A new application of Lagrange-Bürmann expansions. II. Application to unsteady heat conduction problems with radiation
[For Part I see ibid. 34, 697-727 (1983; Zbl 0554.65003).] The problem (P) considered concerns the solution of the equation \[ (*)\quad \partial T(x,t)/\partial t=\partial^ 2T(x,t)/\partial x^ 2\quad (0\leq x,t\leq \infty) \] subject to the conditions \(T(\infty,t)=T(x,0)=0\) and, with \(U(t)=T(0,t)\) and \(V(t)=\partial T(x,t)/\partial x\) when \(x=0\), either (a) \(V(t)=-\{1-U(t)\}^ n\) or (b) \(V(t)=1-U(t)^ n\). Cases (a) and (b) concern body radiation and gas radiation respectively. The value \(n=1\) is associated with a linear law of heat exchange and \(n=4\) with Stefan-Boltzmann radiation. T(x,t) may be exhibited as an indefinite integral whose integrand involves U(t). Use of a series development of U(t) in terms of t in conjunction with this integral expression meets with limited success. With \(t'=t^{-1/2}\) the changes of variables \(\eta =(1/2)xt'\), \(e(t)=t'U(t)\), \(\tau =h(t)=t\{de(t)/dt\}/e(t)\) and \(\theta (\eta,\tau)=T(x,t)/U(t)\) are introduced and (P) is reformulated in terms of the independent variables \(\eta\) and \(\tau\) and the dependent variable \(\theta\) (\(\eta\),\(\tau)\). It is provisionally assumed that a series expansion for e(t) in powers of \(t^{1/2}\) in cases (a,b) with \(n=1\) and case (a) with \(n=4\) or in powers of \(t^ 2\) in case (b) with \(n=4\) is available. A Lagrange-Bürmann expansion for e(t) in powers of \(\tau\) is obtained, using theory developed in Part I of the paper under review. \(\theta\) (\(\eta\),\(\tau)\) is exhibited as a series \(\sum^{\infty}_{i=0}\theta_ i(\eta)\tau^ i\), the coefficients \(\theta_ i(\eta)\) being obtained recursively from difference- differential relationships derived by substituting \(\theta\) (\(\eta\),\(\tau)\) in the equation obtained from (*) and equating corresponding powers of \(\tau\). It is shown how the coefficients in the series assumed for e(t) can be determined as the \(\theta_ i(\eta)\) are obtained.
scientific article; zbMATH DE number 3944397