Closed solutions to a differential-difference equation and an associated plate solidification problem

Two distinct and novel formalisms for deriving exact closed solutions of a class of variable-coefficient differential-difference equations arising from a plate solidification problem are introduced. Thereupon, exact closed traveling wave and similarity solutions to the plate solidification problem are obtained for some special cases of time-varying plate surface temperature.


Background
In a recent paper by Grzymkowski et al. (2013), an analytical method for finding the approximate temperature distribution in a solidifying plate modeled by the one-phase problem was introduced. Equation (1), which models the temperature distribution only in the solid phase of the plate is such that x is the half of the plate thickness, λ its thermal conductivity, κ its latent heat of fusion, γ its mass density and b its diffusivity given by (1a) ∂ t T = b∂ xx T in (�(t),x) × (0, τ ); (1b) �(t) =x − ξ(t), ξ(0) = 0; (1c) T x=x = �(t) for t ∈ [0, τ ]; (1d) T (1e) −λ∂ x T x=�(t) b = λ/(γ κ). Further, �(t) describes the time-dependent position of the solidification front, ξ(t) the time-dependent thickness of the solidified layer, �(t) is the time-dependent temperature of the plate's surface, and τ the duration for complete solidification. The approach taken by Grzymkowski and coauthors relied on the assumption that the unknown temperature field T = T (x, t) takes the form of an exponential generating function with the undetermined sequence A j (t) ∞ 0 and solidification front �(t) [through ξ(t)] satisfying the boundary conditions where in Eq. (3b) and through the rest of this paper, the prime is indicative of an ordinary derivative with respect to the temporal variable t.
This technique, which is plausible provided series (2) is convergent in the interval (�(t), x) for all t ∈ (0, τ ), leads to the reformulation of Problem (1) as that of finding such A j (t) and ξ(t) as would satisfy the variable coefficient differential-difference equation (D�E) Z + is the set of positive integers.
D Es of similar structures to Eq. (4) with both constant and variable coefficients have been studied in the literature by various methods. Numerical and truncation techniques were deployed by were deployed by Barry (1966) in studying a first order constant coefficient first order D E. Feynman-Dyson (Feynman 1951) and Magnus (Blanes et al. 2009) time-ordering techniques have been employed in solving constant and variable coefficient Raman-Nath equations (which constitute a class of first order D Es) in Bosco and Dattoli (1983), Bosco et al. (1984), Dattoli et al. (1984), Dattoli et al. (1985). Alimohomadi et al. (2012) solved a variable coefficient D E using the Wei-Norman Lie-algebraic time ordering method of Wei and Norman (1963), Shang (2012).
The discrete version of Lie-group symmetric reduction was introduced by Levi and Winternitz (1991), Levi and Winternitz (1993). This novelty afforded the study of symmetry reductions of several classes of differential-difference equations: Shen (2007) used a combination of the classical Lie-group method and symbolic computation to solve nonlinear constant coefficient D Es; Shen et al. (2004) derived symmetry reductions of Toda-like lattice equations by a similar method; while Li et al. (2008) and Lv et al. (2011) deployed a synthesis of the Lie group and Harrison-Estabrook geometric techniques in the study of Lie symmetries of some differential-difference equations. The technique employed in this paper derives partly from that of Shen (2005) in which the direct similarity method of Clarkson and Kruskal (1989) was extended to the study of D Es.
Solidification of materials has been extensively studied in the literature, see Kurz and Fisher (1992), Dantzig andRappaz (2009), Glicksman (2011) and the references therein, and still receives continuous attention due to its huge significance to industrial processes. Several mathematical techniques have been developed and exploited in the study of models similar to (1); Chuang and Szekely (1971) proposed a Green's function-based semi-analytical method for studying approximate solutions of a solidification problem; Charach and Zoglin (1985) employed a combination of the heat balance integral method (Goodman 1958;Mitchell and Myers 2010;Layeni and Johnson 2012) and time-dependent perturbation theory to construct approximate solutions for solidification of a finite slab which is valid for small Stefan numbers and uniformly in time; Prud'homme et al. (1989) investigated heat transfer during the solidification of materials in various geometries by the method of strained coordinates; and Gonzalez et al. (2003) developed a computational simulation system for modelling a solidification process during continuous casting.
Grzymkowski et al. resorted to deriving approximate analytical and numerical solutions for Eq. (4), and consequently Eq. (1), primarily due to the difficulty encountered in obtaining closed-form solutions satisfying boundary conditions Eq. (3). Consequent upon the inability of their approach at solving D E (4) exactly and partially for the sake of completeness we revisit the problem, proffering two efficient protocols for solving (4).
The objective of this paper is twofold. The one is to construct exact solutions to D E (4) through two distinct syntheses of the ideas of Shen (2005), Clarkson and Kruskal (1989) or Bateman (1943). The other is to apply the obtained results in establishing exact closed-form solutions to the plate solidification problem (1).
The rest of this paper is organized as follows: The second Section gives the similarity reductions and closed form solutions to D E (4). The third Section gives two distinct classes of solutions to the solidification process courtesy the results of the second, while the last concludes the paper.

