Quartic non-polynomial spline approach to the solution of a system of third-order boundary-value problems

Abstract

Quartic non-polynomial splines are used to develop a new numerical method for computing approximations to the solution of a system of third-order boundary-value problems associated with obstacle, unilateral, and contact problems. It is shown that the new method gives approximations, which are better than those produced by other collocation, finite-difference, and spline methods. Convergence analysis of the method is discussed through standard procedures. A numerical example is given to illustrate the applicability and efficiency of the novel method.

Mid-knot cubic non-polynomial spline for a system of second-order boundary value problems

“…The method is shown to be of order two, and numerical results indicate better accuracy over polynomial spline methods. Higher degree non-polynomial splines have also been used in higher-order boundary value problems, for example quartic non-polynomial spline for third-order boundary value problem [24,26], quintic non-polynomial spline for fourth-order boundary value problem [14] and sextic non-polynomial spline for fifthorder boundary value problem [15]. Out of all these work, only [26] gives the numerical solutions of the third-order boundary value problem at mid-knots of a uniform mesh while the rest obtains the numerical solutions at the knots.…”

Numerical treatment of certain fractional and non-fractional differential equations

“…The method is shown to be of order two, and numerical results indicate better accuracy over polynomial spline methods. Higher degree non-polynomial splines have also been used in higher order boundary value problems, for example quartic non-polynomial spline for third-order boundary value problem [115,118], quintic non-polynomial spline for fourth-order boundary value problem [67] and sextic non-polynomial spline for fifth-order boundary value problem [68]. Out of all these work, only [118] gives the numerical solutions of the third-order boundary value problem at mid-knots of a uniform mesh while the rest obtains the numerical solutions at the knots.…”