Original Differential Problem Research Articles

Finite Difference Method for Solving Parabolic-Type Integro-Differential Equations in Multidimensional Domain with Nonhomogeneous First-Order Boundary Conditions

Introduction. We investigate a multidimensional (in terms of spatial variables) parabolic-type integro-differential equation with nonhomogeneous first-order boundary conditions. The locally one-dimensional finite difference scheme developed herein can be applied to solve various applied problems leading to multidimensional parabolic-type integro-differential equations. Examples include mathematical modelling of cloud processes, addressing the issue of active intervention in convective clouds to prevent hail and artificially enhance precipitation, as well as describing the droplet mass distribution function due to microphysical processes such as condensation, coagulation, fragmentation, and freezing of droplets in convective clouds.Materials and Methods. In this study, an approximate solution to the initial-boundary value problem is constructed using the locally one-dimensional scheme of A.A. Samarsky with a specified order of approximation О(h2 +τ). The primary research method employed is the method of energy inequalities.Results. A priori estimates have been obtained in the discrete interpretation, from which uniqueness, stability, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem follow, with a convergence rate equal to the order of approximation of the difference scheme.Discussion and Conclusions. The research findings can be utilized for further development of boundary value problem theory for parabolic equations with variable coefficients. Additionally, they may find applications in the fields of difference scheme theory, computational mathematics, and numerical modelling.

To boundary value problems for a generalized hyperbolic equation with two fractional differentiation operators

Boundary value problems for a one-space-dimensional hyperbolic equation with two fractional differentiation operators and variable coefficients are considered. The method of finite differences is used for the approximate solution to the problems posed. Difference schemes for each of the problems are constructed, and by the method of energy inequalities for various relations between the orders of the fractional derivative (α, β) a priori estimates in differential and difference interpretations are obtained. The obtained estimates imply the uniqueness and stability of the solution with respect to the right-hand side and initial data, as well as the convergence of the solution of the difference problem to the solution of the original differential problem. Algorithms for the numerical solution of the problems posed are constructed.

Разностная схема повышенного порядка аппроксимации для уравнения Аллера с переменными коэффициентами

The initial boundary value problem for the one-dimensional Aller’s equation with variable coefficients and boundary conditions of the first kind is studied. The problem under study describes the processes of heat transfer in a heterogeneous environment, moisture transfer in soils, and fluid filtration in fractured porous media. To numerically solve the problem posed, a difference scheme of high order of accuracy was constructed - fourth order of accuracy in h and second order of accuracy in τ. Using the method of energy inequalities, an a priori estimate of the solution in a difference treatment is obtained. From this estimate it follows that the solution is unique and stable with respect to the right-hand side and initial data. Under the assumption of the existence of an exact solution to the original differential problem in the class of sufficiently smooth functions, and also due to the linearity of the problem under consideration, the obtained a priori estimate implies that the solution of the constructed difference problem converges to the solution of the original differential problem at a rate equal to the order of approximation of the difference scheme. The goal and scientific novelty of the work is to obtain a new numerical scheme of a higher order of approximation when solving the Dirichlet problem for the Aller’s equation with variable coefficients.

Dynamic Modelling and Mutual Coordination of Electricity and Watershed Networks for Spatio-Temporal Operational Flexibility Enhancement Under Rainy Climates

This paper proposes a watershed-electricity nexus model to unlock the flexibility of watershed networks (WSNs) for supporting the operation of power distribution networks (PDNs) under rainy climates. The proposed model exploits the spatio-tem-poral flexibility of geographically dispersed pump clusters to pro-vide reserve services to PDNs, and a hyperbolic partial differential function derived from Saint-Venant hydrodynamic equations is formed to describe the dynamic processes of river stream flows. Be-sides, a flexibility evaluation method based on a composite sensitiv-ity matrix of water levels with respect to power injections is pre-sented to quantify the time-varying adjustable power domain of pump loads. Then, a multi-stage interactive coordinated scheduling strategy is developed for the mutual operation of WSNs and PDNs, where drainage pumps are jointly optimized to provide flexible power reserves, while an optimal PDN economic dispatch is per-formed to improve the power supply voltage of pump loads. Fur-thermore, an equivalent mixed-integer linear programming refor-mulation method is derived to cope with the original nonlinear par-tial differential optimization problem for computational tractabil-ity improvements. Comparative results have validated the effective-ness of the proposed strategy in eliminating voltage violations and shaving peak loads.

