Abstract
A functionally-fitted Numerov-type method is developed for the numerical solution of second-order initial-value problems with oscillatory solutions. The basis functions are considered among trigonometric and hyperbolic ones. The characteristics of the method are studied, particularly, it is shown that it has a third order of convergence for the general second-order ordinary differential equation, y''=f left( x,y,y' right) , it is a fourth order convergent method for the special second-order ordinary differential equation, y''=f left( x,yright) . Comparison with other methods in the literature, even of higher order, shows the good performance of the proposed method.
Highlights
Second-order differential equations (DEs) whether solving analytically or numerically, can be reduced into an equivalent system of firstorder equations to leverage on methods constructed for first-order systems
According to Onumanyi et al [31] and Awoyemi [4], the approach of reducing a second-order differential equation to a system of first-order differential equations is marred by a large human effort and a more demanding storage memory during implementation
We emphasize that we will introduce a method which approximates the solution of the Initial-Value Problem (IVP) in (1.1) considering that the method is exact when the solution is in the space generated by σ (x) = {1, sin(ωx), cos(ωx), sinh(ωx), cosh(ωx)}
Summary
Second-order differential equations (DEs) whether solving analytically or numerically, can be reduced into an equivalent system of firstorder equations to leverage on methods constructed for first-order systems (see Enright [7], Lambert [28], Brugnano et al [5] and Fatunla [12] for a numerical point of view). In what follows we will consider a Functionally-Fitted Block Numerov’s Method (FFBNM) for the integration of second-order Initial-Value Problem (IVP) systems of the form y = f (x, y, y ), y (x0) = y0, y (x0) = y0,. We emphasize that we will introduce a method which approximates the solution of the IVP in (1.1) considering that the method is exact when the solution is in the space generated by σ (x) = {1, sin(ωx), cos(ωx), sinh(ωx), cosh(ωx)} This functional basis is motivated by its easiness to analyze and its expectations to provide improved approximations for second-order initial-value problems with periodic or oscillatory solutions.
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