Clarkson-Kruskal's similarity reduction and explicit solutions
In this Section, in line with the direct method of Clarkson and Kruskal (1989), we seek solutions to D E (4) which are of the form with ϒ, , V j and z being continuously differentiable functions.
As a consequence of the above, D E (4) has solutions Equations (8) and (9) constitute a Clarkson-Kruskal symmetry reduction of Eq. (4). Following Bateman (1943), a solution to Eq. (9) can be constructed by assuming V j (z(t)) = exp (ipz(t))v j (p), i = √ −1, p ∈ R, thereby yielding the reduction which has the solution C 1 and C 2 being arbitrary constants. In summary, we have the following result.

A variant Clarkson-Kruskal similarity reduction
Equation (6) can be alternatively pictured as the system of uncoupled equations which can be recast as Here we shall only study the special case of Eqs. (14) and (15) for which z(t) = t; rather than Eq. (15b), the following equation is studied Next, we shall furnish a difference equation reformulation of Eq. (16) by employing the variant Clarkson-Kruskal ansatz g j being a continuously differentiable function of t for each j. Substituting ansatz (17) into differential-difference equation Eq. (15b) yields It is clear that our present objective can be attained by imposing conditions on Eq. (18) which simultaneously make ‫ג‬ j (t), j (t) and j (t) constant in t. One such way is to, firstly, endow g j (t) with a recurrence from which it follows that j (t) and j (t) are equivalentand the derivative of g j (t) with respect to t being Layeni et al. SpringerPlus (2016) 5:1273 Secondly, requiring that � ′ (t)� −1 (t) be of a constant proportion to ξ ′ 2 (t), that is converts Eq. (23) into Finally, we shall enforce a set of constraints which makes j (z(t), t) constant with respect to the variable t: We set This yields the following reduction of Eq. (15b):

Incorporating Eqs. (19)-(22) into Eq. (18) transforms it into
where C 9 and C 10 are integration constants. It is worth observing that Eqs. (5), (15a) and (27)  where κ 1 and κ 2 are arbitrary constants. A closed solution to the difference equation can be sought in the sense of Popenda (1987) or Mallik (1997). However the solutions to this class of variable coefficient difference equations are known to be quite cumbersome, and as such for practical purposes the generators of ṽ j ∞ 0 are constructed instead. Suppose that Z(x) is the sought generating function, of the sequence ṽ j ∞ 0 . Then, x 0 exp(y 2 )dy or through the error function Erf as D(x) = √ π 2 exp(−x 2 )Erf(x); H r (x) is rth Hermite polynomial which satisfies the ordinary differential equation 1 F 1 is the Kummer confluent hypergeometric function defined by . In the aforegone, U(p, q), V(p, q) and W(p, q) are, respectively, (32) C 1 and C 2 being arbitrary constants. The above results can be summarized as the following.

Traveling wave solutions
In this subsection, Theorem 1 is applied in solving Eq. (1). This approach is only admissible provided the range of convergence of series (2), per time, contains the spatial domain �(t) < x <x of Eq. (1). Elementary calculations confirm this in the affirmative: In point of fact, the range of convergence of the series is the set of real numbers. Due to the fact that �(t) =x − ξ(t), x =x being the plate's surface, we shall only consider ξ(t) in the form Further, the nature of solution (12) 2 suggests the existence of a traveling wave solution to the Stefan problem (1), one which is derived by substituting it into Eq. (2): up to integration constants ϒ 0 , C 1 and C 2 . Solving Stefan problem (1) up to initial and boundary conditions demands the application of Corollary 1. This leads to the special case of plate solidification process Eq. (1) for which the temperature of the surface of the plate differs from that of the constant temperature T ⋆ of the solidification front by a magnitude (−1 + exp (bt)) : Upon a reflection of the analyses of the previous Section and the preceding paragraph of this Section, it is observed that Eq. (41) has the exact closed travelling wave solution with speed b, the value of the thermal diffusivity.

Similarity solutions
The study of similarity solutions to the solidification problem due the analyses of "Clarkson-Kruskal's similarity reduction and explicit solutions" section is hinged on the nature of the second solution of the D E (4) as given by Theorem 2. Consequent upon Eq. (36), similarity solutions of the form which are self-similar in variables T (x, t)/ exp (q + 1) ξ ′ 2 (t)dt and x/ξ ′ −1 (t), are obtained.