Two-Step Hybrid Block Method for Solving Nonlinear Jerk Equations

In this paper, a block method with one hybrid point for solving Jerk equations is presented. The hybrid point is chosen to optimize the local truncation errors of the main formulas for the solution and the derivative at the end of the block. Analysis of the method is discussed, and some numerical examples show that the proposed method is efficient and accurate.


Introduction
Jerk is the rate of acceleration change in physics; that is, the time derivative of acceleration, and as such the second velocity derivative, or the third time position derivative. The jerk is important in several mechanics and acoustics applications. The Jerk vector is here resolved into tangential-normal and radial-transverse components for planar motion, and the normal component is expressed as an affine differential invariant recognized as the aberrancy. Several geometric properties of the Jerk vector are established for plane motion using known aberrancy properties of curves [1].
Nonlinear third-order differential equations, known as nonlinear Jerk equations, involving the third temporal displacement derivative, are of great interest in analyzing some structures which exhibit rotating and translating movements, such as robots or machine tools, where excessive Jerk leads to accelerated wear of transmissions and bearing elements, noisy operations and large contouring errors in discontinuities (such as corners) in the machining path [2].
Many authors have studied the numerical solutions of the Jerk equation, harmonic balance approach to periodic solutions is used in [3], in [4] they have written the high-order ordinary differential equation in terms of its differential invariants. New algorithm for the numerical solutions of nonlinear third-order differential equations was used jacobi-gauss collocation method in [5], He's variational iteration method was used in [6] for nonlinear Jerk equations. Modified harmonic balance method was used for nonlinear Jerk equations in [7]. In this paper, we consider a Jerk equation of the form where the parameters are , , , α β δ  and γ is constants. The current work is motivated by optimizing local truncation errors in order to find a hybrid point in a two-step block method to have the most accurate solution for Jerk Equation (1). We organize this paper as follows: The next section illustrates the method derivation, Section 3 presents the analysis of the method involving order four, Section 4 presents the numerical examples showing the productivity of the new technique when it is contrasted with the different strategies proposed in the scientific writing.

Characteristics of the Method
This section is presented the basic properties of the main method and analyzed it to establish their validity.

Order of the Method
We can rewrite the hybrid block method in the form where , , , If ( ) z x is a sufficiently differentiable function, the linear difference operator  associated with the implicit two-step block hybrid method is considered in Equation (8), Equation (9), Equation (10), Equation (13), that is given  Note that the proposed method has order 4 p = at least [9].

Zero Stability
We can write the method as a vector As 0 h → in Equation (14), we can write the method as a vector form. Hence the block method is zero -stable [9].

Consistency
The block method has order 4 p = , in case 1 p ≥ , this is a sufficient condition to be consistent with the associated block method [10].

Convergence
We can establish the convergence of the two-step with three points hybrid block method if and only if it is consistent and zero stable [11].

Region of Absolute Stability
As we mentioned earlier, zero-stability is a concept of the numerical method behavior for 0 h → . To decide whether a numerical method will produce good results with a given value of 0 h > , we need a concept of stability that is different from zero-stability. In most numerical methods intended to solve problems of third order, the stability properties are usually analyzed by considering the linear equation given by the Dalquist test [11]. λ ≥ that tend to zero for x → ∞ .
We will define the region where the numerical method reproduces the manner of the exact solutions. Let us explain the procedure for obtaining such a region. Our method has nine equations in which there are four different terms of first derivatives:  [12], We eliminated these terms from the equations system by using mathematica, and get a recurrence equation in the terms The roots of the characteristic equation must be less than 1, for the method to be stable. The stability region for the method has shown in Figure 1   respectively. Example 1 was solved in [13], the comparison be-tween our method and result in [13] is proposed in Table 4 and Table 5. It is seen in Figures 2-4         respectively. Example 2 was solved in [13], the comparison between our method and result in [13] is proposed in Table 9 and Table 10. It is seen in Figures 5-7

Conclusion
A two-step with one hybrid point was proposed and proceeded as a self-starting method for solving nonlinear Jerk equations. We considered one hybrid point and specified for approximation after optimizing local truncation errors related to the main formula. Therefore, our method's good convergent and stability properties make it attractive for the numerical solution of nonlinear problems. The method presented is zero stable, consistent and convergence of four-algebraic order. The numerical results and figures show their efficiency and precision compared to other methods in the literature.