Variable mesh methods for the numerical solution of two-point singular perturbation problems

Spline finite difference methods for singular two point boundary value problems

“…When a,i---1, it reduces to the uniform mesh case. This non-uniform mesh has been successfully used by Jain, Iyengar and Subramanyam [4] in solving two point singular perturbation boundary value problems. Substituting the expressions for xs-1, x~+ 1 and expanding in Taylor's series, we get…”

Geometric Mesh Three-Point Discretization for Fourth-Order Nonlinear Singular Differential Equations in Polar System

“…For the convergence, the coefficients , , and associated with (5) and (15) must be negative (see, [22]), from which we obtain the condition | − √ 5/2| < 1/2. Now applying the methods (5) and (15) to (33) and using the similar technique discussed in Section 3 for singular coefficients ( ) and ( ), we obtain the following system of difference equations:…”

“…The geometric mesh method for self-adjoint singular perturbation problems using 2 Advances in Numerical Analysis finite difference approximations was discussed in [21]. The use of geometric mesh in the context of boundary value problems was studied extensively in [22][23][24]. In this paper, we derive a geometric mesh finite difference method for the solution of fourth-and sixth-order differential boundary value problems with order of accuracy being three.…”

Patching Asymptotics Solution of a Cable with a Small Bending Stiffness

“…In these applications, bending stresses take place along very short zones, compared to the total length, which makes the use of standard finite difference or finite element techniques rather inefficient, as pointed out clearly in Burgess 1993. Some advanced numerical techniques have been developed to cope with these large gradients (Bieniasz 2008;Jain et al 1984;Rao and Kumar 2007;Stynes and Oriordan 1986), but their application is still troublesome as the boundary layer becomes very short.…”