Asymptotic method for solution of oscillatory fractional derivative

In the current paper an asymptotic method to the problem of establishing the optimal program trajectory and optimal control in the movement of a sucker rod pumping unit of oscillatory systems with liquid damper is considered, where the plunger is inside the Newtonian fluid. In this case the mass of the head is large enough. From the properties of the plunger motion, the boundary conditions are taken to be periodic and the transition of motion from one mode to the second is described by impulse systems. By means of expedient transformations, the given equation of motion with fractional derivatives is reduced to the equation of fractional order containing a small parameter (the inverse of the mass of the head). Using the method of discretization of oscillatory systems with liquid dampers, a system of the first-order difference equations is reduced to the two-dimensional system. Using the given static data, the definition of fractional derivatives in subordinate terms is considered, a quadratic functional is constructed and this problem is investigated by the least squares method. Constructing the extended functional the discrete Euler-Lagrange equations are obtained. The control actions and the corresponding optimal program trajectory is found from the obtained system of discrete Euler-Lagrange equations using the Matlab software package and the algorithm of the calculation process is proposed. The results are illustrated with a specific, simple numerical example from practice and the graphs of optimal control and optimal program trajectory are given. Keywords: asymptotic method; Newtonian fluid; Euler-Lagrange equations; algorithm of the calculation.

In this paper, we propose an efficient technique-based optimal homotopy analysis method with Green’s function technique for the approximate solutions of nonlocal elliptic boundary value problems. We first transform the nonlocal boundary value problems into the equivalent integral equations. We then apply the optimal homotopy analysis method for the approximate solution of the considered problems. Several examples are considered to compare the results with the existing technique. The numerical results confirm the reliability of the present method as it tackles such nonlocal problems without any limiting assumptions. We also provide the convergence and the error estimation of the proposed method.

Как известно, в работе А.В. Бицадзе показано, что задача Дирихле для уравнения смешанного типа некорректна. Естественно возникает вопрос: нельзя ли заменить условия задачи Дирихле другими условиями, охватывающими всю границу, которые обеспечивают корректность задачи? Впервые такие краевые задачи (нелокальные краевые задачи) для уравнения смешанного типа были предложены и изучены в работах Ф.И. Франкля при решении газодинамической задачи об обтекании профилей потоком дозвуковой скорости со сверхзвуковой зоной, оканчивающейся прямым скачком уплотнения. Близкие по постановке задачи для уравнения смешанного типа второго рода второго порядка, имеются в работах А.Н. Терехова, С.Н. Глазатова, М.Г. Каратопраклиевой и С.З. Джамалова. В этих работах для уравнения смешанного типа второго рода второго порядка изучены нелокальные краевые задачи в ограниченных областях. Такие задачи для уравнения смешанного типа первого рода в трехмерном случае (в частости, для уравнения Трикоми) в неограниченных областях изучены в работах С.З. Джамалова и Х. Туракулова. Для уравнений смешанного типа второго рода в неограниченных областях нелокальные краевые задачи в многомерном случае практически не исследованы. С этой целью в данной работе в неограниченном параллелепипеде формулируется и изучается нелокальная краевая задача периодического типа для трехмерного уравнения смешанного типа второго рода второго порядка. Для доказательства единственности обобщённого решения используется метод интегралов энергии. Для доказательства существования обобщённого решения сначала используется преобразование Фурье и в результате получается новая задача на плоскости, а для разрешимости этой задачи используется методы «ε-регуляризации»и априорных оценок. Используя эти методы, и равенство Парсеваля, докажем единственность, существование и гладкость обобщённого решения одной нелокальной краевой задачи периодического типа для трехмерного уравнения смешанного типа второго рода второго порядка. As is known, A.V. Bitsadze in his studies pointed out that the Dirichlet problem for a mixed-type equation, in particular for a degenerate hyperbolic-parabolic equation, is ill-posed. The question naturally arises: is it possible to replace the conditions of the Dirichlet problem with other conditions covering the entire boundary, which will ensure the well-posedness of the problem? For the first time, such boundary value problems (nonlocal boundary value problems) for a mixed-type equation were proposed and studied in the works of F.I. Frankl when solving the gas-dynamic problem of subsonic flow around airfoils with a supersonic zone ending in a direct shock wave. Problems close in formulation to a mixed-type equation of the second order were considered in the studies by A.N. Terekhov, S.N. Glazatov, M.G. Karatopraklieva and S.Z. Dzhamalov. In these papers, nonlocal boundary value problems in bounded domains are studied for a mixed-type equation of the second kind of the second order. Such problems for a mixed-type equation of the first kind in the three-dimensional case (in particular, for the Tricomi equation) in unbounded domains are studied in the works of S.Z. Dzhamalov and H. Turakulov. For mixed-type equations of the second kind in unbounded domains, nonlocal boundary value problems in the multidimensional case are practically not studied. In this article, nonlocal boundary value problem of periodic type for a mixed-type equation of the second kind of the second order, is formulated and studied in an unbounded parallelepiped. To prove the uniqueness of the generalized solution, the method of energy integrals is used. To prove the existence of a generalized solution, the Fourier transforms is used and as a result, a new problem is obtained on the plane. And for the solvability of this problem, the methods of “ε-regularization”and a priori estimates are used. The uniqueness, existence, and smoothness of a generalized solution of a nonlocal boundary value problem of periodic type for a three-dimensional mixed-type equation of the second kind of the second order are proved using above-mentioned methods and Parseval equality.

