On the canonical solution of indefinite problem with m turning points of even order

Abstract

We consider the differential equationon a finite interval I , say I = [0, 1], where I contains m turning points, that is here, zeros of φ, under the assumption that one of the turning points is of odd order while the others are of even order. Using of the asymptotic estimates provided in [W. Eberhard, G. Freiling, A. Schneider, Connection formulae for second-order differential equations with a complex parameter and having an arbitrary number of turning points, Math. Nachr. 165 (1994) 205-229] for a special fundamental system of solutions of ( * ) in I , we study the infinite product representation of solutions of ( * ). Such representations are relevant to the inverse spectral problem.

Inverse problem for differential systems having a singularity and turning point of even or odd order

“…(b) Let = 2 0 + 1 and C(t, λ) be the solution of (2.2) with satisfying the initial conditions C(0, λ) = 1, C (0, λ) = 0. From [17], for 0 < t < t 1 ,…”

A Reliable Treatment for Nonlinear Differential Equations

“…In this paper and in our proposed method, taking the Laplace transform from both sides of the equation solves this difficulty. Indeed, in vast majority of cases in differential equations with variable coefficients, we cannot obtain an exact solution, so we must look for approximate solutions, such as asymptotic techniques [9,10], analytical [11][12][13][14], and numerical methods [15][16][17][18][19].…”

Dual Equation and Inverse Problem for an Indefinite Sturm–liouville Problem With M Turning Points of Even Order

“…In most differential equations with variable coefficients it is impossible to obtain an exact solution, so one must resort to various approximation methods of solution. One of the most useful mathematical methods of achieving this, is representing the solution by an asymptotic form [15,16]. But, in methods connected with dual equations, the closed form of the solution is needed.…”