J. Nig. Soc. Phys. Sci. 4 (2022) 706 Journal of the Nigerian Society of Physical Sciences A Higher-order Block Method for Numerical Approximation of Third-order Boundary Value Problems in ODEs Adefunke Bosede Familuaa, Ezekiel Olaoluwa Omoleb,c,∗, Luke Azeta Ukpebord aDepartment of Mathematics and Statistics, First Technical University, Ibadan, Oyo State, Nigeria bDepartment of Mathematics and Statistics, Joseph Ayo Babalola University, Ikeji Arakeji, Osun State, Nigeria cDepartment of Mathematics, Federal University Oye-Ekiti, Ekiti State, Nigeria dDepartment of Mathematics, Ambrose Alli University Edo State, Nigeria Abstract In recent times, numerical approximation of 3rd-order boundary value problems (BVPs) has attracted great attention due to its wide applications in solving problems arising from sciences and engineering. Hence, A higher-order block method is constructed for the direct solution of 3rd-order linear and non-linear BVPs. The approach of interpolation and collocation is adopted in the derivation. Power series approximate solution is interpolated at the points required to suitably handle both linear and non-linear third-order BVPs while the collocation was done at all the multi- derivative points. The three sets of discrete schemes together with their first, and second derivatives formed the required higher-order block method (HBM) which is applied to standard third-order BVPs. The HBM is self-starting since it doesn’t need any separate predictor or starting values. The investigation of the convergence analysis of the HBM is completely examined and discussed. The improving tactics are fully considered and discussed which resulted in better performance of the HBM. Three numerical examples were presented to show the performance and the strength of the HBM over other numerical methods. The comparison of the HBM errors and other existing work in the literature was also shown in curves. DOI:10.46481/jnsps.2022.706 Keywords: Convergence analysis, Linear & nonlinear problems, Ordinary differential equations, Power series basic function, Third-order boundary value problems Article History : Received: 13 March 2022 Received in revised form: 28 May 2022 Accepted for publication: 30 May 2022 Published: 25 August 2022 c© 2022 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license (https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Communicated by: J. Ndam 1. Introduction Numerous common happenings in connection with phys- ical sciences, and engineering are modeled in form of linear and nonlinear BVPs. Although, some modeled problems do not have theoretical solution or closed form. Consequently, numer- ∗Corresponding author tel. no: +2348065086963 Email address: eoomole@jabu.edu.ng, omolez247@gmail.com (Ezekiel Olaoluwa Omole) ical method is employed to solve such class of modeled prob- lems. In this article, the numerical solution of third-order BVPs of the type y′′′(x) = v(x, y, y′, y′′), (1) with the initial conditions, boundary conditions or any other form as follow, y(a) = αa, y ′(a) = βa, y ′(xb) = νb, (2) y(xa) = α0, y ′(a) = βa, y(b) = νb, (3) 1 Familua et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 706 2 y(xa) = αa, y ′(b) = βb, y(b) = νb, (4) The constants parameters in equations (2) - (4) are taken to be continuous functions and v fulfils the condition for the existence and uniqueness of the problem. It follows that (1) is a third-order ordinary differential equations with initial, and boundary conditions (3) - (4). Modeled equation (1) is of great importance to scientists and engineers due to its numerous usage in sciences and engineering. Scholars have developed numerous techniques for solving (1). The conventional methods of solving (1) could be; by reducing it to the system of first-order ordinary differential equations and a suitable