On convergence of difference schemes of high accuracy for one pseudo-parabolic Sobolev type equation

Difference schemes of the finite difference method and the finite element method of high-order accuracy in time and space are proposed and investigated for a pseudo-parabolic Sobolev type equation. The order of accuracy in space is improved in two ways using the finite difference method and the finite element method. The order of accuracy of the scheme in time is improved by a special discretization of the time variable. The corresponding a priori estimates are determined and, on their basis, the accuracy estimates of the proposed difference schemes are obtained with sufficient smoothness of the solution to the original differential problem. Algorithms for the implementation of the constructed difference schemes are proposed.

Устойчивость и сходимость разностной схемы, аппроксимирующей нелокальную краевую задачу для нагруженного уравнения влагопереноса с производными дробного порядка

We study a boundary value problem for a loaded moisture transfer equation with two Gerasimov - Caputo fractional derivatives of different orders (α, β), variable coefficients, and a nonlocal integral boundary condition. On a uniform grid a difference scheme of order of approximation 𝑂(ℎ2 + 𝜏2) for 𝛼 = 𝛽 and 𝑂(ℎ2 + 𝜏2−𝑚𝑎𝑥⁡{𝛼,𝛽}) for 𝛼 ≠ 𝛽 is constructed. The study is carried out by the method of energy inequalities. For different ratios between the orders of fractional derivatives α and β, for solving the posed nonlocal boundary value problem, a priori estimates in the difference interpretation are obtained, from which the uniqueness of the solution of the posed problem, the continuous and uniform dependence of the solution on the input data, and the conv ergence of the solution of the difference problem to the solution of the original differential problem at a rate equal to the order of approximation of the difference scheme.

Finite-difference method for solving a multidimensional pseudoparabolic equation with boundary conditions of the third kind

We study an initial-boundary value problem for a multidimensional pseudoparabolic equation with variable coefficients and boundary conditions of the third kind. The multidimensional pseudoparabolic equation is reduced to an integro-differential equation with a small parameter. It is shown that as the small parameter tends to zero, the solution of the resulting modified problem converges to the solution of the original problem. For an approximate solution of the obtained problem, a locally one-dimensional difference scheme by A. A. Samarsky is constructed. An a priori estimate is obtained by the method of energy inequalities, from which the uniqueness, stability, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem follow. For a two-dimensional problem, an algorithm for the numerical solution of the initial-boundary value problem for a pseudoparabolic equation with conditions of the third kind is developed.

Unified Integral Transforms and Non-Classical Eigenvalue Problems in Heat and Mass Transfer

Abstract The generalized integral transform technique (GITT) is reviewed as a computational–analytical methodology in linear and nonlinear convection–diffusion problems, based on eigenfunction expansions extracted from characteristic differential operators, coefficients, and boundary conditions present in the original partial differential problem formulation. Here, the emphasis is on the employment of nonclassical eigenvalue problems as the expansion basis, which do not fall into the more usual framework of Sturm–Liouville problems. The goal is to enable or improve the eigenfunction expansions convergence, by incorporating more information from the original operators into the chosen eigenvalue problem, while requiring the handling of such a more involved expansion base. In this concern, the proposed differential eigenvalue problem can itself be handled by the GITT, leading to an algebraic eigensystem analysis. Different classes of nonclassical eigenvalue problems are then reviewed and associated with typical applications in heat and mass transfer. Representative test cases are then chosen to illustrate the extended methodology and demonstrate the convergence rates attainable by this enhanced hybrid solution path.

Grid Method for Solving an Initial-Boundary-value Problem for a General Multidimensional Equation of Parabolic Type

The problems concerning the physical process studies that lead to mathematical models based on a parabolic type equation are of great practical importance.When solving the parabolic equation, a transition from the one-dimensional equation to the multidimensional one gives a rise to some difficulties. The trouble is that amount of computations significantly increases in the transition from the one-dimensional problems to the multidimensional ones In this regard a task to build the economical difference schemes for a numerical solution of multidimensional problems is a challenge.A difference scheme that approximates the problem over time is economical, if it is unconditionally stable and the desirable number of arithmetic operations in the transition from layer to layer is proportional to the number of nodes in each time layer.The paper dwells on the construction of a locally one-dimensional (economical) difference scheme for the approximate solution of a parabolic type equation of a general form in a multidimensional domain, the main idea of which is to reduce a complex problem to a sequential solution of boundary value problems of a simpler structure. At the same time, there is a construction of the economical, unconditionally stable difference scheme for each of the intermediate problems. To have a numerical solution of the problem, there is a construction of the locally one-dimensional difference scheme of A.A. Samarsky. Using a method of energy inequalities allows us to obtain a priori estimates in the differential and difference interpretations, whence it follow that there are the uniqueness, stability, as well as the convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem. For a bi-dimensional problem, a numerical solution algorithm is constructed.

