Solution of eighth-order boundary value problems using the non-polynomial spline technique

Abstract

The non-polynomial spline technique is used for the numerical solution of eighth-order linear special case boundary value problems. The method presented in this paper has also been proven to be secondorder convergent. To compare the method developed in this paper with those developed by Inc and Evans, and Siddiqi and Twizell, two examples are considered and it is observed that our method is more efficient.

Adomian Decomposition Method with Green’s Function for Solving Tenth-Order Boundary Value Problems

“…Comparing them with the DTM [9], EDSM [10], NPSM [11] and VIT [12] results, it can be noticed that the result obtained by the present method (SAwGF) is very superior to that obtained by the other mentioned methods. Table 4 exhibits the maximum residual error of the SAwGF and MTwGF for n φ .…”

“…A few of them are: Tenth degree spline method [7], Modified Decomposition Method with the inverse operator (MDM) [8], Differential Transform Method (DTM) [9], Eleventh Degree Spline Method (EDSM) [10], Non-Polynomial Spline Method (NPSM) [11], Variational Iteration Technique (VIT) [12] and Homotopy Perturbation Method (HPM) [13].…”

An improved adaptation of homotopy analysis method

“…Inc and Evans [22] used Adomian decomposition method to approximate solutions of eighth order boundary value problems. Siddiqi and Akram [23][24][25][26][27] presented the solutions of 5th, 6th, 8th, 10th, and 12th order boundary value problems using nonpolynomial spline techniques. Hassan and Erturk [28] applied differential transformation method to obtain the solution of some linear and nonlinear higher order boundary value problems.…”

Numerical solution for solving special eighth-order linear boundary value problems using Legendre Galerkin method

“…A selective review for getting the numerical solution of the eighth-order boundary value problems is presented here. Boutayeb and Twizell [18] used finite difference methods, Akram and Siddiqi [19,20] used nonic and non-polynomial spline functions, respectively, Akram and Rehman [21] developed reproducing kernel space, Viswanadham and Ballem [22] used Galerkin method with quintic B-spline, Inc and Evans [23] constructed Adomian decomposition method, Wazwaz [24] developed modified Adomian decomposition method, Siddiqi and Iftikhar [25] used homotopy analysis method, and Ballem and Viswanadham [26] presented the Galerkin method with septic B-splines, whereas Abbasbandy and Shirzadi [27] developed variational iteration method.…”