Equations Of Parabolic Type Research Articles

New energy effect in non-cylindrical domains with a thermally insulated moving boundary

Objectives. To develop mathematical model representations of the energy effect in non-cylindrical domains having a thermally insulated moving boundary; to introduce a new boundary condition for thermal insulation of a moving boundary both for locally equilibrium heat transfer processes in the framework of classical Fourier phenomenology, as well as for more complex locally non-equilibrium processes in the framework of Maxwell–Cattaneo–Lykov–Vernott phenomenology, taking into account the finite rate of heat propagation into analytical thermophysics and applied thermomechanics; to consider an applied problem of analytical thermophysics according to the theory of thermal shock for a domain with a moving thermally insulated boundary free from external and internal influences; to obtain an exact analytical solution of the formulated mathematical models for hyperbolic type equations; to investigate the solutions obtained using a computational experiment at various values of the parameters included in it; to describe the wave nature of the kinetics of the processes under consideration.Methods. Methods and theorems of operational calculus, Riemann–Mellin contour integrals are used in calculating the originals of complex images with two branch points. A new mathematical apparatus for the equivalence of functional constructions for the originals of the obtained operational solutions, which considers the computational difficulties in finding analytical solutions to boundary value problems for equations of hyperbolic type in the domain with a moving boundary, is developed.Results. Developed mathematical models of locally nonequilibrium heat transfer and the theory of thermal shock for equations of hyperbolic type in a domain with a moving thermally insulated boundary are presented. It is shown that, despite the absence of external and internal sources of heat, the presence of a thermally insulated moving boundary leads to the appearance of a temperature gradient in the domain and, consequently, to the appearance of a temperature field and corresponding thermoelastic stresses in the domain, which have a wave character. A stochastic analysis of this energy effect forms the basis for a proposed transition of the kinetic energy of a moving thermally insulated boundary into the thermal energy of the domain. The presented model representations of the indicated effect confirmed the stated assumption.Conclusions. Mathematical models for locally nonequilibrium heat transfer processes and the theory of thermal stresses are developed and investigated on the basis of constitutive relations of the theory of thermal shock for equations of hyperbolic type in a domain with a thermally isolated moving boundary. A numerical experiment is presented to demonstrate the possibility of transiting from one form of analytical solution of a thermophysical problem to another equivalent form of a new type. The described energy effect manifests itself both for parabolic type equations based on the classical Fourier phenomenology, as well as for hyperbolic type equations based on the generalized Maxwell–Cattaneo–Lykov–Vernott phenomenology.

Bounded on the semi-axis multiperiodic solution of a linear finite-hereditarity integro-differential equation of parabolic type

The question of the existence of a solution of linear integro-differential systems of parabolic type limited on the semiaxis in a spatial variable and multiperiodic in time variables was considered. Sufficient conditions of multiperiodic oscillations in time variables in a linear homogeneous equation with a boundary condition and in a linear inhomogeneous equation were established. A linear homogeneous and inhomogeneous finitehereditarity integro-differential equation of convective-diffusion type were investigated.

On the time-optimal control problem for a heat equation

In previous works, we have considered some control problems for parabolic type equations, namely, control problems for parabolic type equations were studied as boundary value problems of the first type, and the weight function was expanded into a Fourier series by sines. In this paper, we consider boundary control problem for a heat equation on the interval. In the part of the bound of the given domain it is given value of a solution and it is required to find a control to get the average value of the solution. By the mathematical-physics methods it is proved that like this control exists and the estimate of a minimal time for achieving the given average temperature over some domain is found.

Stochastic control methods for optimization problems in Ornstein-Uhlenbeck spread models

We develop stochastic optimal control methods for spread financial models defined by the Ornstein–Uhlenbeck (OU) processes. To this end, we study the Hamilton - Jacobi - Bellman (HJB) equation using the Feynman - Kac (FK) probability representation. We show an existence and uniqueness theorem for the classical solution of the HJB equation, a quasi-linear partial derivative equation of parabolic type. Then we show a special verification theorem and, as a consequence, construct optimal consumption/investment strategies for power utility functions. Moreover, using fixed point tools we study the numeric approximation for the HJB solution and we establish the convergence rate which, as it turns out in this case, is super geometric, i.e., more rapid than any geometric one. Finally, we illustrate numerically the behavior of the obtained strategies.