Mathematical Modeling and Short-Term Forecasting of the COVID-19 Epidemic in Bulgaria: SEIRS Model with Vaccination

Data from the World Health Organization indicate that Bulgaria has the second-highest COVID-19 mortality rate in the world and the lowest vaccination rate in the European Union. In this context, to find the crucial epidemiological parameters that characterize the ongoing pandemic in Bulgaria, we introduce an extended SEIRS model with time-dependent coefficients. In addition to this, vaccination and vital dynamics are included in the model. We construct an appropriate Cauchy problem for a system of nonlinear ordinary differential equations and prove that its unique solution possesses some biologically reasonable features. Furthermore, we propose a numerical scheme and give an algorithm for the parameters identification in the obtained discrete problem. We show that the found values are close to the parameters values in the original differential problem. Based on the presented analysis, we develop a strategy for short-term prediction of the spread of the pandemic among the host population. The proposed model, as well as the methods and algorithms for parameters identification and forecasting, could be applied to COVID-19 data in every single country in the world.

Разностные методы решения интегро-дифференциального уравнения конвекции-диффузии с граничными условиями третьего рода

An initial-boundary value problem for a multidimensional integro-differential convection-diffusion equation with variable coefficients and boundary conditions of the third kind is investigated. For the numerical solution of the multidimensional problem, a locally one-dimensional difference scheme is constructed. It is shown that the approximation error of the scheme is 𝑂(|ℎ|2+𝜏), where |ℎ|2=ℎ12+ℎ22+...+ℎ𝑝2. Using the method of energy inequalities, we obtain an a priori estimate in the 𝐿2 - norm, which implies uniqueness, stability with respect to the right-hand side and initial data, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem at a rate equal to the approximation error. Numerical calculations are carried out on a test example, illustrating the theoretical calculations obtained in this work.

Конечно-разностные методы решения нелокальной краевой задачи для многомерного параболического уравнения с граничными условиями интегрального вида

The article considers a non-local boundary value problem for a multidimensional parabolic equation with integral boundary conditions. To solve the problem, we obtain an a priori estimate in differential form, which implies the uniqueness and stability of the solution with respect to the right-hand side and initial data on the layer in the $L_2$-norm. For the numerical solution of a nonlocal boundary value problem, a locally one-dimensional (economical) difference scheme by A.A. Samarskii with the order of approximation $O(h^2+\tau)$, the main idea of which is to reduce the transition from layer to layer to the sequential solution of a number of one-dimensional problems in each of the coordinate directions. Using the method of energy inequalities, a priori estimates are obtained, which imply uniqueness, stability, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem in the $L_2$-norm at a rate equal to the order of approximation of the difference scheme. An algorithm for the numerical solution is constructed.

The Euler–Lagrange Approximation of the Mean Field Game for the Planning Problem

The paper presents a finite-difference analogue of the differential problem formulated in terms of the theory of ‘‘Mean Field Games’’ for solving the planning problem of convey to a given state. Here optimization problem is formulated as coupled pair of parabolic partial differential equations of the Kolmogorov (Fokker–Planck) and Hamilton–Jacobi–Bellman type. The proposed Euler–Lagrange finite-difference analogue inherits the basic properties of an optimization differential problem at a discrete level. As a result, it can serve as an approximation of the original differential problem when the discretization steps tend to zero, or as a self-contained optimization task with a finite set of participants. For the proposed analogue, the algorithm of monotonous minimization of the value functional is constructed and illustrated on a model economic task.

Study of a Numerical Method for Solving a Boundary Value Problem for a Differential Equation with a Fractional Time Derivative

Recently, there has been an increased interest in the problem of numerical implementation of multiphase filtration models due to its enormous economic importance in the oil industry, hydrology, and nuclear waste management. In contrast to the classical models of filtration, filtration models in highly porous fractured formations with the fractal geometry of wells are not fully understood. The solution to this problem reduces to solving a system of differential equations with fractional derivatives. In the paper, a finite-difference scheme is constructed for solving the initial-boundary value problem for the convection-diffusion equation with a fractional time derivative in the sense of Caputo-Fabrizio. A priori estimates are obtained for solving a difference problem under the assumption that there is a solution to the problem in the class of sufficiently smooth functions that prove the uniqueness of the solution and the stability of the difference scheme. The convergence of the solution of the difference problem to the solution of the original differential problem with the second order in time and space variables is shown. The results of computational experiments confirming the reliability of theoretical analysis are presented.

