Accurate Four-Step Hybrid Block Method for Solving Higher-Order Initial Value Problems

: This paper focuses on developing a self-starting numerical approach that can be used for direct integration of higher-order initial value problems of Ordinary Differential Equations. The method is derived from power series approximation with the resulting equations discretized at the selected grid and off-grid points. The method is applied in a block-by-block approach as a numerical integrator of higher-order initial value problems. The basic properties of the block method are investigated to authenticate its performance and then implemented with some tested experiments to validate the accuracy and convergence of the method.


Introduction:
The article considers higher-order initial value problems of the form: …(1) Equation (1) applies in applied science and engineering to model problems with linear and nonlinear IVPs. Most of the resulting problems of these phenomena have no analytical solutions. Thus, the quest is to provide numerical methods as numerical integrators to the resulting analytical solutions.
A computational solution to various differential equations creates the need to develop methods to solve such equations. These approaches provide more novel behaviour to the field of science and engineering. The authors considered developing and using the mathematical functions and terms to model phenomena. In the pursuit of using these concepts such as Hilbert Space, Fredholm operators, to solve fractional, Bagley-Torvik and Painlevé, Riccati and Bernoulli, Volterra integro-differential equations within the Atangana-Baleanu fractional approach [1][2][3][4] . The developed numerical methods were a powerful tool to generate virtuous universally charming numerical solutions proficient in treating various fractional equations. This has helped in scientific forecasts through mathematical modelling and numerical simulations [1][2][3][4] Conventionally, higher-order differential equations are reduced to a system of first-order equations, which are solved by an appropriate numerical method for first-order. However, this approach has some shortcomings as it requires evaluation of too many functions and much computational effort 5 -6 . Therefore, 7 -8 developed block methods for the direct integration of higherorder to reduce the computational burden experienced in the reduction approach with better accuracy and a lower convergence rate.
The block approach is a method that generates approximations at the different grid and off-grid points within the proposed space of integration without overlying the sub-intervals to overcome the difficulties encountered in the reduction process and the predictor-corrector approach as discussed by ( 9 -16 ). Likewise, 17 developed a block method with generalized equidistant points, which allow varying the choice of the points to produce a five-step block method. The resulting family of implicit schemes considered a general block form with the incorporation of Taylor series expansion to produce solutions to fourth-order initial value problems at all grid points. The good of the block method is that the solution will be assessed at more than one point at each use of the method. The points vary on the construction of the block method, which improves the accuracy and efficiency of the method (18). In a recent publication, (19) developed a method that is capable of handling two different orders of differential equations. This was used to solve second and third-order differential equations.
Thus, our intension in this paper is to propose a four-step hybrid block method that can directly integrate three different higher-order (third, fourth and fifth-orders) ordinary differential equations to solve initial value problems with better accuracy when compared to existing methods and validated with their basic properties.

Analysis of the Properties of the Block Method
The properties of the Block Method are analyzed as follows:

Order and Error Constant
According to (20), our discrete scheme is a Linear Multistep Method (LMM) of the form: 3 4 5 0 0 0 j n j j n j j n j j n j j j o j j Thus, the block in equation (4) is of order p if 0,0,0,0,0,1 r  Hence, the block method is zero stable since the root r is simple.

Consistency
A block method is said to be consistent if it has an order 1 p  (20). Hence, the block method is consistent, since 10. p 

Convergence of the Block Method
According to (20), a method is said to be convergent, if 1 p  and zero stable. Hence, the method is convergent since it satisfies these conditions. It follows that a necessary requirement for a linear multistep method to be convergent is that the solution n y generated by the method converges to the exact solution   yxas the step length h tends to zero. Thus, the Order of Accuracy measures the rate of convergence of the numerical approximation produced from the developed method to the exact solution. This is exemplified by the error generated from the computation. In this wise, the method is said to be accurate since the error produced, which is ms respectively. Equation (10) is applied to (5) and simplified in the form x  is the region of periodicity and   iy  is the imaginary part of the complex plane. The resulting values are used to find the locus of all points, which is mapped in the MATLAB environment to produce the required stability region of the block method, as shown in Fig. 1. Definition: A numerical method is said to be Astable if its region of absolute stability contains the whole of the left-hand half-plane Re 0 h  (see (20) for details).

Figure 1. RAS of the four-point Hybrid Block Approach
From the definition, the figure above shows that the method is A-Stable since its area of periodicity lies within     , 0 .
x   

Numerical Experiments
To validate the proficiency of the developed method, three test experiments were considered. The ATFF Block method was applied to solve the following experiments.   (13) and (16).  (6) and (15).

Open Access
Baghdad Science Journal

Discussion of Results:
The numerical results displayed in Tables  (1 -3) show the results generated for an experiment (1 -3) using the proposed method to solve third, fourth, and fifth-order problems. The new method in terms of errors outperforms the errors in the methods of (13) and (16). (13) Table 1 for details). In the same vein, the proposed method generates more accurate results when compared to the methods of (9) and (11) as observed in Table 2 Table 3). Despite the performance of the proposed method to (5), as seen in table 2, the proposed method has an advantage over (5) because of its abilities to handle three different orders of differential equations which [5] is incapable of handling. Method (5) was designed to solve only fourth-order IVPs, but the proposed method is designed to solve the third, fourth, and fifth-order IVPs.

Conclusion:
The proposed method, which is a four-point hybrid block approach, was used to integrate higher-order problems with initial values. The convergence and the accuracy of the method, as reflected in the results and errors presented to show the efficiency of the block with enhanced precision when matched with some previous works. Thus, the new method can be considered as novel and a good substitute to solve Higher-Order differential equations with initial values.

Funding Statement:
The source of finance for this research has been paid by the authors themselves.