By Mikhail V. Fedoryuk (auth.)
In this publication we current the most effects at the asymptotic idea of standard linear differential equations and structures the place there's a small parameter within the greater derivatives. we're excited about the behaviour of options with appreciate to the parameter and for giant values of the self sufficient variable. The literature in this query is massive and broadly dispersed, however the equipment of proofs are sufficiently related for this fabric to be prepare as a reference e-book. we now have limited ourselves to homogeneous equations. The asymptotic behaviour of an inhomogeneous equation will be acquired from the asymptotic behaviour of the corresponding primary procedure of strategies by means of employing equipment for deriving asymptotic bounds at the proper integrals. We systematically use the concept that of an asymptotic enlargement, info of that may if helpful be present in [Wasow 2, Olver 6]. via the "formal asymptotic answer" (F.A.S.) is known a functionality which satisfies the equation to some extent of accuracy. even supposing this idea isn't really accurately outlined, its that means is often transparent from the context. We additionally notice that the time period "Stokes line" utilized in the publication is akin to the time period "anti-Stokes line" hired within the physics literature.
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Additional info for Asymptotic Analysis: Linear Ordinary Differential Equations
Then Pl,2(X) E COO(I) and there are linearly independent eigenvectors el(x), e2(x) of A(x) of class COO(I). The matrix T = (el(x), e2(x)) reduces A(x) to diagonal form, that is T-l(x)A(x)T(x) Let et(x), e~(x) = A(x) = diag(Pl(x), P2(X)). be the rows of T-l(x); then § 4. Systems Containing a Large Parameter ej(x)ek(x) = bjk' 43 ejA(x) = pj(x)ej(x). S. \S(x) L 00 y A-k h(x) . (3) k=O Substituting this into (1), we obtain the recurrence system of equations (A(x) - 8'(x)I)fo(x) = 0, (4) k=O, 1, ...
The matrix T = (el(x), e2(x)) reduces A(x) to diagonal form, that is T-l(x)A(x)T(x) Let et(x), e~(x) = A(x) = diag(Pl(x), P2(X)). be the rows of T-l(x); then § 4. Systems Containing a Large Parameter ej(x)ek(x) = bjk' 43 ejA(x) = pj(x)ej(x). S. \S(x) L 00 y A-k h(x) . (3) k=O Substituting this into (1), we obtain the recurrence system of equations (A(x) - 8'(x)I)fo(x) = 0, (4) k=O, 1, ... (A(x)-8'(x)I)fk+I(X)=-f~(x), It follows from the first equation that 8'( x) is an eigenvalue and fo (x) is an eigenvector of A(x).
2. 1 Asymptotic Behaviour of the Solutions. We consider the equation y" + Ap(X)Y' + A2q(x)y = 0 (16) on the interval I = [a, b], where p(x), q(x) E COCCI). jD(x)) , D(x) = p2(x) - 4q(x). The point Xo E I is called a turning point of (16) if the roots of the characteristic equation coincide for x = Xo. Consequently, the turning points are the roots of the equation D( x) == p2 (x) - 4q( x) = 0 . (17) We introduce the conditions: 1) Equation (16) has no turning points, that is ®J: 0 for x E I. jD(x) such that Re (yD(x)) ~ 0 for x E I.
Asymptotic Analysis: Linear Ordinary Differential Equations by Mikhail V. Fedoryuk (auth.)