numerical methods for first-order ODEs would be applied to solve the system of equations. The shooting method, and finite different method. The limitations of these methods have been discussed by numerous scholars [1–4]. For instance, The reduction approach have been reported by various literatures to have alot of limitations such computational burden, lots of human efforts, requires lots of time for the computation, and complexity in the computer computation which affects the accuracy and efficiency of the method in terms of error and time of execution. On the other hand, the shooting method suffers inaccuracy and instability while the finite different method is very demanding and does not give a satisfactory results. Other numerical methods for solving ordinary differential equations also exist in literature [5–15]. In other to improve the limitations and the weakness associated with the conventional methods, we present a higher-order block method capable of solving (1) directly, accurately and reduces computational time. The new method namely HBM is expected to improve the accuracies of the existing methods in the literature. 2. Methodology In this section, a new method capable of handing (1) is constructed. We considered an approximate equation as the power series of the form, y(x) = 3k+1∑ z=0 mz x z (5) where the step-number (k = 3) is taken into consideration, 3k+1 is equivalent to r + 2s − 1, where r is the interpolation points and s is the collocation points. The third and fourth derivative of (5) yields y′′′(x) = 3k+1∑ z=3 z(z − 1)(z − 2)mz x z−3 y(iv)(x) = 3k+1∑ z=4 z(z − 1)(z − 2)(z − 3)mz x z−4 (6) Now, interpolating (5) at xn+r, r = 0(1)k − 1 and collocating both the third and fourth derivatives in (6) at xn+s, s = 0(1)k to generate the system of ten by ten equations written in matrix form as X1 A1 = B1 (7) X1 =  1 xn x2n x 3 n x 4 n x 5 n · · · x 9 n x 10 n 0 1 2xn 3x2n 4x 3 n 5x 4 n · · · 9x 8 n 10x 9 n 0 0 2 6xn 12x2n 20x 3 n · · · 72x 7 n 90x 8 n 0 0 0 6 24xn 60x2n · · · 504x 6 n 720x 7 n 0 0 0 6 24xn+1 60x2n+1 · · · 504x 6 n+1 720x 7 n+1 0 0 0 6 24xn+2 60x2n+2 · · · 504x 6 n+2 720x 7 n+2 0 0 0 6 24xn+3 60x2n+3 · · · 504x 6 n+3 720x 7 n+3 0 0 0 0 24 120xn · · · 3024x5n 5040x 6 n 0 0 0 0 24 120xn+1 · · · 3024x5n+1 5040x 6 n+1 0 0 0 0 24 120xn+2 · · · 3024x5n+2 5040x 6 n+2 0 0 0 0 24 120xn+3 · · · 3024x5n+3 5040x 6 n+3  B1 = (yn, yn+1, yn+2, vn, vn+1, vn+2, vn+3, wn, wn+1, wn+2, wn+3) T A1 = (m0, m1, m2, m3, m4, m5, m6, m7, m8, m9, m10) T Solving the system of equations above, we obtained the values of the coefficients mn, n = 0, ..., 10 as follows. After obtaining the values of these coefficients and changing the variable, x = xn + lh using the appropriate transformation, the polynomial in (5) may be written as, y (l) = a0yn + a1yn+1 + a2yn+2 + h 3 (b0vn + b1vn+1 + b2vn+2 + b3vn+3) +h4(c0wn + c1wn+1 + c2wn+2 + c3wn+3) (8) 2 Familua et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 706 3 where a0 = de 2 , a1 = −el, a2 = dl 2 , d = (l − 1), e = (l − 2) b0 = h2 544320 de(77 l7 − 1059 l6 + 5699 l5 − 14625 l4 + 15749 l3 + 3165 l2 − 22003 l + 18381) b1 = lh3 20160 de(7 l7 − 79 l6 + 304 l5 − 370 l4 − 206 l3 + 122 l2 + 778 l + 2090) b2 = − lh3 20160 de(7 l7 − 89 l6 + 409 l5 − 755 l4 + 319 l3 + 199 l2 − 41 l − 521) b3 = − lh3 544320 de(77 l7 − 789 l6 + 2864 l5 − 4230 l4 + 1574 l3 + 1086 l2 + 110 l − 1842) c0 = lh4 181440 de(7 l7 − 99 l6 + 559 l5 − 1581 l4 + 2245 l3 − 1191 l2 − 503 l + 873) c1 = lh4 20160 de(7 l7 − 89 l6 + 424 l5 − 878 l4 + 550 l3 + 382 l2 + 46 l − 626) c2 = lh4 20160 de(7 l7 − 79 l6 + 319 l5 − 517 l4 + 205 l3 + 137 l2 + l − 271) c3 = lh4 181440 de(7 l7 − 69 l6 + 244 l5 − 354 l4 + 130 l3 + 90 l2 + 10 l − 150) (9) Remark 1. The variable coefficients functions in (8) are continuous and differentiable within the interval of solution of [a, b] with a step