Abstract
A cubic B-spline collocation approach is described and presented for the numerical solution of an extended system of linear and nonlinear second-order boundary-value problems. The system, whether regular or singularly perturbed, is tackled using a spline collocation approach constructed over uniform or non-uniform meshes. The rate of convergence is discussed theoretically and verified numerically to be of fourth-order. The efficiency and applicability of the technique are demonstrated by applying the scheme to a number of linear and nonlinear examples. The numerical solutions are contrasted with both analytical and other existing numerical solutions that exist in the literature. The numerical results demonstrate that this method is superior as it yields more accurate solutions.
Highlights
We study a generalized nonlinear system of second-order boundary value problems given by a0(x)u + a1(x)u + a2(x)u + a3(x)v + a4(x)v + a5(x)v + g1(x, u, v, u, v ) = f1(x), b0(x)u + b1(x)u + b2(x)u + b3(x)v + b4(x)v + b5(x)v + g2(x, u, v, u, v ) = f2(x)
The spline collocation method, which was first introduced by Christara and Ng [2] and [5], has been unified with an adaptive technique to solve the nonlinear system under consideration on uniform and non-uniform meshes via mesh redistribution [2] and manipulating an iterative scheme arising from Newton’s method by mapping uniform node points to non-uniform ones such that the errors are reduced
The scheme has been successfully applied for both regular systems as well as ones which are singularly perturbed that possess boundary layers
Summary
The spline collocation method, which was first introduced by Christara and Ng [2] and [5], has been unified with an adaptive technique to solve the nonlinear system under consideration on uniform and non-uniform meshes via mesh redistribution [2] and manipulating an iterative scheme arising from Newton’s method by mapping uniform node points to non-uniform ones such that the errors are reduced This collocation approach has been employed by Khuri and Sayfy for the numerical solution of a spectrum of problems, including a boundary layer problem [12], a generalized nonlinear Klein-Gordon equation [10], a generalized parabolic problem subject to non-classical conditions [13], and Troesch’s problem [11]. A conclusion is given that summarizes the outcomes of the simulations
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