Phase-plane solutions to some singular perturbation problems

Nonlinear Boundary Value Problems Describing Mobile Carrier Transport in Semiconductor Devices

“…Since α(y(t, ε)) = C ε ⊃ (E 1 (E 0 ), 0) and the point (E 1 (E 0 ), 0) is a unique saddle point with a separatrix C ε , Theorem 4.1 can be obtained by the reasoning used in the 5-th proposition of R. O'Malley's theorem [7]. Proof of Theorem 4.2 completely repeats that of Theorem 4.1.…”

“…The proofs of these theorems follow from some results of R. O'Malley [7] and the next three propositions. It appears that the asymptotic behaviour of the solutions of the problem (11) depends essentially on the properties of such differential equations asÿ…”

“…Therefore it can be obtained from Proposition 5 (see theorem from [7]) that the equation (13) has a solution y 0 (t) with the properties mentioned above.…”

Shooting Method for Nonlinear Singularly Perturbed Boundary-Value Problems

“…In (1.5), Q(u) = − V (u), where V (u) is a double well potential with wells of equal depth located at the preferred phases u = s − and u = s + . The problem of finding asymptotic behavior of the solutions to (1.1) & (1.2) or (1.1) & (1.3) has been studied earlier by O'Malley [5], using a phase-plane analysis. Although his approach provides useful qualitative information about the solutions with internal layer behavior, it does not give quantitative information such as asymptotic formulas for the solutions.…”

The Numerical Solution of Boundary Value Problems for Stiff Differential Equations

“…While the trivial solution y(x) = 0 does not satisfy our hypotheses, it is also valid for these boundary conditions. In addition, O'Malley [31] shows that there are denumerably many solutions of this problem switching back and forth between ± 1. The results of calculations corresponding to the limiting solution z(x) = 1 obtained by the asymptotic method and Pearson's method are presented in the last two columns of Table 7 and in Figure 12.…”