Differentiability and its asymptotic analysis for nonlinear singularly perturbed boundary value problem
The authors consider a two-point singularly perturbed boundary value problem on the interval \(t\in [0,1]\) for a system of differential equations \[ \varepsilon\frac{dz}{dt}=f(z,y,t),\qquad \frac{dy}{dt}=g(z,y,t), \] where \(z\) and \(f\) are both \(M\)-dimensional vectors, \(y\) and \(g\) are both \(m\)-dimensional vectors. Boundary conditions for unknown vector-function \(y\) are given at the left interval's end only (\(y(0,\varepsilon)=y^0\)), while boundary conditions for unknown vector-function \(z\) are really two-point boundary condition (\(a z(0,\varepsilon)+b z(1,\varepsilon)=z^0\), with \(a\) and \(b\) being constant matrices). It is well-known that under certain general assumptions referred as the case of conditional stability, the above-formulated problem has a solution with an asymptotics given by boundary function method. The main result of this paper is that the latter solution is continuously differentiable with respect to the boundary data \(y^0\) and \(z^0\). Moreover, every derivative with respect to the boundary data is well-approximated by the corresponding derivative of the asymptotics obtained via boundary function method.
scientific article; zbMATH DE number 5324195