Pseudodifferential Equations Research Articles

A Semi-Analytical Analysis of Gas Slippage Effect in Pressure Transient Behavior of Non-Condensate Gas Reservoirs with the Different Boundary Condition

The conventional models utilized to study flow behavior in a Non-Condensate gas reservoir did not consider the effects of gas slippage and different outer boundary conditions. For gas wells with a constant production flow rate in a bounded oil or gas reservoir, commonly, the outer boundary conditions are infinite boundary conditions or zero flux. In this study, the dimensionless pseudo pressure and dimensionless derivative of pseudo pressure in the presence of 4 different outer boundary conditions (infinite reservoir, constant pressure, exponential, and power-law) besides effects of gas slippage, wellbore storage, and skin factor are analyzed. To do this, the dimensionless pseudo pressure partial differential equation of radial flow was derived from the combination of continuity equation with Darcy’s law, the equation of state, compressibility equation, and dimensionless parameters. Then the derived partial differential equation is solved analytically in the Laplace domain. The obtained results of this work have important significance to understand the effects of different conditions on the transient pressure behavior of Non-Condensate gas reservoirs.

On a Method of Solution of Systems of Fractional Pseudo-Differential Equations.

This paper is devoted to the general theory of linear systems of fractional order pseudo-differential equations. Single fractional order differential and pseudo-differential equations are studied by many authors and several monographs and handbooks have been published devoted to its theory and applications. However, the state of systems of fractional order ordinary and partial or pseudo-differential equations is still far from completeness, even in the linear case. In this paper we develop a new method of solution of general systems of fractional order linear pseudo-differential equations and prove existence and uniqueness theorems in the special classes of distributions, as well as in the Sobolev spaces.

Efficient asymptotics of solutions to the Cauchy problem with localized initial data for linear systems of differential and pseudodifferential equations

Начальные данные в задаче Коши мы называем локализованными, если они задаются функциями, сосредоточенными в окрестности подмногообразия положительной коразмерности, причeм размер окрестности зависит от малого параметра и вместе с ним стремится к нулю. Хотя решения линейных дифференциальных и псевдодифференциальных уравнений с локализованными начальными данными составляют относительно узкий подкласс множества всех решений, они очень важны с точки зрения физических приложений. Такие решения возникают во многих разделах математической физики. Они описывают распространение возмущений различной природы (будь то волны цунами, вызванные подводным землетрясением, или электромагнитные волны, излучаемые антеннами), и их исследованию (в том числе асимптотическому) посвящена обширная литература. Эффективными естественно называть асимптотики, позволяющие исследовать задачу достаточно быстро и с достаточно скромными вычислительными затратами. Понятие эффективности зависит от доступного вычислительного инструментария и значительно изменилось с появлением программных систем Wolfram Mathematica, MatLab и им подобных, обеспечивающих принципиально новые возможности оперативной реализации и визуализации математических построений, но и предъявляющих к конструкции асимптотик новые требования. В статье даeтся обзор современных методов построения эффективных асимптотик в задачах с локализованными начальными данными. Рассматриваемый класс уравнений и систем включает уравнения Шрeдингера и Дирака, уравнения Максвелла, линеаризованные уравнения газо- и гидродинамики, уравнения линейной теории волн на воде, теории упругости, акустики и т. д. Библиография: 109 названий.

Canonical Operator on Punctured Lagrangian Manifolds

A version of Maslov’s canonical operator on Lagrangian manifolds with singularities of special form (the so-called punctured manifolds) is described. These manifolds naturally arise in the construction of asymptotic solutions for a wide class differential and pseudodifferential equations; in particular, for Petrovsky hyperbolic systems or pseudo-differential equations of water waves.

THE MAXIMUM PRINCIPLE FOR THE EQUATION OF LOCAL FLUCTUATIONS OF RIESZ GRAVITATIONAL FIELDS OF PURELY FRACTIONAL ORDER

The parabolic pseudodifferential equation with the Riesz fractional differentiation operator of α ∈ (0; 1) order, which acts on a spatial variable, is considered in the paper. This equation naturally summarizes the known equation of fractal diffusion of purely fractional order. It arises in the mathematical modeling of local vortices of nonstationary Riesz gravitational fields caused by moving objects, the interaction between the masses of which is characterized by the corresponding Riesz potential. The fundamental solution of the Cauchy problem for this equati- on is the density distribution of the probabilities of the force of local interaction between these objects, it belongs to the class of Polya distributions of symmetric stable random processes. Under certain conditions, for the coefficient of local field fluctuations, an analogue of the maximum principle was established for this equation. This principle is important in particular for substantiating the unity of the solution of the Cauchy problem on a time interval where the fluctuation coefficient is a non-decreasing function.

