Matched Asymptotic Expansions: Ideas and Techniques

Second order phase field asymptotics for multi-component systems

“…with respect to the L 2 -product over R. Differentiating (22) with respect to z we obtain that ∂ z Φ 0 lies in the kernel of L(Φ 0 ). Since Φ 0 (−z) = 1 − Φ 0 (z) we get with the help of Assumption C that ∂ z Φ 0 and h (Φ 0 ) are even, hence (25) allows for an even solution and in the following we will assume that Φ 1 is even.…”

“…δ = 10 −3 ) remains away from the outer boundary during the evolution. The function Φ 0 is the solution to (22) with the boundary conditions Φ 0 (z) → 0, 1 as z → ∞, −∞. Initial values for u were obtained by matching outer and inner solution to leading and first order obtained from the asymptotic expansions (see e.g.…”

“…Initial values for u were obtained by matching outer and inner solution to leading and first order obtained from the asymptotic expansions (see e.g. [22])…”

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“…We notice that since expansion (4.11) is our hypothetical p, the outer expansion u must be contained in w when both written in terms of x = r]~1^. Consequently we should expect to find unknown ^ 's from Equations (4.16) while searching for unknown tjk s. The sequence of equations for tjk s has the following form (see Equations .., (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) with the successive equations containing other Aj's which enter the formulas for PjkS. At first glance, the task of finding Aj's and tjk s using Equations (4.19) seems to be daunting.…”

“…The expansion for w, w (x,e) = w0 (x) + ewi (x) + e2 w2 (x) + ..., (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) implies that…”