Method of Green’s Potentials for Elliptic PDEs in Domains with Random Apertures

Problems with topological uncertainties appear in many fields ranging from nano-device engineering to the design of bridges. In many of such problems, a part of the domains boundaries is subjected to random perturbations making inefficient conventional schemes that rely on discretization of the whole domain. In this paper, we study elliptic PDEs in domains with boundaries comprised of a deterministic part and random apertures, and apply the method of modified potentials with Green's kernels defined on the deterministic part of the domain. This approach allows to reduce the dimension of the original differential problem by reformulating it as a boundary integral equation posed on the random apertures only. The multilevel Monte Carlo method is then applied to this modified integral equation and its optimal $\epsilon^{-2}$ asymptotical complexity is shown. Finally, we provide the qualitative analysis of the proposed technique and support it with numerical results.

Approximation of the Multidimensional Optimal Control Problem for the Heat Equation (Applicable to Computational Fluid Dynamics (CFD))

This work is devoted to finding an estimate of the convergence rate of an algorithm for numerically solving the optimal control problem for the three-dimensional heat equation. An important aspect of the work is not only the establishment of convergence of solutions of a sequence of discrete problems to the solution of the original differential problem, but the determination of the order of convergence, which plays a very important role in applications. The paper uses the discretization method of the differential problem and the method of integral estimates. The reduction of a differential multidimensional mixed problem to a difference one is based on the approximation of the desired solution and its derivatives by difference expressions, for which the error of such an approximation is known. The idea of using integral estimates is typical for such problems, but in the multidimensional case significant technical difficulties arise. To estimate errors, we used multidimensional analogues of the integration formula by parts, Friedrichs and Poincare inequalities. The technique used in this work can be applied under some additional assumptions, and for nonlinear multidimensional mixed problems of parabolic type. To find a numerical solution, the variable direction method is used for the difference problem of a parabolic type equation. The resulting algorithm is implemented using program code written in the Python 3.7 programming language.

New Properties of Modified Second Kind Chebyshev Polynomials

Modified second kind Chebyshev polynomials for solving higher order differential equations are presented in this paper. This technique, along with some new properties of such polynomials, will reduce the original differential equation problem to the solution of algebraic equations with a straightforward and computational digital computer. Some illustrative examples are included. The modified second kind Chebyshev polynomial is calculated using only a small number of the modified second kind Chebyshev polynomials, which leads to attractive results.

Eigenvibrations of an elastic bar with mechanical resonator

The differential eigenvalue problem governing eigenvibrations of an elastic bar with fixed first end and mechanical resonator attached to second end is investigated. This problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. We introduce limit differential eigenvalue problems and derive the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problems as a resonator parameter tending to infinity. The original differential eigenvalue problem is approximated by the finite difference method on a uniform mesh. Error estimates for approximate eigenvalues and eigenfunctions are established. Theoretical results are illustrated by numerical experiments for model problems. Investigations of this paper can be generalized for the cases of more complicated and important problems on eigenvibrations of beams, plates and shells with attached resonators.

Quadrature finite element method for the problem on eigenvibrations of a bar with elastic support

The differential eigenvalue problem describing eigenvibrations of a bar with fixed ends and with elastic support at an interior point is investigated. This problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. We formulate a limit differential eigenvalue problem and prove the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problem as stiffness coefficient tending to infinity. The original differential eigenvalue problem is approximated by the quadrature finite element method of arbitrary order on a uniform grid. Error estimates for approximate eigenvalues and eigenfunctions are established. Theoretical results are illustrated by numerical experiments for a model problem. Investigations of this paper can be generalized for the cases of more complicated and important problems on eigenvibrations of beams, plates and shells with elastic support.

Finite difference approximation of eigenvibrations of a bar with oscillator

The second-order ordinary differential spectral problem governing eigenvibrations of a bar with attached harmonic oscillator is investigated. We study existence and properties of eigensolutions of formulated bar-oscillator spectral problem. The original second-order ordinary differential spectral problem is approximated by the finite difference mesh scheme. Theoretical error estimates for approximate eigenvalues and eigenfunctions of this mesh scheme are established. Obtained theoretical results are illustrated by computations for a model problem with constant coefficients. Theoretical and experimental results of this paper can be developed and generalized for the problems on eigenvibrations of complex mechanical constructions with systems of harmonic oscillators.