An Algorithm for Solving Boundary Value Problems

One-point pseudospectral collocation for the one-dimensional Bratu equation

“…Other methods like Adomian decomposition [27,26,12] implicitly use Taylor expansions and are subject to the same difficulty.…”

“…Bratu's own article appeared in 1914 [9]; generalizations are sometimes called the ''Liouville-Gelfand'' or ''Liouville-Gelfand-Bratu'' problem in honor of Gelfand [15] and the nineteenth century work of the great French mathematician Liouville. In recent years, it has been a popular testbed for numerical and perturbation methods [1,17,16,27,21,26,20,12,22,12]. In this note, we apply very low order spectral methods.…”

A novel method for nonlinear two-point boundary value problems: Combination of ADM and RKM

“…. , n. The numerical results compared with variational iteration method (VIM) [25], ADM [26] and modified homotopy perturbed method (MHPM) [27] are shown in Table 1. From Table 1, it is shown that the present method is more effective than methods in [25][26][27], but it is also difficult to obtain accurate approximations.…”

“…To exhibit the applicability of this method to more difficult problems, consider the following nonlinear twopoint BVP, Troeschs problem[23][24][25][26][27]: Absolute errors uðxÞ À U 6 4 ðxÞ ; uðxÞ À U 11 4 ðxÞ for Example 3.1.…”

Numerical solution of Troesch’s problem by simple shooting method

“…This results in the problem being very difficult to solve by the shooting method and this difficulty increases as n increases. Although, several iterative approximate methods such as Adomian decomposition method [6,7], variational iteration method [8], and modified homotopy perturbation method [9] fail to solve this problem for n > 1, other iterative or numerical methods such as differential transform method [10], multipoint shooting method combined with continuation and perturbation technique [11], invariant imbedding method [12], inverse shooting method [13] and simple shooting method combined with modified Newton's method, overflow trap or parameter mapping technique [14][15][16] have been successfully applied to this problem for n > 5 and yielded results varying in accuracy.…”