Differential Equations Of Parabolic Type Research Articles

Mixed problem for a linear differential equation of parabolic type with nonlinear impulsive conditions

Mixed problem for a linear differential equation of parabolic type with nonlinear impulsive conditions

Quasi Solution of an Inverse Fractional Stochastic Nonlinear Partial Differential Equation of Parabolic Type

Quasi Solution of an Inverse Fractional Stochastic Nonlinear Partial Differential Equation of Parabolic Type

Invariant manifolds for stochastic delayed partial differential equations of parabolic type

The aim of this paper is to prove the existence and smoothness of stable and unstable invariant manifolds for a stochastic delayed partial differential equation of parabolic type. The stochastic delayed partial differential equation is firstly transformed into a random delayed partial differential equation by a conjugation, which is then recast into a Hilbert space. For the auxiliary equation, the variation of constants formula holds and we show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov–Perron method. Subsequently, we prove the smoothness of these invariant manifolds under appropriate spectral gap condition by carefully investigating the smoothness of auxiliary equation, after which, we obtain the invariant manifolds of the original equation by projection and inverse transformation. Eventually, we illustrate the obtained theoretical results by their application to a stochastic single-species population model.

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.

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.

Adaptive techniques for solving chaotic system of parabolic-type

Time-dependent partial differential equations of parabolic type are known to have a lot of applications in biology, mechanics, epidemiology and control processes. Despite the usefulness of this class of differential equations, the numerical approach to its solution, especially in high dimensions, is still poorly understood. Since the nature of reaction-diffusion problems permit the use of different methods in space and time, two important approximation schemes which are based on the spectral and barycentric interpolation collocation techniques are adopted in conjunction with the third-order exponential time-differencing Runge-Kutta method to advance in time. The accuracy of the method is tested by considering a number of time-dependent reaction-diffusion problems that are still of current and recurring interests in one and high dimensions.

Convergence to a self-similar solution for a one-phase Stefan problem arising in corrosion theory

Steel corrosion plays a central role in different technological fields. In this article, we consider a simple case of a corrosion phenomenon which describes a pure iron dissolution in sodium chloride. This article is devoted to prove rigorously that under rather general hypotheses on the initial data, the solution of this iron dissolution model converges to a self-similar profile as $t\rightarrow +\infty$ . We will do so for an equivalent formulation as presented in the book of Avner Friedman about parabolic equations (Friedman (1964) Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, NJ.). In order to prove the convergence result, we apply a comparison principle together with suitable upper and lower solutions.

Efficient numerical techniques for computing the Riesz fractional-order reaction-diffusion models arising in biology

In this work, the solution of Riesz space fractional partial differential equations of parabolic type is considered. Since fractional-in-space operators have been applied to model anomalous diffusion or dispersion problems in the area of mathematical physics with success, we are motivated in this paper to model the standard Brownian motion with the fractional order operator in the sense of the Riesz derivative. We formulate two viable, efficient and reliable high-order approximation schemes for the Riesz derivative which incorporated both the left- and right-hand sides of the Riemann-Liouville derivatives. The proposed methods are analyzed for both stability and convergence. Finally, the methods are used to explore the dynamic richness of pattern formation in two important fractional reaction-diffusion equations that are still of recurring interest. Experimental results for different values of the fractional parameters are reported.

Mathematical modeling of the gettering process for a cylindrical region

The paper considers a mathematical model of the technical process of external gettering of silicon rods. The purpose of this process is to remove unwanted impurities from silicon. The model of this process is an initial-boundary value problem for a linear partial differential equation of parabolic type. A feature of this problem is the presence of a nonlocal in time boundary condition, which reflects the law of conservation of mass (matter). For numerical modeling, an algorithm based on difference approximations of differential operators of the 1st and 2nd order is proposed. The classical sweep method is used to find a solution to a discrete problem.

Spectral semi-discretization algorithm for a class of nonlinear parabolic PDEs with applications

This manuscript deals with a novel hybrid spectral collocation approach to find the approximate solutions of a class of nonlinear partial differential equations of parabolic type pertaining to various important physical models in mathematical biology. The problem under consideration arises in the study of propagation of gene and transmission of nerve impulses. A second-order time discretization algorithm based on Taylor series expansion is first employed to tackle the underlying nonlinearity of the problem. Then, the spectral collocation approach based on the alternative Laguerre polynomials (with positive coefficients) is adopted for approximation of the resulting semi-discrete ordinary differential equations. The convergence of the spectral technique is established. Three numerical test examples are given to demonstrate the applicability and efficiency of the proposed approach. The computed results are compared with the results obtained by other available methods in order to show the advantage of our method.

