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We consider an approximate solution for the one-dimensional semilinear singularly-perturbed boundary value problem, using the previously obtained numerical values of the boundary value problem in the mesh points and the representation of the exact solution using Greens function. We present an $varepsilon$-uniform convergence of such gained the approximate solutions, in the maximum norm of the order $mathcal{O}left(N^{-1}right)$ on the observed domain. After that, the constructed approximate solution is repaired and we obtain a solution, which also has $varepsilon$--uniform convergence, but now of order $mathcal{O}left(ln^2N/N^2right)$ on $[0,1].$ In the end a numerical experiment is presented to confirm previously shown theoretical results.
We consider smooth systems limiting as $epsilon to 0$ to piecewise-smooth (PWS) systems with a boundary-focus (BF) bifurcation. After deriving a suitable local normal form, we study the dynamics for the smooth system with $0 < epsilon ll 1$ using a c
Boundary equilibria bifurcation (BEB) arises in piecewise-smooth systems when an equilibrium collides with a discontinuity set under parameter variation. Singularly perturbed BEB refers to a bifurcation arising in singular perturbation problems which
Fluid flows containing dilute or dense suspensions of thin fibers are widespread in biological and industrial processes. To describe the motion of a thin immersed fiber, or to describe the forces acting on it, it is convenient to work with one-dimens
An explicit solution of the stationary one dimensional half-space boundary value problem for the linear Boltzmann equation is presented in the presence of an arbitrarily high constant external field. The collision kernel is assumed to be separable, w
For a stationary and axisymmetric spacetime, the vacuum Einstein field equations reduce to a single nonlinear PDE in two dimensions called the Ernst equation. By solving this equation with a {it Dirichlet} boundary condition imposed along the disk, N