The simulation of non-stationary gas pressure in a pipeline with pumping and injection

In this work, based on the fundamental research of I. A. Charny on the unsteady movement of a real fluid in pipes, an equation that describes the pressure dynamics in a complex section of a gas pipeline was compiled. The use of the Dirac impulse function made it possible to describe the dynamics of unsteady pressure in a single equation in the case of point pumping and injection at given points. Linearization of the model by averaging the velocity of the gas made it possible to bring the equation to a hyperbolic form. If we neglect the forces of inertia compared to the forces of resistance, then the mathematical model will represent a partial differential equation of the second order parabolic type. The dynamics of pressure at specific points of pumping and injection under boundary conditions of the second kind (rates are given) was obtained using the finite cosine Fourier transform. The working formula that allows you to determine the pressure at any point at a given point in time is a trigonometric series. The series are rapidly convergent, so there is no difficulty in working with them. Particular cases are considered (without pumping and injection). The fulfillment of the boundary conditions is checked. It is easy to rebuild the working formula for pumping and / or injection at several points in a given area. Calculations of the mass flow and average velocity are made. In practice, the pressure is usually set at the start and end points of the area under study. In this case, it becomes possible to go to the costs (to derivatives) at the ends of the section under consideration.

Nonlinear optics problem with transformation of a spatial variable and an oblique derivative

In this paper, we consider a functional differential equation of parabolic type on a strip with a transformation of a spatial variable and boundary conditions with an oblique derivative. Using the Laplace and Fourier transforms, we obtain a representation of the problem under consideration in the form of a nonlinear integral equation. A special case of this representation is considered. The proved statements make it possible to implement iterative methods for obtaining approximate solutions of nonlinear partial differential equations taking into account the given conditions. The results show that the presented method is promising for solving similar problems.

Local Polya fluctuations of Riesz gravitational fields and the Cauchy problem

We consider a pseudodifferential equation of parabolic type with a fractional power of the Laplace operator of order $\alpha\in(0;1)$ acting with respect to the spatial variable. This equation naturally generalizes the well-known fractal diffusion equation. It describes the local interaction of moving objects in the Riesz gravitational field. A simple example of such system of objects is stellar galaxies, in which interaction occurs according to Newton's gravitational law. The Cauchy problem for this equation is solved in the class of continuous bounded initial functions. The fundamental solution of this problem is the Polya distribution of probabilities $\mathcal{P}_\alpha(F)$ of the force $F$ of local interaction between these objects. With the help of obtained solution estimates the correct solvability of the Cauchy problem on the local field fluctuation coefficient under certain conditions is determined. In this case, the form of its classical solution is found and the properties of its smoothness and behavior at the infinity are studied. Also, it is studied the possibility of local strengthening of convergence in the initial condition. The obtained results are illustrated on the $\alpha$-wandering model of the Lévy particle in the Euclidean space $\mathbb{R}^3$ in the case when the particle starts its motion from the origin. The probability of this particle returning to its starting position is investigated. In particular, it established that this probability is a descending to zero function, and the particle "leaves" the space $\mathbb{R}^3$.

Stochastic differential equations with critically irregular drift coefficients

This paper is concerned with stochastic differential equations (SDEs for short) with irregular coefficients. By utilising a functional analytic approximation approach, we establish the existence and uniqueness of strong solutions to a class of SDEs with critically irregular drift coefficients in a new critical Lebesgue space, where the element may be only weakly integrable in time. To be more precise, let b:[0,T]×Rd→Rd be Borel measurable, where T>0 is arbitrarily fixed and d⩾1. We consider the following SDEXt=x+∫0tb(s,Xs)ds+Wt,t∈[0,T],x∈Rd, where {Wt}t∈[0,T] is a d-dimensional standard Wiener process. For p,q∈[1,+∞), we denote by C[q]([0,T];Lp(Rd)) the space of all Borel measurable functions f such that t1qf(t)∈C([0,T];Lp(Rd)). If b=b1+b2 such that |b1(T−⋅)|∈C[q]([0,T];Lp(Rd)) with 2/q+d/p=1 and ‖b1(T−⋅)‖C[q]([0,T];Lp(Rd)) is sufficiently small, and that b2 is bounded and Borel measurable, then we show that there exists a weak solution to the above equation, and if in addition limt↓0⁡‖t1qb(T−t)‖Lp(Rd)=0, the pathwise uniqueness holds. Furthermore, we obtain the strong Feller property of the semi-group and the existence of density associated with the above SDE. Besides, we extend the classical results concerning partial differential equations (PDEs) of parabolic type with Lq(0,T;Lp(Rd)) coefficients to the case of parabolic PDEs with L[q]∞(0,T;Lp(Rd)) coefficients, and derive the Lipschitz regularity for solutions of second order parabolic PDEs (see Theorem 3.1). Our results extend Krylov-Röckner and Krylov's profound results of SDEs with singular time dependent drift coefficients [20,23] to the critical case of SDEs with critically irregular drift coefficients in a new critical Lebesgue space.

