Abstract
In the present paper, we introduce a non-polynomial quadratic spline method for solving third-order boundary value problems. Third-order singularly perturbed boundary value problems occur frequently in many areas of applied sciences such as solid mechanics, quantum mechanics, chemical reactor theory, Newtonian fluid mechanics, optimal control, convection-diffusion processes, hydrodynamics, aerodynamics, etc. These problems have various important applications in fluid dynamics. The procedure involves a reduction of a third-order partial differential equation to a first-order ordinary differential equation. Truncation errors are given. The unconditional stability of the method is analysed by the Von-Neumann stability analysis. The developed method is tested with an illustrated example, and the results are compared with other methods from the literature, which shows the applicability and feasibility of the presented method. Furthermore, a graphical comparison between analytical and approximate solutions is also shown for the illustrated example.
Highlights
We introduce a non-polynomial quadratic spline method for solving third-order boundary value problems
Some approaches for solving nonlinear partial differential equations have been addressed in recent literature; the most prominent of these were the non-polynomial spline methods
The non-polynomial spline used for solving nonlinear partial differential equations was employed by many researchers
Summary
Some approaches for solving nonlinear partial differential equations have been addressed in recent literature; the most prominent of these were the non-polynomial spline methods. The non-polynomial spline used for solving nonlinear partial differential equations was employed by many researchers. In [4] [5], the criteria for deriving stability conditions of the different methods were considered for the numerical solution of a third-order linear dispersive equation. Research by Tirmizi et al (2008) used Quartic non-polynomial spline functions to develop a class of numerical methods for solving self-adjoint singularly perturbed problems [6]. In 2018, Sultana et al presented a new three-level implicit method, which was developed to solve linear and nonlinear third-order dispersive partial differential equations [12]. A novel approach, based on using non-polynomial splines to solve a third-order dispersive partial differential equation is proposed.
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