Dynamics of solutions of nonlinear functional differential equation of parabolic type

Purpose of this work is to study the initial-boundary value problem for a parabolic functional-differential equation in an annular region, which describes the dynamics of phase modulation of a light wave passing through a thin layer of a nonlinear Kerr-type medium in an optical system with a feedback loop, with a rotation transformation (corresponds the involution operator) and the Neumann conditions on the boundary in the class of periodic functions. A more detailed study is made of spatially inhomogeneous stationary solutions bifurcating from a spatially homogeneous stationary solution as a result of a bifurcation of the “fork” type and time-periodic solutions of the “traveling wave” type. Methods. To represent the original equation in the form of nonlinear integral equations, the Green’s function is used. The method of central manifolds is used to prove the theorem on the existence of solutions of the indicated equation in a neighborhood of the bifurcation parameter and to study their asymptotic form. Numerical modeling of spatially inhomogeneous solutions and traveling waves was carried out using the Galerkin method. Results. Integral representations of the considered problem are obtained depending on the form of the linearized operator. Using the method of central manifolds, a theorem on the existence and asymptotic form of solutions of the initial-boundary value problem for a functional-differential equation of parabolic type with an involution operator on an annulus is proved. As a result of numerical modeling based on Galerkin approximations, in the problem under consideration, approximate spatially inhomogeneous stationary solutions and time-periodic solutions of the traveling wave type are constructed. Conclusion. The proposed scheme is applicable not only to involutive rotation operators and Neumann conditions on the boundary of the ring, but also to other boundary conditions and circular domains. The representation of the initial-boundary value problem in the form of nonlinear integral equations of the second kind allows one to more simply find the coefficients of asymptotic expansions, prove existence and uniqueness theorems, and also use a different number of expansion coefficients of the nonlinear component in the right-hand side of the original equation in the neighborhood of the selected solution (for example, stationary). Visualization of the numerical solution confirms the theoretical calculations and shows the possibility of forming complex phase structures.

Evolution equations with nonlocal initial conditions and superlinear growth

We carry out an analysis of the existence of solutions for a class of nonlinear partial differential equations of parabolic type. The equation is associated to a nonlocal initial condition, written in general form which includes, as particular cases, the Cauchy multipoint problem, the weighted mean value problem and the periodic problem. The dynamic is transformed into an abstract setting and by combining an approximation technique with the Leray-Schauder continuation principle, we prove global existence results. By the compactness of the semigroup generated by the linear operator, we do not assume any Lipschitzianity, nor compactness on the nonlinear term or on the nonlocal initial condition. In addition, the exploited approximation technique coupled to a Hartman-type inequality argument, allows to treat nonlinearities with superlinear growth. Moreover, regarding the periodic case, we are able to show the existence of at least one periodic solution on the half line.

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ БИОКОРРОЗИИ БЕТОНА В ЖИДКИХ СРЕДАХ

Microbiological corrosion poses a serious danger to building structures in direct contact with an aquatic medium. The destruction of cement concrete during biocorrosion is determined by the processes of mass transfer. The article presents a physical and mathematical model of diffusion processes in the “cement concrete - biofilm - liquid” system. A model of mass transfer in an unbounded two-layer plate is calculated in the form of differential equations of parabolic type in partial derivatives. The author presents the results of the numerical experiment to study the influence of the mass conductivity and mass transfer on the kinetics and dynamics of the process. A numerical experiment has been carried out to illustrate the effect of similarity criteria on the intensity of the process of corrosive mass transfer.

The Problem of Minimax Estimation of Functionals for Non-Stationary Diffusion Processes

