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Boundary value problem for ordinary second order differential equations

Authors
Journal
USSR Computational Mathematics and Mathematical Physics
0041-5553
Publisher
Elsevier
Publication Date
Volume
4
Issue
5
Identifiers
DOI: 10.1016/0041-5553(64)90138-7
Disciplines
  • Mathematics

Abstract

Abstract The investigation of some equations of mathematical physics leads to the examination of the following boundary value problems for ordinary second order differential equations. 1. 1. To find the solutions y = y( x) of the equation y″ + 2 x − y + y n = 0, n > 0, x ⩾ 0 , (1.1) which satisfy the conditions y (0) = y 0 < ∞, y′ (0) = 0, y (∞) = 0, (1.2) where y 0 is an unknown positive parameter. 2. 2. To find the solutions η = η( x) of the equation η″ = η − η n x n−1 , n > 0, x ⩾ 0 , (1.3) satisfying the conditions η (0) = 0, η′ (0) = α < ∞, η ( t8) = 0, (1.4) where a is an unknown positive parameter. Equation (1.3) is obtained from equation (1.1) by the substitution η( x) = xy( x). The solutions of equation (1.1) with initial conditions y(0) = y 0, y′(0) = 0 corresponds to the solution of the equation (1.3) with initial conditions η (0) = 0, η′ (0) = α = y 0. Problems (1.1)–(1.2), (1.3)–(1.4) with n = 2 and n = 3 arise in nonlinear field theory in the study of the interaction between elementary particles. With n = 3 2 equation (1.1) represents an equation of the Thomas-Fermi type and the corresponding problem (1.1)–(1.2) arises in the statistical theory of the nucleus. The problem (1.3)–(1.4) for n = 3 was considered in [1]–[4], where some considerations concerning the existence and properties of the solution of the given boundary value problem were put forward and the results of computer calculations were given. In [4] five values of the initial derivative a were given with a defined accuracy, corresponding to five different solutions of the problem. In [5] problem (1.3)–(1.4) is reduced to the boundary value problem for the linear equation y″ = y − g3 n−1 x n−1y , ε > 0 , by linearizing the nonlinear equations. For this linear equation some idea of the possible behaviour of its solution was put forward without proof. Unfortunately the basis of the method of linearization was not satisfactory. Finally, in [6], recently published, the existence of positive solutions of problem (1.1)–(1.2) was established for any n if 1 < n ⩽ 4. These solutions were found on points of a set of functions where the minimum of the special functional is reached. The aim of the present paper is to give a quantitative picture of the behaviour of the solution of equations (1.1) and (1.3) for any real n from some interval. In addition the solvability of problems (1.1)–(1.2) and (1.3)–(1.4) is investigated. Section 1 of this paper is concerned with the proof of the almost obvious Theorem 1 about the absence in problems (1.1)–(1.2) and (1.3)–(1.4) of positive solutions for the case 0 < n ⩽ 1. In Section 2 the problem (1.3)–(1.4) for all n > 1 is replaced by the equivalent problem (2.1)–(2.2). Equation (2.1) is the same as equation (1.3) in the region η ⩾ 0. For the equation (2.1) with any real n > 1 the properties of the solutions of the Cauchy problem are investigated. Lemma 1 establishes the existence of a solution, determined for all x > 0 and satisfying the given initial conditions at the point x = 0. Lemma 2 shows the uniqueness of such a solution. Lemma 3 establishes the presence of the continuous dependence of the solutions of equation (2.1) on the variation of the initial derivatives at the point x = 0. In Section 3 we examine problems (1.1)–(1.2) and (1.3)–(1.4) for all real n in the interval 1 < n ⩽ 3. In Theorem 2 the existence of positive solutions of the given problems is proved for any n E . (1, 3].

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