size given by the function h = b−aN . It follows that N is the number of sub-interval of the solution. The continuous function (8) and its first y′(x) and second derivatives y′′(x) were used to generate the main and auxiliary methods which produces a sum of nine equations jointed together to supply the entire approximations on the interval for the direct solution of third-order BVPs of the type (1). Furthermore, by evaluating (8), its first and second derivatives at xn+l, l = 0(1)k, The following nine equations or otherwise called HBM were acquired. yn+1 = yn + y ′ nh + 1 2 y′′n h 2 + 62387 544320 h3vn + 89 3360 h3vn+1 + 439 20160 h3vn+2 + 1031 272160 h3vn+3 + 1879 181440 h4wn − 359 10080 h4wn+1 − 13 960 h4wn+2 − 17 18144 h4wn+3 yn+2 = yn + 2y ′ nh + 2 y ′′ n h 2 + 5048 8505 h3vn + 164 315 h3vn+1 + 4 21 h3vn+2 + 244 8505 h3vn+3 + 172 2835 h4wn − 4 15 h4wn+1 − 34 315 h4wn+2 − 4 567 h4wn+3 (10) yn+3 = yn + 3y ′ nh + 9 2 y′′n h 2 + 657 448 h3vn + 2187 1120 h3vn+1 + 2187 2240 h3vn+2 + 117 1120 h3vn+3 + 351 2240 h4wn − 729 1120 h4wn+1 − 729 2240 h4wn+2 − 27 1120 h4wn+3 y′n+1 = y ′ n + h y ′′ n + 19519 68040 h2vn + 1301 10080 h2vn+1 + 181 2520 h2vn+2 + 3329 272160 h2vn+3 + 371 12960 h3wn − 313 2520 h3wn+1 − 89 2016 h3wn+2 − 137 45360 h3wn+3 y′n+2 = y ′ n + 2h y ′′ n + 5731 8505 h2vn + 296 315 h2vn+1 + 109 315 h2vn+2 + 344 8505 h2vn+3 + 206 2835 h3wn − 20 63 h3wn+1 − 52 315 h3wn+2 − 4 405 h3wn+3 (11) 3 Familua et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 706 4 y′n+3 = y ′ n + 3h y ′′ n + 603 560 h2vn + 2187 1120 h2vn+1 + 729 560 h2vn+2 + 27 160 h2vn+3 + 27 224 h3wn − 243 560 h3wn+1 − 243 1120 h3wn+2 − 9 280 h3wn+3 y′′n+1 = y ′′ n + 6893 18144 hvn + 313 672 hvn+1 + 89 672 hvn+2 + 397 18144 hvn+3 + 1283 30240 h2wn − 851 3360 h2wn+1 − 269 3360 h2wn+2 − 163 30240 h2wn+3 y′′n+2 = y ′′ n + 223 567 hvn + 20 21 hvn+1 + 13 21 hvn+2 + 20 567 hvn+3 + 43 945 h2wn − 16 105 h2wn+1 − 19 105 h2wn+2 − 8 945 h2wn+3 (12) y′′n+3 = y ′′ n + 93 224 hvn + 243 224 hvn+1 + 243 224 hvn+2 + 93 224 hvn+3 57 1120 h2wn − 81 1120 h2wn+1 + 81 1120 h2wn+2 − 57 1120 h2wn+3 3. The Properties of the HBM In this section, It is important to examine and discuss the convergence analysis of the HBM such as the order & the error constants, convergence, zero stability, and convergence. 3.1. Order & Error Constant of the HBM According to [16–18], the linear difference operator L in respect to equation (10) is defined by L [ y (x) ; h ] = ∑k j=0 { a jy (xn + jh) − h3v jy′′′ (xn + jh) − h4w jy′′′′ (xn + jh) } (13) y (x) is assumed to be continuously differentiable function. Therefore function (13) can be maximize in taylor series about x to obtain L (y (x) ; h) = D0y (x) + D1hy ′ (x) + D2h 2y′′ (x) + ... + Dqh qy(q) (x) (14) where Dq, q = 1, 2, ... are constants in such that, D0 = D1 = ... = Dq = 0, Dp+3 , 0 (15) The method (10) is of uniform order 11 with the following error constants Dq+3 = (−5.143239 × 10−10,−1.312531 × 10−09,−2.394622 × 10−09)T . 3.2. Consistency As stated by [18–20], a LMM of the form (10) is said to be consistent if it has order greater than or equals to one. The HBM satisfies the condition for consistency since its order, is 11 which is greater than one. 3.3. Zero-stability of the HBM Taking into consideration method (10) could be written in matrix difference form as , A(0)Ym = A (1)Ym−1 + h 3 [ C(0)Gm + C (3)Gm−3 ] + h4 [ D(0) Hm + D (4) Hm−3 ] (16) The matrix parameter A(0), A(1), C(0), C(3), D(0), D(4), H(0), H(1) are the square matrices whose arrays are the coefficients (10) and are defined as below. 