On a Mixed Problem for a Parabolic Type Equation with General form Constant Coefficients Under Inhomogeneous Boundary Conditions

On a Mixed Problem for a Parabolic Type Equation with General form Constant Coefficients Under Inhomogeneous Boundary Conditions

Letter to the Editor: Correction to the “Kernel determination problem in an integro-differential equation of parabolic type with nonlocal condition” [Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2023, vol. 33, issue 1, pp. 90-102

In this note we give a complete correct formulation and proof of Lemma 1 of [1].

Asymptotic behavior of solutions for a new general class of parabolic Kirchhoff type equation with variable exponent sources

In this paper, we study the following problem:{ut−M(∫Ω|∇u|p(x)p(x)dx)Δp(x)u=|u|m(x)−2u,(x,t)∈Ω×(0,T),u(x,t)=0,(x,t)∈∂Ω×(0,T),u(x,0)=u0(x),x∈Ω, where Ω⊂RN is a bounded domain with smooth boundary ∂Ω, the functions p,m:Ω¯→R and the Kirchhoff function M:[0,∞)→[0,∞) are specified below. We give a new class of general Kirchhoff function which covers both non-degenerate and degenerate cases, and investigate its effects on the existence and non-existence of global weak solutions to the problem in case the initial energy is subcritical. Moreover, we also give the decay estimate for the energy functional in the former case and an upper bound for the maximal existence time in the latter case. This is a continue works by Nhan et al. (2020) [27] and also an affirmative answer for the comments by Guo et al. (2022) [13].

On the Adjoint Problem in a Domain with Deviation Out from the Characteristic for the Mixed Parabolic-Hyperbolic Equation with the Fractional Order Operator

В настоящей статье доказана классическая, сильная разрешимость и вольтерровость сопряженной задачи c отходом от характеристики для уравнения смешанного параболо-гиперболического типа с оператором дробного порядка в смысле Герасимова-Капуто. Целью исследования является решение сопряженной задачи для уравнения смешанного параболо-гиперболического типа дробного порядка. Учитывая, свойств операторов дробного порядка найдены сопряженный оператор и применены постановки сопряженной задачи. Для исследования поставленной задачи в параболической частью смешанной области решается первой краевой задачи для уравнения параболического типа дробного порядка в смысле Герасимова-Капуто. Используя, свойств функции Райта получено функциональное соотношение на линии перехода. Точно также решая, задачи Коши гиперболической частью смешанной области находим функциональное соотношение. Следовательно, поставленная задача эквивалентным образом сводится к интегральному уравнению Вольтерра второго рода со слабой особенностью. Согласно теории интегральных уравнений Вольтерра второго рода доказывается однозначной разрешимость полученного уравнения. Кроме того, используя методы операторов интегро — дифференцирования дробного порядка, теории специальных функций, априорных оценок, теория интегральных уравнений доказываются теоремы единственности, существования и вольтерровость сопряженной задачи в области с отходом от характеристики для уравнения смешанного типа дробного порядка. Полученные результаты новые и отличаются от результатов М.А. Садыбекова и А.С. Бердышева. In this article, it was proved the classical, strong solvability and Volterra property of the adjoint problem with departure from the characteristic for an equation of mixed parabolic-hyperbolic type with a fractional order operator in the sense of Gerasimov-Caputo. The aim of the research is to solve the conjugate problem for the equation of a mixed parabolic-hyperbolic type of fractional order. Taking into account the properties of fractional order operators, the adjoint operator is found and the statements of the adjoint problem are applied. To study the formulated problem in the parabolic part of the mixed domain, the first boundary value problem for a parabolic type equation of fractional order in the sense of Gerasimov-Caputo is solved. Using the properties of the Wright function, a functional relation is obtained on the transition line. In the same way, solving the Cauchy problem with the hyperbolic part of the mixed domain, we find a functional relation. Consequently, the problem posed reduces in an equivalent way to a Volterra integral equation of the second kind with a weak singularity. According to the theory of Volterra integral equations of the second kind, the unique solvability of the resulting equation is proved. In addition, using the methods of integro-differentiation operators of fractional order, the theory of special functions, a priori estimates, the theory of integral equations, uniqueness, existence and Volterra theorems for the adjoint problem in a domain with deviation out of the characteristic for a mixed-type equation of fractional order are proved. The results obtained are new and differ from the results of M. A. Sadybekov and A. S. Berdyshev.

