An asymptotic-numerical method for a model system with turning points
A study is made of the asymptotic behavior of solutions to a boundary value problem for a singularly perturbed nonlinear differential equation of fourth order having a turning point. The fourth order system considered has the form \((i)\quad \epsilon u^{(4)}+f(u,u')u^{(3)}=0,\quad 0<x<1,\) \((ii)\quad u(0)=u(1)=0,\quad u^{(2)}(0)=A,\quad u^{(2)}(1)=B,\) where f is specialized to be (1) \(f=u\) or (2) \(f=u'.\) Following \textit{K. E. Barrett} [Quest. J. Mech. Appl. Math. 27, 57-68 (1974; Zbl 0285.65053) and Z. Angew. Math. Phys. 26, 807-817 (1975; Zbl 0326.76035)] the method for determining the asymptotic behavior is to replace (i), (ii) with a spline approximation obtained by solving a sequence of constant coefficient problems on abutting subintervals of [0,1] and determining the asymptotic behavior of the approximation. The asymptotic behavior of solutions varies according to the sign and relative magnitude of A and B. The results presented both concur and extend results obtained by \textit{F. W. Dorr}, \textit{S. V. Parter} and \textit{L. F. Shampine} [SIAM Review 15, 43-88 (1973; Zbl 0293.34086)]. New results are that if \(A<0\), \(B>0\), and (1) then \(u^{(2)}(x,\epsilon)\to 0\) on [\(\delta\),1-\(\delta\) ], \(\delta >0\), and also if \(A>0\), \(B<0\) and (1) then \(u^{(2)}(x,\epsilon)\to A\), \(0\leq x\leq c-\delta\), \(u^{(2)}(x,\epsilon)\to B\), \(c+\delta \leq x\leq 1\), with \(c=(-B)^{1/3}/A^{1/3}+(-B)^{1/3}\).
singularly perturbed nonlinear differential equation of fourth order
scientific article; zbMATH DE number 4155789
scientific article