In this paper, a nonlocal boundary value problem for the Benjamin-Bona-Mahony-Burgers equation is studied in a rectangular domain. By introducing new functions, the nonlocal boundary value problem for a nonlinear third-order partial differential equation is reduced to a boundary value problem for a second-order hyperbolic equation with a mixed derivative and functional relations. Before using the approximate method, the nonlinear problem under consideration is examined for the presence of solutions, it is necessary to clarify where these solutions are located, that is, to find the region of isolation of solutions. The isolation area of the solution in our case is a ball in which there is a unique solution to the problem. Next, an algorithm for finding a solution to a nonlocal boundary value problem is proposed. In terms of the initial data, conditions for the convergence of the algorithms are established, which simultaneously ensure the existence and isolation of a solution to a nonlinear nonlocal boundary value problem. Estimates between the exact and approximate solutions of the problem under consideration are obtained. The results obtained are of a theoretical nature and can be used in the construction of computational algorithms for solving nonlocal boundary value problems for the Benjamin-Bona-Mahony-Burgers equation.

Pseudo-differential operators and -toroidal symbols generated by a non-local boundary value problem are investigated. A formula for compositions with pseudo-differential operators generated by a non-local boundary value problem is derived. In the Hilbert space concepts of the Fourier transform and convolution generated by a non-local boundary value problem are introduced.

We consider a non-local boundary value problem for the linear third order equation with hyperbolic operator in the main part. Sufficient conditions were stated to coefficients of the equation and to given functions in order that this non-local boundary value problem has a unique solution. For the proof, we use the Riemann's method.

The nonlocal boundary value problems for differential operator equations of second order with dependent coefficients are studied. The principal parts of the differential operators generated by these problems are non‐selfadjoint. Several conditions for the maximal regularity and the Fredholmness in Banach‐valued Lp‐spaces of these problems are given. By using these results, the maximal regularity of parabolic nonlocal initial boundary value problems is shown. In applications, the nonlocal boundary value problems for quasi elliptic partial differential equations, nonlocal initial boundary value problems for parabolic equations, and their systems on cylindrical domain are studied.

On the numerical solution of some nonlinear and nonlocal boundary value problems

For elliptic boundary value problems of nonlocal type in Euclidean space, the well posedness has been studied by several authors and it has been well understood. On the other hand, such kind of problems on manifolds have not been studied yet. Present article considers differential equations on smooth closed manifolds. It establishes the well posedness of nonlocal boundary value problems of elliptic type, namely Neumann-Bitsadze-Samarskii type nonlocal boundary value problem on manifolds and also DirichletBitsadze-Samarskii type nonlocal boundary value problem on manifolds, in H¨older spaces. In addition, in various H¨older norms, it establishes new coercivity inequalities for solutions of such elliptic nonlocal type boundary value problems on smooth manifolds.

The limit behaviour of solutions to a class of non-linear and non-local elliptic boundary value problems, with respect to the shrinking of the equivalued surface, is obtained by means of the theory of boundary value problems for second order linear elliptic equations.

Nonlocal conjugate type boundary value problems of higher order

An m-point nonlocal boundary value problem is posed for quasi-linear differential equations of first order on the plane. Nonlocal boundary value problems are investigated using the algorithm of reducing nonlocal boundary value problems to a sequence of Riemann-Hilbert problems for a generalized analytic function. The conditions for the existence and uniqueness of a generalized solution in the space C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sub> (G) are considered. A algorithm and numerical method of the solution of an nonlocal problem by means of the Mathcad package is presented.

We consider nonlocal boundary value problems which includes elliptic differential operator equations of the second order with discontinuous coefficients and nonlocal boundary conditions together with transmission conditions. We establish Fredholmness for these nonlocal boundary value problems.

In three-dimensional domain, an identification problem of the source function for Hilfer type partial differential equation of the even order with a condition in an integral form and with a small positive parameter in the mixed derivative is considered. The solution of this fractional differential equation of a higher order is studied in the class of regular functions. The case, when the order of fractional operator is 0 &lt; α &lt; 1, is studied. The Fourier series method is used and a countable system of ordinary differential equations is obtained. The nonlocal boundary value problem is integrated as an ordinary differential equation. By the aid of given additional condition, we obtained the representation for redefinition (source) function. Using the Cauchy–Schwarz inequality and the Bessel inequality, we proved the absolute and uniform convergence of the obtained Fourier series.

The classical Burgers equation is well studied and is often used in problems of hydrodynamics and nonlinear acoustics. In recent years, there has been a significantly increasing interest in mathematical models that take into account the power-law memory effect of the medium. Such models are described by equations in which the time derivative is replaced by a fractional derivative. The study object is a generalized Burgers equation with a fractional Riemann-Liouville time derivative. A memory effect of the medium is assumed to be small, so a small parameter, with respect to which the fractional derivative is expanded into a series, is extracted from the fractional-order differentiation. As a result, the initial fractional differential generalization of the Burgers equation is approximated by an equation with a small parameter. The aim of the work is to study the symmetry properties of such a partial differential equation with a small parameter and to construct conservation laws for it. To achieve the goal, methods of modern group analysis are used, as well as widely known methods of integrating systems and partial differential equations of first order. The group classification of the equation under study is carried out according to the function standing at the first derivative with respect to the spatial variable. It is shown that if this function is of arbitrary form, then the admissible approximate group of point transformations is three-parameter. For power-law and linear functions, the admissible approximate group of point transformations extends to five- and seven-parameter groups, respectively. Examples of approximately invariant solutions for some admissible operators are constructed. It is proved that the studied equation with a small parameter is approximately nonlinearly self-adjoint. Based on the principle of nonlinear self-adjointness, conservation laws are constructed for each group generator. It is shown that all conservation laws are either trivial or have the form of the original equation. The results develop the theory of approximate transformation groups for fractional differential equations. The obtained symmetries can be used to construct approximate invariant solutions of the equation in question.