A New L-Stable Third Derivative Hybrid Method for Solving First Order Ordinary Differential Equations

In this paper, an L-stable third derivative multistep method has been proposed for the solution of stiff systems of ordinary differential equations. The continuous hybrid method is derived using interpolation and collocation techniques of power series as the basis function for the approximate solution. The method consists of the main method and an additional method which are combined to form a block matrix and implemented simultaneously. The stability and convergence properties of the block were investigated and discussed. Numerical examples to show the efficiency and accuracy of the new method were presented.

A Family of Functionally-Fitted Third Derivative Block Falkner Methods for Solving Second-Order Initial-Value Problems with Oscillating Solutions

One of the well-known schemes for the direct numerical integration of second-order initial-value problems is due to Falkner (Falkner, 1936. Phil. Mag. S. 7, 621). This paper focuses on the construction of a family of adapted block Falkner methods which are frequency dependent for the direct numerical solution of second-order initial value problems with oscillatory solutions. The techniques of collocation and interpolation are adopted here to derive the new methods. The study of the properties of the proposed adapted block Falkner methods reveals that they are consistent and zero-stable, and thus, convergent. Furthermore, the stability analysis and the algebraic order conditions of the proposed methods are established. As may be seen from the numerical results, the resulting family is efficient and competitive compared to some recent methods in the literature.

On the best interval quadrature formula for the class of functions $W^3 H^{\omega}$

The optimal interval quadrature formula in the class of functions with convex majorant of the modulus of continuity of third derivative is obtained.

Acceleration and Jerk After a Jump Stabilization Task in Individuals With and Without Chronic Ankle Instability

Studies have demonstrated that individuals with chronic ankle instability (CAI) have diminished dynamic stability. Jerk-based measures have been utilized to examine dynamic balance because of their ability to quantify changes in acceleration and may provide an understanding of the postural corrections that occur during stabilizing following a jumping task. The purpose of this study was to compare acceleration and jerk following a jump stabilization task between individuals with CAI and the uninjured controls. Thirty-nine participants volunteered to participate in this case control study. Participants completed a jump stabilization task requiring them to jump off 2 feet, touch a marker set at 50% of their maximal vertical jump height, land on a single limb, and maintain balance for 3 seconds. Acceleration was calculated as the second derivative, and jerk was calculated as the third derivative of the displacement of the resultant vector position. Participants with CAI had greater acceleration (mean difference = 55.6 cm/s2; 95% confidence interval, 10.3 to 100.90; P = .017) and jerk compared with the uninjured controls (mean difference = 1804.5 cm/s3; 95% confidence interval, 98.7 to 3510.3; P = .039). These results suggest that individuals with CAI made faster and more frequent active postural control corrections to regain balance following a jump compared with the uninjured controls.