MTM2016-75140-P, co-financed by the European Community fund FEDER and by Xunta de Galicia, project ED431C 2019/02 (Spain)S

On the Existence of Eigenvalues of a Boundary Value Problem with Transmitting Condition of the Integral Form for a Parabolic-Hyperbolic Equation

In the paper, we investigate a local boundary value problem with transmitting condition of the integral form for mixed parabolic-hyperbolic equation with non-characteristic line of type changing. Theorem on strong solvability of the considered problem has been proved and integral representation of the solution is obtained in a functional space. Using Lidskii Theorem on coincidences of matrix and spectral traces of nuclear operator and Gaal’s formula for evaluating traces of nuclear operator, which is represented as a product of two Hilbert-Schmidt operators, we prove the existence of eigenvalues of the considered problem

A Priori Estimates for the Solution of an Initial Boundary Value Problem of Fluid Flow through Fractured Porous Media

The paper studies a model of fluid flow in a fractured porous medium in which fractures are distributed uniformly over the volume. This model includes a nonlinear equation containing several terms with fractional derivatives in the sense of Caputo of order belonging to the interval 1,2. The relevance of studying this problem is determined by its practical significance in the oil industry, since most of the world’s oil reserves are in these types of reservoirs. The uniqueness of the solution to the problem in a differential form and its dependence on the initial data and the right-hand side of the equation is proved. A numerical method is proposed based on the use of the finite difference approximation for integer and fractional time derivatives and the finite element method in the spatial direction. A change of variables is introduced to reduce the order of the fractional derivatives. Furthermore, the fractional derivative is approximated by using the L1-method. The stability and convergence of the proposed numerical method are rigorously proved. The theoretical order of convergence is confirmed by the results of numerical tests for a problem of fluid flow in fractured porous media with a known exact solution

Convergence Analysis of a Numerical Method for a Fractional Model of Fluid Flow in Fractured Porous Media

The present paper is devoted to the construction and study of numerical methods for solving an initial boundary value problem for a differential equation containing several terms with fractional time derivatives in the sense of Caputo. This equation is suitable for describing the process of fluid flow in fractured porous media under some physical assumptions, and has an important applied significance in petroleum engineering. Two different approaches to constructing numerical schemes depending on orders of the fractional derivatives are proposed. The semi-discrete and fully discrete numerical schemes for solving the problem are analyzed. The construction of a fully discrete scheme is based on applying the finite difference approximation to time derivatives and the finite element method in the spatial direction. The approximation of the fractional derivatives in the sense of Caputo is carried out using the L1-method. The convergence of both numerical schemes is rigorously proved. The results of numerical tests conducted for model problems are provided to confirm the theoretical analysis. In addition, the proposed computational method is applied to study the flow of oil in a fractured porous medium within the framework of the considered model. Based on the results of the numerical tests, it was concluded that the model reproduces the characteristic features of the fluid flow process in the medium under consideration

Boundary value problems for fourth-order mixed type equation with fractional derivative

In this work we study direct and inverse problems for fourth-order mixed type equations with the Caputo fractional derivative. Applying method of separation of variables we prove unique solvability of these problems

Solvability Issues of a Pseudo-Parabolic Fractional Order Equation with a Nonlinear Boundary Condition

This paper is devoted to the fundamental problem of investigating the solvability of initial-boundary value problems for a quasi-linear pseudo-parabolic equation of fractional order with a sufficiently smooth boundary. The difference between the studied problems is that the boundary conditions are set in the form of a nonlinear boundary condition with a fractional differentiation operator. The main result of this work is establishing the local or global solvability of stated problems, depending on the parameters of the equation. The Galerkin method is used to prove the existence of a quasi-linear pseudo-parabolic equation’s weak solution in a bounded domain. Using Sobolev embedding theorems, a priori estimates of the solution are obtained. A priori estimates and the Rellich–Kondrashov theorem are used to prove the existence of the desired solutions to the considered boundary value problems. The uniqueness of the weak generalized solutions of the initial boundary value problems is proved on the basis of the obtained a priori estimates and the application of the generalized Gronwall lemma. The need to consider and study such initial boundary value problems for a quasi-linear pseudo-parabolic equation follows from practical requirements, such as solving fractional differential equations that simulate physical processes that occur during the study of liquid filtration processes, etc