Mathematical aspects of fluid–multiferroic solid interaction problems

In the paper, we consider a three‐dimensional model of fluid–solid interaction when a thermo‐electro‐magneto‐elastic body occupying a bounded region Ω+ is embedded in an inviscid fluid occupying an unbounded domain . In this case, we have a six‐dimensional thermo‐electro‐magneto‐elastic field (the displacement vector with three components, electric potential, magnetic potential, and temperature distribution function) in the domain Ω+, while we have a scalar acoustic pressure field in the unbounded domain Ω−. The physical kinematic and dynamic relations are described mathematically by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations, we prove the uniqueness and existence theorems for the corresponding boundary‐transmission problems in appropriate Sobolev‐Slobodetskii and Hölder continuous function spaces.

A residual power series method for solving pseudo hyperbolic partial differential equations with nonlocal conditions

AbstractThis article will give the residual power series method (RPSM) for solving pseudo hyperbolic partial differential equations with nonlocal conditions, RPSM is essentially based on general formula of Taylor series with residual error function. A new analytical solution is investigated. The analytical solution is designed to find the approximation solutions by RPSM and compare the obtained results from the current method with the exact solution that detects the precision, reliability, and rapid convergence of the proposed method. Finally at different times through the graphical representation of obtained results are given.

A solution to a fractional order semilinear equation using variational method

We will discuss how we obtain a solution to a semilinear pseudo-differential equation involving fractional power of laplacian by using a method analogous to the direct method of calculus of variations. More precisely, we will discuss the existence of a minimizer of a suitable energy type functional whose Euler-Lagrange equation is the given semilinear pseudo-differential equation, and also discuss the regularity of such a minimizer so that it will be a solution to the semilinear equation.

Boundary Value Problems for Elliptic Pseudodifferential Equations in a Multidimensional Cone

We consider model boundary value problems for elliptic pseudodifferential equations in multidimensional cones. A result on the unique solvability and representation of solutions of some boundary value problems in suitable Sobolev–Slobodetskii spaces is obtained. A priori estimates of solutions are given.

Anisotropic regularity of linearized compressible vortex sheets

We consider supersonic vortex sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients, Morando et al. recently derived a pseudo-differential equation that describes the time evolution of the discontinuity front of the vortex sheet. In agreement with the classical stability analysis, the problem is weakly stable if [Formula: see text], and the well-posedness holds in standard weighted Sobolev spaces. Our aim in this paper is to improve this result, by showing the existence in functional spaces with additional weighted anisotropic regularity in the frequency space.

Stress and Displacement Intensity Factors of Cracks in Anisotropic Media

A relation connecting stress intensity factors (SIF) with displacement intensity factors (DIF) at the crack front is derived by solving a pseudodifferential equation connecting stress and displacement discontinuity fields for a plane crack in an elastic anisotropic medium with arbitrary anisotropy. It is found that at a particular point on the crack front, the vector valued SIF is uniquely determined by the corresponding DIF evaluated at the same point.

Mixed boundary–transmission problems for composite layered elastic structures

We investigate mixed type boundary–transmission problems of the generalized thermo‐electro‐magneto‐elasticity theory for complex elastic anisotropic layered structures containing interfacial cracks. This type of problems is described mathematically by systems of partial differential equations with appropriate transmission and boundary conditions for six dimensional unknown physical field (three components of the displacement vector, electric potential function, magnetic potential function, and temperature distribution function). We apply the potential method and the theory of pseudodifferential equations and prove uniqueness and existence theorems of solutions to different type mixed boundary–transmission problems in appropriate Sobolev spaces. We analyze smoothness properties of solutions near the edges of interfacial cracks and near the curves where different type boundary conditions collide.

Solving Nonlinear p-Adic Pseudo-differential Equations: Combining the Wavelet Basis with the Schauder Fixed Point Theorem

Recently theory of p-adic wavelets started to be actively used to study of the Cauchy problem for nonlinear pseudo-differential equations for functions depending on the real time and p-adic spatial variable. These mathematical studies were motivated by applications to problems of geophysics (fluids flows through capillary networks in porous disordered media) and the turbulence theory. In this article, using this wavelet technique in combination with the Schauder fixed point theorem, we study the solvability of nonlinear equations with mixed derivatives, p-adic (fractional) spatial and real time derivatives. Furthermore, in the linear case we find the exact solution for the Cauchy problem. Some examples are provided to illustrate the main results.

Nonlocal in time problem for evolution pseudodifferential equations with analytic symbols

Nonlocal in time problem for evolution pseudodifferential equations with analytic symbols

Unusual bifurcation of a Neumann boundary value problem

Boundary layer behaviour of a family of second order nonlinear differential equations with Neumann boundary condition arising from an order-reduction of a pseudo-differential equation in fluid dynamics is analysed. The problem is considered with two perturbation parameters μ,δ. Using asymptotic and numerical methods it is shown that by perturbing δ the problem in the limit μ→0 changes from a homogeneous problem with symmetric solutions (including outer solutions) of a simplified equation, to a non-homogeneous problem which, in general, does not have a non-zero outer limit solution (with boundary layers). By studying the bifurcation diagrams as μ and δ vary it is shown that the main branch of solutions will ‘tear’ for μ=n−2 where n is an even integer. Blow-up regions, isolated islands of solution, and for μ≪1 “exponentially small” regions, all occur: a structure that is difficult for conventional path following algorithms to find.

