The Asymptotics of Solutions of a Singularly Perturbed Equation with a of Fractional Turning Point
We develop the classical Vishik – Lyusternik – Vasil’eva – Imanaliev boundary-value method for constructing uniform asymptotic expansions of solutions of singularly perturbed equations with singular points. In this paper, by modernizing the classical method of boundary functions, uniform asymptotic expansions of solutions of singularly perturbed equations with a fractional turning point are constructed. As we know, problems with turning points are encountered in the Schrodinger equation for the tunnel junction, problems with a classical oscillator, problems of continuum mechanics, the problem of hydrodynamic stability, the Orr – Sommerfeld equation, and also in the determination of heat to a pipe, etc. Determination of the behavior of solving similar problems with aspiration small (large) parameter to zero (to infinity) is an actual problem. We study the Cauchy and Dirichlet problems for singularly perturbed linear inhomogeneous ordinary differential equations of the first and second order, respectively. Here it is proved that the principal terms of the asymptotic expansions have negative fractional powers with respect to a small parameter. As practice shows, solutions to most singularly perturbed equations with singular points have this property. The constructed decompositions of the solutions are asymptotic in the sense of Erdey, when the small parameter tends to zero. Estimates for the remainder terms of the asymptotic expansions are obtained. The asymptotic expansions are justified. The idea of modifying the method of boundary functions is realized for ordinary differential equations, but it can also be used in constructing the asymptotic of the solution of singularly perturbed partial differential equations with singularities.
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