Troesch’s problem: A B-spline collocation approach

Numerical Solution of a Class of Nonlinear System of Second-Order Boundary-Value Problems: A Fourth-Order Cubic Spline Approach

“…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]. For further details and applications of the technique see [4,5,9,10,11,12,13] and the references therein.…”

“…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]. For further details and applications of the technique see [4,5,9,10,11,12,13] and the references therein. The convergence analysis is deliberated and the method is verified to be of fourth-order rate of convergence which is then conformed numerically using the double-mesh principle.…”

Solution of Troesch’s problem using fixed-point iteration with embedded green’s functions

“…In another work, he applied simple shooting technique and variable transformation to obtain the solutions for up to ( ) [2]. Khuri andSayfy (2011) were successful in getting acceptable results for ( ) byusing the adaptive spline collocation approach over a non-uniform mesh [5]. Temimi (2012) solved the problem for ( ) by a discontinuous Galerk infinite element method [7].…”

Numerical Solution of Troesch’s Problem by Sinc-Collocation Method

“…This problem has been studied extensively. Troesch found its numerical solution in [7] using the shooting method, in [8] using the decomposition technique, in [9][10][11] using the variational iteration method, in [12] using a combination of the multipoint shooting method with the continuation and perturbation technique, in [13] using the quasilinearization method, in [14] using the method of transformation groups, in [15] the invariant imbedding method, in [16] using the inverse shooting method, in [17] using the modified homotopy perturbation method, in [18] using sinc-Galerkin method, in [19] using B-spline method, in [20] using the differential transform method and in [21] using chebychev collocation method. The purpose of this paper is to introduce a novel approach based on sinc function for the numerical solution of the class of nonlinear boundary value problems given in (1)- (2).…”

“…In Table 3, the numerical solution obtained by the sinc-collocation method for 5   is compared with the numerical approximation of the exact solutions given by a Fortran code called TWPBVP and the numerical solution obtained by B-spline collocation method [19].…”