Abstract
Cubic Hermite collocation method is proposed to solve two point linear and nonlinear boundary value problems subject to Dirichlet, Neumann, and Robin conditions. Using several examples, it is shown that the scheme achieves the order of convergence as four, which is superior to various well known methods like finite difference method, finite volume method, orthogonal collocation method, and polynomial and nonpolynomial splines and B-spline method. Numerical results for both linear and nonlinear cases are presented to demonstrate the effectiveness of the scheme.
Highlights
Since time immemorial mathematicians are ardently pursuing the solution of linear or nonlinear two point boundary value problems (BVPs) of the type: d2y dx2 + α1 (x) dy dx α2 y = f (x), x ∈ [a, b] (1)subject to Dirichlet, Neumann, and Robin’s boundary conditions.Such BVPs have wide application in astronomy, biology, boundary layer theory, deflection in cables, diffusion process, electromagnetism, heat transfer, and other topics
It is well known that closed form analytical solution of such problems cannot be obtained in many cases; numerical techniques such as collocation method [1, 2], Bspline interpolation [3], Hermite cubic collocation [4,5,6], finite difference method [7,8,9], nonlinear shooting method [10], geometric Hermite interpolation [11], quintic B-spline collocation method [12], polynomial and nonpolynomial spline approaches [13,14,15], quartic spline solution [16], cubic spline collocation method [17], and finite volume element method [18] are frequently used
The results reported by [9] for 10 elements using finite difference method are matching with exact solution up to 3 decimal places, whereas using cubic Hermite collocation method (CHCM) the results are matching up to 9 decimal places
Summary
Since time immemorial mathematicians are ardently pursuing the solution of linear or nonlinear two point boundary value problems (BVPs) of the type: d2y dx. It is well known that closed form analytical solution of such problems cannot be obtained in many cases; numerical techniques such as collocation method [1, 2], Bspline interpolation [3], Hermite cubic collocation [4,5,6], finite difference method [7,8,9], nonlinear shooting method [10], geometric Hermite interpolation [11], quintic B-spline collocation method [12], polynomial and nonpolynomial spline approaches [13,14,15], quartic spline solution [16], cubic spline collocation method [17], and finite volume element method [18] are frequently used. Different linear and nonlinear differential equations are solved subject to Dirichlet, Neumann, and Robin boundary conditions using the present method.
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