Abstract
Many problems of mechanics and physics are posed in unbounded (or infinite) domains. For solving these problems one typically limits them to bounded domains and find ways to set appropriate conditions on artificial boundaries or use quasi‐uniform grid that maps unbounded domains to bounded ones. Differently from the above methods we approach to problems in unbounded domains by infinite system of equations. In this paper we present starting results in this approach for some one‐dimensional problems. The problems are reduced to infinite system of linear equations. A method for obtaining approximate solution with a given accuracy is proposed. Numerical experiments for several examples show the effectiveness of the offered method.
Highlights
Many problems of mechanics and physics are posed in unbounded or infinite domains, for example, heat transport problems in infinite or semi-infinite bar, aerosol propagation in atmosphere, problem of ocean pollution, wave propagation in unbounded media, and problem of computing the potential distribution due to a source of current in or on the surface of the Earth
Instead of transferring boundary condition on infinity without changes one tries to set appropriate conditions on artificial boundary. This is a direction of researches that attracts the attention of many specialists in the fields of mathematics, mechanics, and physics see 1–4
We construct a different scheme for the problem in unbounded domain and suggest a method for treating the infinite system in order to obtain an approximate solution with a given accuracy
Summary
Many problems of mechanics and physics are posed in unbounded or infinite domains, for example, heat transport problems in infinite or semi-infinite bar, aerosol propagation in atmosphere, problem of ocean pollution, wave propagation in unbounded media, and problem of computing the potential distribution due to a source of current in or on the surface of the Earth For solving these problems one usually restricts oneself to treat the problem in a bounded domain and try to use available efficient methods for finding exact or approximate solutions in the restricted domain. We construct a different scheme for the problem in unbounded domain and suggest a method for treating the infinite system in order to obtain an approximate solution with a given accuracy. Some numerical examples demonstrate the efficiency of the proposed method and its advantage over the quasi-uniform grid method
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