Power series solutions to some nonlinear diffusion problems (Q1312130)

The paper under review deals with the problem of dopant diffusion through a semiconductor. Let \(c= c(x,t)\) denote the concentration of the dopant at the point \(x>0\) and time \(t>0\); then the dopant diffusion can be described in dimensionless form as follows: \[ c_ t= (c^ m c_ x)_ x \qquad (x>0,\;t>0) \tag{1} \] and the boundary and initial conditions are: \[ c(0,t)=1, \quad t>0, \qquad c(x,0)=0, \quad x>0. \tag{2} \] By means of the function \(u(\eta)\) introduced by \textit{J. R. King} [IMA J. Appl. Math. 38, 87-95 (1987; Zbl 0812.35101)]: \[ u(\eta)= \begin{cases} c^ m(\eta), \quad &\eta\leq\eta_ 0\\ 0, &\eta> \eta_ 0\end{cases}, \] where \(\eta= x/t^{1/2}\) and \(\eta= \eta_ 0\) denotes the free boundary of the considered problem. (1)--(2) can be reduced to \[ -{\textstyle {1\over 2}} \eta u'= uu''+ {\textstyle {1\over m}} u'{}^ 2 \quad (\eta\in [0,\eta_ 0 ]), \qquad u(0)=1, \tag{3} \] and by introducing the \(\eta= \eta_ 0 (1- s)\) and \(u(\eta)= \eta_ 0^ 2 mv(s)\), the problem (3) is reduced to \[ mvv''+ (v')^ 2= {\textstyle{1\over 2}} (1-s)v'\quad (s\in [0,1]), \qquad v(0)=0, \quad \lim_{s\to 0^ +} (v') (s)= {\textstyle {1\over 2}}.\tag{4} \] It turns out that the solution of the problem (4) can be represented by the power series \[ v(s)= \sum_{n=1}^ \infty a_ n (m)s^ n \tag{5} \] with a recurrence relation for the coefficients. The main result of the paper is the convergence proof for the power series solution (5) and the statement that if \(m\geq 1\), the radius of convergence is greater than 2.46.