Softcode of Multi-Processing Milne’s Device for Estimating First-Order Ordinary Differential Equations

MATERIALS AND METHODS

Softcode for multi-processing Milne’s device is a collection of multi-processing predicting-correcting scheme of Adams type. This requires Adams-Bashforth-Adams-Moulton (multi-processing predicting-correcting of ilk range) scheme. This involves u-length multi-processing predicting scheme and u-1-length multi-processing correcting scheme of ilk range. This compendium is established as:

(3)

(4)

Equation 3 and 4 determines the multi-processing predicting-correcting scheme of multi-processing Milne’s device. Remarking g(t_m+i)≈g_m+i, g(x_m+i, g_m+i)≈z_m+i having j = 0, 1, 2. To attain Eq. 3 and 4, the approximative function is penned below to evaluate the analytical resolution g(t) on clear-cut time intervals of [t_n, t_n-j] by way of interpolation of the form:

(5)

Revising Eq. 5 in softcode format produces the softcode approximate function as:

(6)

where, x₀, x₁, x₂ and x₃ are parameters required to be settle in a special manner. Presuming that Eq. 6 corresponds with the precise result at approximately selected definite length of time interval t_n, t_n-j to yield approximation as:

(7)

Taking that the approximating function (Eq. 6) gratifies (Eq. 1) at more or less chosen points t_n+j, j = 0, 1, 2 to obtain the following approximates as:

(8)

Merging Eq. 7 and 8 will generate quadruplet formations which produces At = b:

(9)

(10)

Figuring out the systems of equation applying Mathematica 9 kernel, softcodes gives x_j, j = 0, 1, 2, 3 and putting back values of x_j's into Eq. 6 will generates the uninterrupted multi-processing prediction scheme and multi-processing correcting scheme of Milne’s device as:

(11)

(12)

Assessing the uninterrupted multi-processing prediction scheme and multi-processing correcting scheme of Milne’s device at some favourable grids, t_n+j, j = 1, 2, 3 will originate the multi-processing prediction Milne’s device and multi-processing correcting Milne’s device as:

(13)

where, β₁, β₂, β₃, μ₁, μ₂ and μ₃ are parametric quantity^{1,4,11,12,16,17,23-25,27} for more details.

Devising bounds of convergence for multi-processing Milne’s device: To set in motion numeric operation of multi-processing Milne’s device, the r-length multi-processing predicting scheme and r-1-length multi-processing correcting scheme are put to use as multi-processing predicting-correcting scheme owns alike range Locate^{11,12,16,17,19-22} for more. Uniting Lambert^11,12, Dormand¹⁶, Faires and Burden¹⁷, Oghonyon et al.^19-21 and Ascher and Petzold²², it is workable to find approximative chief local truncation error of multi-processing predicting-correcting scheme in absentia of higher order differential coefficients, g(t). What is more, p₁ = c₁ where, p₁ and c₁ represents range of multi-processing predicting and correcting schemes. Straightaway, scheme of range p₁, taking apart multi-processing predicting r-length gives rise to the chief principal local truncation errors:

(14)

Likewise, looking into multi-processing correcting scheme r-1-step brings forth chief local truncation errors as:

(15)

where, continues as classified quantity of length h₁ and g(t) behave as analytic resolution to higher derived function conforming to the initial stipulation g(t_n)≈g_n. Look into Lambert^11,12, Dormand¹⁶, Faires and Burden¹⁷, Oghonyon et al.^19-21 and Ascher and Petzold²² more items.

Further advancement for less precondition measures of length h₁ is reached g⁽⁴⁾ (t_n)≈g⁽⁴⁾ (t_n) and the potency of multi-processing Milne’s device trusts instantly on this presumption stated over.

Reducing in advance the chief the principal local truncation errors of Eq. 14 and 15 over besides dismissing considerations of range O(h^p1+5). Thus, introduces no concern achieving the numerical formulation of chief local truncation errors of the multi-processing Milne’s device:

(16)

Referring the avouchment that and are named predicting and correcting estimations founded thru multi-processing Milne’s device of order p₁, even though and are each separately called chief local truncation errors. τ₁, τ₂ and τ₃ are bounds of convergency of the multi-processing Milne’s device.

Advancing forward, these approximates of the chief local truncation error (Eq. 16) is utilized to make decision on acceptance or rejection thereby iterating with less or smaller varying length. The length is sustain free-based on a try out laid down by Eq. 16 ^{11,12,16,17,19-22} for more details. The chief local truncation errors (Eq. 16) is the bounds of convergence of the multi-processing Milne’s, device denoted differently as multi-processing Milne’s device for adjusting to convergence.

Numerical problems: Two problems tested are worked with MPMD. The bounds of convergency considered includes; 10^–4, 10^–6, 10^–8, 10^–10, 10^–11 and 10^–14. Find Sunday et al.¹³, Rufai et al.²⁸ and Sunday et al.²⁹ for more actions. A computer programming codes on MPMD is written utilizing Mathematica 9 kernel. The act of accomplishment is carried out in a multi-processing manner via MPMD (Appendix).

Test problem 1: Consider the nonlinear IVP, g'(t) = -10(g(t)-1)^{^}2, g(0) = 2.

Analytical result:

Test problem 2: Consider Prothero-Robinson periodic vibration ODE, g'(t) = L(g(t)-sin(t))+cos (t), L = -1, g(0) = 0.

Analytical result: g(t) = sin (t).

RESULTS AND DISCUSSION

Under this section, the computational output shows the execution of MPMD for solving first-order ODEs. The final output supplied were obtained with the aid of Mathematica 9 Kernel 64 on Microsoft windows (64 bit) to demonstrate the efficiency and accuracy of the first-order ODEs^13,28,29.

Table 1 demonstrates the numerical results of problems 1 and 2 using MPMD equated with existing methods.

Table 1 presents a summary of the result displayed and items considered. This includes; method utilized, computed max errors and bounds of convergency. Again, shows the comparison with other existing results and justifies MPMD as a preferable proficiency in terms of the computed max errors:

Table 1:	Summary of results

Softcodes algorithm rule: A well written algorithmic rule that will execute MMD and assess the computed max errors of MPMD in the family of P(EC)^j or P(EC)^j E style, conditionally, when the style is implemented as many times to ensure convergence. Check out²³:

CONCLUSION

The computed results displayed MPMD is reached utilizing the bounds of convergency. This bounds of convergency examine the acceptance or rejection of the looping with a smaller length. The mathematical outputs establish the performance of MPMD is remarked to showcase a more acceptable computed max error at all bounds of convergency. This is made possible by seeking a suitable/changing length, determining the bounds of convergency as compare to subsisting schemes implemented without these features. This proficiency for a better result is executed at all examined bounds of convergency such as 10^–4, 10^–6, 10^–8, 10^–10, 10^–11 and 10^–14. Thence, it will be concluded that MPMD is worthy for estimating ODEs. Furthermore, MPMD is better and preferred to schemes such as block predictor-corrector methods, block implicit method, block hybrid method because their applications are based on fixed step size, no bounds of convergency and always implemented in predictor-corrector method. Continuous research can be carried out to increase the order of MPMD for examining performance.

SIGNIFICANCE STATEMENT

The significant of this study is as follows:

ACKNOWLEDGMENT

The authors would like to appreciate Covenant University for financing this research work.

Appendix:	The softcodes of problem 1 and 2 implemented via MPMD is given below