Розробка алгоритму розрахунку стійких розв'язків рівняння сен-венана за допомогою протипотокової неявної різницевої схеми

The problem of numerical determination of Lyapunov-stable (exponential stability) solutions of the Saint-Venant equations system has remained open until now. The authors of this paper previously proposed an implicit upwind difference splitting scheme, but its practical applicability was not indicated there. In this paper, the problem is solved successfully, namely, an algorithm for calculating Lyapunov-stable solutions of the Saint-Venant equations system is developed and implemented using an upwind implicit difference splitting scheme on the example of the Big Almaty Canal (hereinafter BAC). As a result of the proposed algorithm application, it was established that: 1) we were able to perform a computational calculation of the numerical determination problem of the water level and velocity on a part of the BAC (10,000 meters) located in the Almaty region; 2) the numerical values of the water level height and horizontal velocity are consistent with the actual measurements of the parameters of the water flow in the BAC; 3) the proposed computational algorithm is stable; 4) the numerical stationary solution of the system of Saint-Venant equations on the example of the BAC is Lyapunov-stable (exponentially stable); 5) the obtained results (according to the BAC) show the efficiency of the developed algorithm based on an implicit upwind difference scheme according to the calculated time. Since we managed to increase the values of the difference grid time step up to 0.8 for calculating the numerical solution according to the proposed implicit scheme.Проблема численного определения устойчивых по Ляпунову (экспоненциальная устойчивость) решений системы уравнений Сен-Венана до сих пор оставалась открытой. Авторами данной статьи ранее была предложена неявная противопоточная разностная схема расщепления, однако не была указана её практическая применимость. В данной работе эта проблема успешно решена, а именно разработан, а также реализован алгоритм расчета устойчивых по Ляпунову решений системы уравнений Сен-Венана с помощью неявной противопоточной разностной схемы расщепления на примере Большого Алматинского канала (далее БАК). В результате применения предложенного алгоритма было установлено, что: 1) нам удалось провести вычислительный расчет задачи численного определения уровня и скорости воды на части БАК (10000 метров), расположенного в Алматинской области; 2) численные значения высоты уровня и горизонтальной скорости воды согласуются с фактическими измерениями параметров потока воды в БАК; 3) предложенный вычислительный алгоритм устойчив; 4) численное стационарное решение системы уравнений Сан-Венана на примере БАК устойчиво по Ляпунову (экспоненциально устойчиво); 5) полученные результаты (по БАК) по расчетному времени показывают эффективность разработанного алгоритма на основе неявной противопоточной разностной схемы. Поскольку значения шага разностной сетки по времени удалось увеличить до 0.8 для расчета численного решения по предложенной неявной схеме.Проблема чисельного визначення стійких за Ляпуновим (експоненціальна стійкість) розв'язків системи рівнянь Сен-Венана досі залишалася відкритою. Авторами цієї статті раніше була запропонована неявна протипотокова різницева схема розщеплення, проте не була вказана її практична застосовність. У даній роботі ця проблема вирішена успішно, а саме розроблений, а також реалізований алгоритм розрахунку стійких за Ляпуновим розв'язків системи рівнянь Сен-Венана за допомогою протипотокової неявної різницевої схеми розщеплення на прикладі Великого Алматинського каналу (далі ВАК). В результаті застосування запропонованого алгоритму було встановлено, що: 1) нам вдалося провести обчислювальний розрахунок задачі чисельного визначення рівня і швидкості води на частині ВАК (10000 метрів), розташованого в Алматинській області; 2) чисельні значення висоти рівня і горизонтальної швидкості води узгоджуються з фактичними вимірами параметрів водного потоку в ВАК; 3) запропонований обчислювальний алгоритм стійкий; 4) чисельне стаціонарне рішення системи рівнянь Сан-Венана на прикладі ВАК стійке за Ляпуновим (експоненціально стійке); 5) отримані результати (по ВАК) за розрахунковим часом показують ефективність розробленого алгоритму на основі неявної протипотокової різницевої схеми. Оскільки, значення кроку різницевої сітки за часом вдалося збільшити до 0.8 для розрахунку чисельного рішення за запропонованою неявною схемо