Application of the Chebyshev collocation method to solve boundary value problems of heat conduction

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Abstract

For one-dimensional inhomogeneous (with respect to the spatial variable) linear parabolic equations, a combined approach is used, dividing the original problem into two subproblems. The first of them is an inhomogeneous one-dimensional Poisson problem with Dirichlet-Robin boundary conditions, the search for a solution of which is based on the Chebyshev collocation method. The method was developed based on previously published algorithms for solving ordinary differential equations, in which the solution is sought in the form of an expansion in Chebyshev polynomials of the 1st kind on Gauss-Lobatto grids, which allows the use of discrete orthogonality of polynomials. This approach turns out to be very economical and stable compared to traditional methods, which often lead to the solution of poorly defined systems of linear algebraic equations. In the described approach, the successful use of integration matrices allows complete elimination of the need to deal with ill-conditioned matrices. The second, homogeneous problem of thermal conductivity is solved by the method of separation of variables. In this case, finding the expansion coefficients of the desired solution in the complete set of solutions to the corresponding Sturm-Liouville problem is reduced to calculating integrals of known functions. A simple technique for constructing Chebyshev interpolants of integrands allows to calculate the integrals by summing interpolation coefficients.

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1. Introduction Many important physics problems that involve two or more independent variables are solved using mathematical models that include partial differential equations, not limited to models based on ordinary differential equations. Models of this kind also include the description of heat propagation in solids using the heat equation with various boundary and initial conditions. An important method for solving partial differential equations known as the separation of variables method will be discussed below. Its essential feature is the reduction of the original partial differential equation to a system of simpler ordinary differential equations, which can be successfully solved based on given initial or boundary conditions. The desired solution to a partial differential equation is expressed as an infinite series, which is the sum of solutions of individual ordinary differential equations. In many cases, it is convenient to represent the required solutions in the form of a series of sines, cosines, or polynomial functions, for example Chebyshev polynomials. This approach allows an effective use of the collocation method - a projection method for solving both integral and differential equations. Therefore, the first part of the present paper is devoted to a discussion of the Chebyshev interpolation method for solving the one-dimensional heat equation. The Chebyshev collocation method has proven itself in solving a wide class of problems [1-5]. Particularly, in [6-9] its effectiveness was demonstrated in solving ODEs and problems of restoring functions from known first- and second-order derivatives. In Ref. [10] stable spectral methods for © Lovetskiy K. P., Sergeev S. V., Kulyabov D. S., Sevastianov L. A., 2024 This work is licensed under a Creative Commons Attribution 4.0 International License https://creativecommons.org/licenses/by-nc/4.0/legalcode solving the Poisson equation with Dirichlet-Dirichlet, Dirichlet-Neumann and Neumann-Neumann boundary conditions were analyzed in detail. The present paper describes an algorithm for solving the 1-D Poisson equation with Dirichlet-Robin boundary conditions. The algorithm is based on the developed new method of spectral collocation and illustrates the effectiveness of the variable separation method in solving inhomogeneous heat conduction problems. Spectral methods have proven themselves to be excellent in solving homogeneous boundary value problems for a wide class of partial differential equations using the method of separation of variables. In cases of inhomogeneous problems, methods for separating variables are not directly applicable. However, in this paper we show how the Chebyshev collocation method can be effectively applied in a two-stage solution scheme for a certain class of inhomogeneous boundary value problems for a 1-D linear parabolic equation. 2. Mathematical model of heat conduction Thermal conductivity is the property of a material to conduct heat, which is assessed primarily from the point of view of Fourier’s law of thermal conductivity. Heat conduction, also called diffusion, is the direct microscopic exchange of kinetic energy of particles (such as molecules) or quasiparticles (such as lattice waves) across a boundary between two systems. On a microscopic scale, thermal conduction occurs when hot, fast-moving, or vibrating atoms and molecules interact with neighboring atoms and molecules, transferring some of their energy (heat) to those neighboring particles. In other words, heat is transferred by conduction when neighboring atoms vibrate relative to each other or when electrons move from one atom to another. When an object has a different temperature than another body or its surroundings, heat flows so that the body and surroundings reach the same temperature, at which point they are in thermal equilibrium. This spontaneous transfer of heat always occurs from a region of high temperature to another region of lower temperature, as postulated by the second law of thermodynamics. Thermodynamic and mechanical heat transfer are calculated using the heat transfer coefficient - the proportionality between heat flow and the thermodynamic driving force of heat flux. Heat flux is a quantitative vector representation of the movement of heat through a surface [11]. In an engineering context, the term “heat” is perceived as synonymous with thermal energy. The heat conduction equation models diffusion processes [12], including thermal energy in solids, solutes in liquids, and biological populations. We will consider the heat conduction equation describing the temperature change in a one-dimensional rod of a finite length. Let us also consider several possible types of boundary conditions that can be used when modeling temperature changes. A commonly used method for solving the heat conduction equation is the complete separation of variables method, which results in the solution of two ordinary differential equations generated by the separation of variables method. To solve one of the emerging subproblems, a uniform approach to solving the heat conduction equation for almost any of the frequently used (Dirichlet- Neumann-Robin) sets of boundary conditions is considered. A technique is proposed for constructing a general solution to the inhomogeneous Poisson equation - the heat conduction equation - regardless of the type of boundary conditions. Concretization of the solution - the determination of a pair of missing coefficients of expansion of the solution according to the selected polynomial basis occurs at the second stage, considering the specified (distinct types and combinations) boundary conditions. 3. Inhomogeneous boundary value problems Let us consider the solution of an inhomogeneous initial-boundary value problem for a onedimensional parabolic equation, including a time-independent inhomogeneous part of the equation and time-independent boundary conditions. Two-sided Dirichlet-Dirichlet conditions
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About the authors

Konstantin P. Lovetskiy

RUDN University

Email: lovetskiy-kp@rudn.ru
ORCID iD: 0000-0002-3645-1060

Candidate of Sciences in Physics and Mathematics, Associate Professor of Department of Computational Mathematics and Artificial Intelligence

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Stepan V. Sergeev

RUDN University

Email: 1142220124@rudn.ru
ORCID iD: 0009-0004-1159-4745

PhD student of Department of Computational Mathematics and Artificial Intelligence

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Dmitry S. Kulyabov

RUDN University; Joint Institute for Nuclear Research

Email: kulyabov-ds@rudn.ru
ORCID iD: 0000-0002-0877-7063

Professor, Doctor of Sciences in Physics and Mathematics, Professor at the Department of Probability Theory and Cyber Security of Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University); Senior Researcher of Laboratory of Information Technologies, Joint Institute for Nuclear Research

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation; 6 Joliot-Curie St, Dubna, 141980, Russian Federation

Leonid A. Sevastianov

RUDN University; Joint Institute for Nuclear Research

Author for correspondence.
Email: sevastianov-la@rudn.ru
ORCID iD: 0000-0002-1856-4643

Professor, Doctor of Sciences in Physics and Mathematics, Professor at the Department of Computational Mathematics and Artificial Intelligence of Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University), Leading Researcher of Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation; 6 Joliot-Curie St, Dubna, 141980, Russian Federation

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Copyright (c) 2024 Lovetskiy K.P., Sergeev S.V., Kulyabov D.S., Sevastianov L.A.

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