Nonexistence results of solutions of parabolic-type equations and systems on the Heisenberg group

In this paper, we present necessary conditions for the existence of weak solution of the parabolic-type equations and systems on the Heisenberg group. The main technique for proving the results relies on the method of test functions.

Kernel determination problem in an integro-differential equation of parabolic type with nonlocal condition

In this paper, an inverse problem for a one-dimensional integro-differential heat equation is investigated with nonlocal initial-boundary and integral overdetermination conditions. We use the Fourier method and the Schauder principle to investigate the solvability of the direct problem. Further, the problem is reduced to an equivalent closed system of integral equations with respect to unknown functions. Existence and uniqueness of the solution of the integral equations are proved using a contractive mapping. Finally, using the equivalency, the existence and uniqueness of the classical solution is obtained.

Rapid Multicomponent Alloy Solidification with Allowance for the Local Nonequilibrium and Cross-Diffusion Effects.

Motivated by the fast development of various additive manufacturing technologies, we consider a mathematical model of re-solidification of multicomponent metal alloys, which takes place after ultrashort (femtosecond) pulse laser melting of a metal surface. The re-solidification occurs under highly nonequilibrium conditions when solutes diffusion in the bulk liquid cannot be described by the classical diffusion equation of parabolic type (Fick law) but is governed by diffusion equation of hyperbolic type. In addition, the model takes into account diffusive interaction between different solutes (nonzero off-diagonal terms of the diffusion matrix). Numerical simulations demonstrate that there are three main re-solidification regimes, namely, purely diffusion-controlled with solute partition at the interface, partly diffusion-controlled with weak partition, and purely diffusionless and partitionless. The type of the regime governs the final composition of the re-solidified material, and, hence, may serve as one of the main tools to design materials with desirable properties. This implies that the model is expected to be useful in evaluating the most effective re-solidification regime to guide the optimization of additive manufacturing processing parameters and alloys design.

Solvability of Mixed Problems for a Fourth-Order Equation with Involution and Fractional Derivative

In the present work, two-dimensional mixed problems with the Caputo fractional order differential operator are studied using the Fourier method of separation of variables. The equation contains a linear transformation of involution in the second derivative. The considered problem generalizes some previous problems formulated for some fourth-order parabolic-type equations. The basic properties of the eigenfunctions of the corresponding spectral problems, when they are defined as the products of two systems of eigenfunctions, are studied. The existence and uniqueness of the solution to the formulated problem is proved.

A Note on the Uniqueness and Existence of the Solution to the Problem of Boundary Values for Some Partial Differential Equations of Second Order

Most physical mathematical problems, when solved, turn into one or more partial differential equations with imposed initial conditions or boundary conditions . This is known as the boundary value problems of differential equations .This research studies the solution of the set of Partial Differential Equations of parabolic and hyperbolic-hyperbolic type with boundary conditions imposed in different regions of the plane x o y This research has been proving the theorem of uniqueness and the existence of the solution.

Dirichlet’s problem for a third-order parabolic hyperbolic type equation of the second kind

Equations of mixed type, the degeneracy line of which is the envelope of a family of characteristics, therefore, is itself also a characteristic, in the literature it is customary to call equations of mixed type of the second kind, which causes additional difficulties in the study of boundary value problems for equations of the second kind. In the present work, a boundary value problem for a homogeneous equation of parabolic-hyperbolic type of the third order of the second kind is investigated. Necessary and sufficient conditions for the existence and uniqueness of a generalized solution of the problem are found. In some special cases, the representation of the solution of the problem under study is written out explicitly.

Limited different schemes for mutual diffusion problems

In the article, the problems of mutual diffusion, represented by a system of nonlinear differential equations of parabolic type, were modeled following numerical solving methods. The developed software was tested using various examples and obtained positive results. Mutual diffusion processes were visualized.

On a boundary control problem for a pseudo-parabolic equation

<abstract><p>Previously, boundary control problems for parabolic type equations were considered. A portion of the thin rod boundary has a temperature-controlled heater. Its mode of operation should be found so that the average temperature in some region reaches a certain value. In this article, we consider the boundary control problem for the pseudo-parabolic equation. The value of the solution with the control parameter is given in the boundary of the interval. Control constraints are given such that the average value of the solution in considered domain takes a given value. The auxiliary problem is solved by the method of separation of variables, and the problem under consideration is reduced to the Volterra integral equation. The existence theorem of admissible control is proved by the Laplace transform method.</p></abstract>