Abstract
The aim of the study is an approximate construction with a given accuracy of solutions of boundary value problems for eigenvalues under various types of boundary conditions. It is shown that the problem of finding approximate large eigenvalues of boundary value problems is reduced to the analysis and solution of singularly perturbed differential equations with variable coefficients. Methods used: asymptotic diagram method developed to construct the asymptotic behavior of solutions of singularly perturbed differential equations and systems; methods of numerical integration of boundary value problems. The main results obtained are: the asymptotics of the required accuracy are constructed in the analytical form for the eigenvalues and eigenfunctions of the boundary value problems under various boundary conditions; analysis of the computational capabilities of the practical use of the constructed asymptotics in comparison with the results of numerical integration.
Highlights
The need for an approximate solution of boundary value problems often arises in the study of a wide variety of applied problems
The processes of heat and mass transfer can be described on the basis of differential equations of continuum mechanics, which reflect the objective laws of conservation of mass, momentum and energy
In the formula (23), the numbers D1 and D2 so far are unknown, they must be determined from the boundary conditions (4) and the requirements that the solution (23) be nontrivial
Summary
The need for an approximate solution of boundary value problems often arises in the study of a wide variety of applied problems. The creation and solution of adequate mathematical models of these processes, where boundary value problems for eigenvalues for differential equations often arise, allows to reduce the amount of experimental research that needs to be carried out, for example, to achieve the required parameters of heat exchangers and systems. F (t ) 0 at t 0;1 , Ξ» is a large parameter Another example is the natural vibrations of an absolutely flexible, heavy, inextensible, uniformly loaded rope of length l0 suspended at one end, which satisfy the following boundary value problem (2): Ξ΅y2( 0 )G 0,pgF (l0 x) y '
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