Purpose. To model the technological process of analysis of energy sources that use random interference, it is necessary to apply special methods from the theory of minimax estimation and optimal control. The article proposes a method for solving the problem of minimax estimation of functionalities for the systems with distributed parameters with incomplete data for the process of neutron diffusion in a nuclear reactor. Methodology. In practice, in the study of non-stationary controlled processes of functioning of different energy sources there are measurement errors. As a rule, the exact values of errors are unknown, and therefore the desired solution of the equations in partial derivatives describing these processes is determined ambiguously. Therefore, it is advisable to set the task of calculating such an optimal estimate, which would best approximate the unknown value, taking into account the known information about the measurement errors. The best estimate can be achieved by applying a minimax approach to estimating functionals from the solutions of the partial differential equations of parabolic type. Findings. For a mathematical model of the neutron diffusion process in a nuclear reactor, the proposed method allows solving the problem of minimax estimation of the functional determined during the solution of the system describing this process. Since in real conditions of reactor operation there are always random obstacles (both in the equation describing the process and the function observed), the method allows finding a minimax estimate of the functional. The problem is reduced to the problem of optimal control with a given quality functionality, which is successfully solved. Originality. Using the methods of minimax estimation and optimal control of systems with distributed parameters, the best a priori estimation of the quality functional of the minimax estimation problem for the mathematical model of neutron diffusion in a nuclear reactor is obtained. Practical value. The method of minimax estimation of functionalities for differential equations of parabolic type proposed in the article allows reducing the problem to the problem of optimal control of the systems with distributed parameters, which can be implemented in Maple package using known algorithms.

A Quasi Solution for a Nonlinear Inverse Stochastic Partial Differential Equation of Parabolic Type

A Quasi Solution for a Nonlinear Inverse Stochastic Partial Differential Equation of Parabolic Type

Modeling and numerical simulation of a parabolic trough collector using an HTF with temperature dependent physical properties

This paper describes the mathematical modeling of a parabolic trough collector where the heat transfer fluid has temperature dependent physical properties for which we have developed our own mathematical expressions. For a specific location and date, the direct solar radiation and other sun–earth parameters are computed on a Julian day basis. Also, we have used certain known and recent correlations for the heat transfer coefficients appearing in this model. The mathematical model is governed by three nonlinear partial differential equations of parabolic type with initial/boundary value conditions. We describe a numerical algorithm in order to obtain the approximate solution of this model. Then, we carry out some numerical simulations which yield valuable information for the performance and efficiency of a solar PTC plant at the specific location.

On nonlinear problems of parabolic type with implicit constitutive equations involving flux

We study systems of nonlinear partial differential equations of parabolic type, in which the elliptic operator is replaced by the first-order divergence operator acting on a flux function, which is related to the spatial gradient of the unknown through an additional implicit equation. This setting, broad enough in terms of applications, significantly expands the paradigm of nonlinear parabolic problems. Formulating four conditions concerning the form of the implicit equation, we first show that these conditions describe a maximal monotone [Formula: see text]-coercive graph. We then establish the global-in-time and large-data existence of a (weak) solution and its uniqueness. To this end, we adopt and significantly generalize Minty’s method of monotone mappings. A unified theory, containing several novel tools, is developed in a way to be tractable from the point of view of numerical approximations.

Mathematical modeling of an optimal oncotherapy for malignant tumors

The paper presents a mathematical model of the optimal treatment for malignant neoplasms. The neoplasm is considered as a distributed parameter object. The scheme for an optimal oncotherapy using a system of partial differential equations of parabolic type is analyzed. The authors propose a solution to the problem using Bellman optimization and the method of adjustable parameters. The optimal control law of the oncotherapy mode is derived. The main results include a scheme for the formation of the Bellman optimal strategy for regulation of control parameters and dynamic parameters, under which the target conditions are guaranteed over time. The work describes an optimization criterion that reflects the total costs of the control system for the oncological treatment. Simulation results demonstrate the efficiency of the optimal control of treatment process. The results of this work can be used in modern clinical practice at the stage of predictive selection of the most effective treatment strategy.

On the solvability of a mixed problem for partial differential equations of parabolic type with involution

Mixed partial differential equation of parabolic type with involution is considered. The sufficient conditions of the existence and uniqueness of the solution for the parabolic type equations with involution is obtained. The Fourier method is used to find a classical solution of the mixed problem for the transformed parabolic type equation with involution.

Mixed Problem for the Differential Equation of Parabolic Type with Measures

We propose a scheme for the solution of a mixed problem for a parabolic differential equation with coefficients that are generalized derivatives of functions of bounded variation. We seek the solution of this problem by the method of reduction. According to this method, the solution of the proposed problem is reduced to the solution of two problems: (i) a quasistationary boundary-value problem with input boundary conditions and (ii) a mixed problem with trivial boundary conditions. The first of these problems is solved by introducing the quasiderivative. For the solution of the second problem, we use the Fourier method and the expansion in eigenfunctions of a certain boundary-value problem for a quasidifferential equation of the second-order. The obtained results can be used, in particular, for the investigation of the processes of heat transfer in multilayer plates, hollow cylinders, and spheres.