Abstract
Abstract In this article, a fast algorithm is developed for the numerical solution of twelfth-order boundary value problems (BVPs). The Haar technique is applied to both linear and nonlinear BVPs. In Haar technique, the twelfth-order derivative in BVP is approximated using Haar functions, and the process of integration is used to obtain the expression of lower-order derivatives and approximate solution for the unknown function. Three linear and two nonlinear examples are taken from literature for checking the convergence of the proposed technique. A comparison of the results obtained by the present technique with results obtained by other techniques reveals that the present method is more effective and efficient. The maximum absolute and root mean square errors are compared with the exact solution at different collocation and Gauss points. The convergence rate using different numbers of collocation points is also calculated, which is approximately equal to 2.
Highlights
Boundary value problems (BVPs) with higher order arises in the field of astrophysics, hydrodynamics and hydromagnetic stability, fluid dynamics, astronomy, beam and long wave theory, and applied physics and engineering
We developed a collocation method based on Haar wavelet for the numerical simulation of twelfth-order BVPs
The maximum absolute errors for distant number of discrete collocation point (CP) and Gauss points (GPs) are shown for each example in tables
Summary
Boundary value problems (BVPs) with higher order arises in the field of astrophysics, hydrodynamics and hydromagnetic stability, fluid dynamics, astronomy, beam and long wave theory, and applied physics and engineering. A tenth degree spline was used by Twizell et al for the numerical solution of tenth-order BVPs and faced some problems in getting results near boundaries of the interval in ref. We developed a collocation method based on Haar wavelet for the numerical simulation of twelfth-order BVPs. The following nonlinear problem of twelfth order will be considered in this article: u(12) (t) = F(t, u, u(1), u(2), u(3), u(4), u(5), u(6), u(7), (1) u(8), u(9), u(10), u(11) ) 0 ≤ t ≤ 1, where F is given function, whereas in case of linear, the following general form is considered: u(12) (t) + f (t)u(t) = g(t), t ∈ [0, 1],.
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