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A one-step 3-D method of the form:

Figure 1 A one-step 3-D method of the form:

Related Figures (15)

The zero stability of the method concerns the stability of the difference system (15) as h > 0. Thus, as (15) tends to The coefficients of the methods in (15) and (16) are orovided as follows: The characteristics polynomial Q(y)) is given by
The zero stability of the method concerns the stability of the difference system (15) as h > 0. Thus, as (15) tends to The coefficients of the methods in (15) and (16) are orovided as follows: The characteristics polynomial Q(y)) is given by
Substituting (21) and (22) into (15) and (16) respectively, we obtain
Substituting (21) and (22) into (15) and (16) respectively, we obtain
TABLE I: MAXERR (maz;|y(x:) — y:)|) for Example 1 Table I displays the maximum error and computation time at different grid points (2) on the closed interval [0,1], ob- tained using the OSMDHBM and compared with the seventh- order linear multi-step method (SOLMM) [6] and the block single-step hybrid algorithms based on Bhaskara approxima- where 7 AF is the maximum absolute error and h is the step size.
TABLE I: MAXERR (maz;|y(x:) — y:)|) for Example 1 Table I displays the maximum error and computation time at different grid points (2) on the closed interval [0,1], ob- tained using the OSMDHBM and compared with the seventh- order linear multi-step method (SOLMM) [6] and the block single-step hybrid algorithms based on Bhaskara approxima- where 7 AF is the maximum absolute error and h is the step size.
TABLE II: Maximum relative error, rate of convergence and number of function evaluations for OSMDHBM in Example
TABLE II: Maximum relative error, rate of convergence and number of function evaluations for OSMDHBM in Example
tion points (BSSHA-BAP) [16] with five intra-step points. It observed that the OSMDHBM achieves superior convergence with a few intra-step points (three) used. However, as the number of partitions R increases, the computation time for OSMDHBM convert to four decimal places also increases. In Table II, the maximum relative error, rate of convergence and number of function evaluations of the OSMDHBM are presented at different grid points (A) on the closed interval [0,1].
tion points (BSSHA-BAP) [16] with five intra-step points. It observed that the OSMDHBM achieves superior convergence with a few intra-step points (three) used. However, as the number of partitions R increases, the computation time for OSMDHBM convert to four decimal places also increases. In Table II, the maximum relative error, rate of convergence and number of function evaluations of the OSMDHBM are presented at different grid points (A) on the closed interval [0,1].
TABLE IV: Maximum relative error, rate of convergence and number of function evaluations for OSMDHBM in Example
TABLE IV: Maximum relative error, rate of convergence and number of function evaluations for OSMDHBM in Example
Fig. 4: Numerical vs Exact solution for y; and yz when w = 3 ath= a TABLE III: Absolute error for Example 2 when 7 = 3 at ha-=-
Fig. 4: Numerical vs Exact solution for y; and yz when w = 3 ath= a TABLE III: Absolute error for Example 2 when 7 = 3 at ha-=-
Table II presents the absolute error at selected points obtained using the OSMDHBM with h = 0.01 and twenty iterations on the closed interval [0,1]. The results were compared with the block two-step hybrid methods based on Bhaskara points (BTSHM-BP) [7] and OTSBHM [7 when ~~ = 5. The OSMDHBM, which uses the fewest intra-step points (three), is superior to BTSHM-BP and OTSBHM. Table IV illustrates the maximum relative error rate of convergence and number of function evaluations o OSMDHBM for different step sizes on the closed interva >
Table II presents the absolute error at selected points obtained using the OSMDHBM with h = 0.01 and twenty iterations on the closed interval [0,1]. The results were compared with the block two-step hybrid methods based on Bhaskara points (BTSHM-BP) [7] and OTSBHM [7 when ~~ = 5. The OSMDHBM, which uses the fewest intra-step points (three), is superior to BTSHM-BP and OTSBHM. Table IV illustrates the maximum relative error rate of convergence and number of function evaluations o OSMDHBM for different step sizes on the closed interva >
Table V shows the computation time and absolute errors obtained using the step size of h = 0.1 at selected points within the closed interval [0,1]. The results, as shown in Table V, were compared with the adomian decomposition method (ADM) [17]. These results indicate that the OS- MDHBM performs better than ADM in terms of accuracy, yielding smaller errors. The maximum relative error, rate of convergence and number of function evaluations of OSMD- HBM for different step sizes within the closed interval [0,1] are displayed in Table VI.
Table V shows the computation time and absolute errors obtained using the step size of h = 0.1 at selected points within the closed interval [0,1]. The results, as shown in Table V, were compared with the adomian decomposition method (ADM) [17]. These results indicate that the OS- MDHBM performs better than ADM in terms of accuracy, yielding smaller errors. The maximum relative error, rate of convergence and number of function evaluations of OSMD- HBM for different step sizes within the closed interval [0,1] are displayed in Table VI.
TABLE V: Absolute error for Example 3 when 7 = 3 at _ i h=
TABLE V: Absolute error for Example 3 when 7 = 3 at _ i h=
TABLE VII: MAXERR (maz;|y(«;) — y;)|) for Example 4
TABLE VII: MAXERR (maz;|y(«;) — y;)|) for Example 4
In this study, a OSMDHBM of order ten has been successfully developed. The method incorporates 3-Ds to directly solve both 2-order L-IVPs and N-IVPs. The N- IVPs are linearized using a MPIM. The analysis confirms that the OSMDHBM exhibits zero stability, consistency, and convergence. To verify the accuracy and efficiency of the method, it has been applied to solve some standard IVPs. The outcomes of these solutions were then compared with those obtained using existing methods (see Table I-VII and Figure 2-6). The outcomes strongly affirm the method’s efficiency and accuracy.
In this study, a OSMDHBM of order ten has been successfully developed. The method incorporates 3-Ds to directly solve both 2-order L-IVPs and N-IVPs. The N- IVPs are linearized using a MPIM. The analysis confirms that the OSMDHBM exhibits zero stability, consistency, and convergence. To verify the accuracy and efficiency of the method, it has been applied to solve some standard IVPs. The outcomes of these solutions were then compared with those obtained using existing methods (see Table I-VII and Figure 2-6). The outcomes strongly affirm the method’s efficiency and accuracy.
Fig. 6: Convergence graph for y; and yo when wy = 3 at R = 219 Fig. 5: Numerical vs Exact solution for y; and y2 when yw = 3 at R = 219
Fig. 6: Convergence graph for y; and yo when wy = 3 at R = 219 Fig. 5: Numerical vs Exact solution for y; and y2 when yw = 3 at R = 219
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