Умови коректності крайових задач

The article deals with the issues of mathematical modeling of technological systems that contain physical fields’ sources. It is believed that in the case of a simple spatial form of the object under study, the boundary value problems will be correct. The interest lies in mathematical models for nonlinear, multilayer objects under the influence of load sources, for which, using the traditional theory of existence and unity, it is impossible to guarantee the correctness of boundary value problems. The author considers boundary value problems for systems of differential and pseudo differential equations in a multilayer medium which describe the state of the studied systems under the action of discrete load sources. The correctness of such problems is proven using the theory of distributions over the space of generalized functions. The object of research is boundary value problems for systems of differential and pseudo differential equations in a multilayer medium. The aim of the research is to build correct boundary value problems, which underlie the calculated mathematical models of the process of action of physical fields on multilayer objects. The necessary and sufficient conditions for the correctness of the parabolic boundary value problem in the space of generalized functions are obtained in the article. It is shown that its solution is infinitely differentiated by a spatial variable. The results of the research can be used to obtain the conditions for the correctness of the boundary value problem for differential equations with variable coefficients. Note that, in some cases, the correctness of the calculated mathematical models determines the correctness of applied optimization mathematical models. The application of the author's research is possible when proving the correctness of boundary value problems for a number of technological processes. The universality of the research allows to widely usage of the results obtained in this work to improve the quality of technological processes.

Non-Archimedean analysis and a wave-type pseudodifferential equation on finite adèles

In this work the ring of finite adeles $${\mathbb {A}}_f$$ of the rational numbers $${\mathbb {Q}}$$ is obtained as a completion of $${\mathbb {Q}}$$ with respect to a certain non-Archimedean metric related to the second Chebyshev function, which allows us to represent any finite adele as a convergent series, generalizing m-adic analysis. This polyadic analysis allows us to introduce a novel pseudodifferential operator $$D^{\alpha }$$ on $$L^2({\mathbb {A}}_f)$$ of fractional differentiation, similar to the Vladimirov operator on the p-adic numbers. The operator $$D^{\alpha }$$ is a positive selfadjoint unbounded operator whose spectrum $$\sigma (D^{\alpha })$$ is essential and it consists of a countable number of eigenvalues, which converges to zero, and zero. Moreover, a sort of multiresolution analysis on $${\mathbb {A}}_f$$ provides us with a wavelet basis which is an orthonormal basis of eigenfunctions of $$D^{\alpha }$$ as well. The Cauchy problem of a wave-type pseudodifferential equation $$\begin{aligned} u_{tt}(x,t)+D^{\alpha }_x u(x,t) =F(x,t), \qquad (x \in {\mathbb {A}}_f), \end{aligned}$$ with appropriate initial conditions $$u(x,0)=f(x), u_t(x,0)=g(x),$$ and external force F(x, t), is solved separating variables and using the Fourier expansion of functions in $$L^2({\mathbb {A}}_f)$$ , with respect to the wavelet basis.

Pseudodifferential operators and Markov processes on certain totally disconnected groups

This article describes a class of invariant Markov processes on certain totally disconnected groups. An invariant pseudodifferential operator on these groups, similar to the Vladimirov operator on the p-adic line, allows us to state an L2-abstract Cauchy problem for a homogeneous heat-type pseudodifferential equation. The fundamental solutions of these parabolic-type pseudodifferential equations give transition functions of time and space homogeneous Markov processes on these groups. Particularly interesting examples are polyadic rings, such as the ring of m-adic numbers, and the ring of finite adèles of the rational numbers.

On Certain 3-Dimensional Limit Boundary Value Problems

The paper is devoted to studying limit behavior for a solution of model elliptic pseudo-differential equation with some integral boundary condition in 4-wedge conical canonical 3D singular domain with two parameters. It is shown that the solution of such boundary value problem can have a limit with respect to endpoint values of the parameters in appropriate Sobolev–Slobodetskii space if the boundary function is a solution of a special functional singular integral equation.

Extremal constant sign solutions and nodal solutions for the fractional p-Laplacian

We study a pseudo-differential equation driven by the degenerate fractional p-Laplacian, under Dirichlet type conditions in a smooth domain. First we show that the solution set within the order interval given by a sub-supersolution pair is nonempty, directed, and compact, hence endowed with extremal elements. Then, we prove existence of a smallest positive, a biggest negative and a nodal solution, combining variational methods with truncation techniques.