4 Familua et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 706 5 A(0) =  1 0 0 0 1 0 0 0 1  , A(1) =  0 0 1 0 0 1 0 0 1  , C(0) =  89 3360 439 20160 1031 272160 164 315 4 21 244 8505 2187 1120 2187 2240 117 1120  C(3) =  0 0 62387544320 0 0 50488505 0 0 657448  , D(0) =  − 359 10080 − 13 960 − 17 18144 − 4 15 − 34 315 − 4 567 − 729 1120 − 729 2240 − 27 1120  D(4) =  0 0 1879181440 0 0 1722835 0 0 3512240 , The limit of (16) is taken as h → 0, to obtain the difference system A(0)Ym − A (1)Ym−1 = 0 (17) The first characteristics of (17) is given by ρ (F) = det ( F A(0) − A(1) ) = F2 (F − 1) = 0 (18) Hence, F = 0, 0, 1 The block method of the form (10) is said to be zero stable if as ρ (F) = 0, then ∣∣∣F j∣∣∣ ≤ 1, j = 0, 1, ... for those roots with∣∣∣F j∣∣∣ = 1, the multiplicity does not exceed 1 [21–26]. Also the block method (10) is consistent since p>1. Since (10) is consistent and zero stable, it is also convergence [27]. 3.4. Convergence In respect to the claim of Lambert [18] which also cor- roborates with proof of Adogbe & Omole, and Adogbe et al. [28, 29] that any numerical method belonging to a class of LMM must satisfies the fundamental and adequate conditions. It follows that for such class of method to be convergent it must be consistent and zero-stable. consequently, the HBM satisfies the conditions for consistency and zero-stability, So it is con- vergent. 4. Implementation Tactics In this part, we present the comprehensive procedure for the implementation of the new proposed method tagged Higher- order Block Method (HBM). The HBM is implemented in block method together with the aid of Newton-Raphson approach via a Mathematica 11.0 code which uses f-solve for linear and find- root for non-linear to simultaneously generate the solution at the initial point to the terminal point while adjusting for bound- ary conditions. Meanwhile, each block integrators in (10), (11) and (12) forms a system of equations which is applied along with the Newton’s method. The starting values in the application of the Newton’s Raphson method which are considered as the approximations provided by the Taylor series expansion formulas yn+i = yn + ihy ′ n + ( (ih)2 2 ) y′′n + ( (ih)3 6 ) vn + ( (ih)4 24 ) wn, i = 0 (1) ..., k y′n+i = y ′ n + ihy ′′ n + ( (ih)2 2 ) vn + ( (ih)3 6 ) wn, i = 0 (1) ..., k y′′n+i = y ′′ n + ihvn + ( (ih)2 2 ) wn, i = 0 (1) ..., k. (19) The wn+i, i = 0 (1) ..., k appearing in (19) connotes the fourth derivative at xn+i, i = 0 (1) ..., k. In other to get a closed form solution of (20) which is expanded in (19), it is important to calculate the values of wn+i, i = 0 (1) ..., k. For more clarifica- tion, y(iv)(x) = ( dv(x, y, y′, y′′, y′′′) d x ) , i = 0 (1) ..., k. (20) y(iv)(x) = dv d x + dv dy y′ + dv dy′ y′′ + dv dy′′ y′′′ + v dv dy′′′ v, i = 0 (1) ..., k. (21) Consequently, the solution of (1) is simultaneously generate on the entire interval of integration by using the main method (10) for n = 0, 1, ..., N2 to obtain 3N −2 equations and the additional method (11) and (12) which are employed to complete the set of equations needed to simultaneously solve 3N by 3N system of equations for solving (1) directly. 5. The Numerical Experiments It is very important to test the accuracy and usefulness of the HBM by applying the HBM to solve modeled problems. Three standard third-order BVPs varying from linear to non- linear was computed and the results were presented and dis- cussed extensively. All computations and programming were carried out using Maple 18.0 and Mathematica 11.0. 5.1. Test problem 1 First test problem, third-order linear boundary value prob- lem solved by Ahmed [30]. y′′′−xy+(x2−2x2−5x−3)ex, y(0) = 0, y′(0) = 1, y′(1) = −e(22) with theoretical solution as y(x) = x(1 − x)ex (23) 5 Familua et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 706 6 Table 1. Numerical results of HBM, AE in HBM and AE in [30] for Problem 1 using N = 10 or h = 0.1 x y-Exact solution y-Computed solution AE in HBM AE in [30] 0.1 0.09946538262680829 0.09946538262680911 8.18789 × 10−16 1.36200 × 10−10 0.2 0.19542444130562720 0.19542444130563258 5.38458 × 10−15 1.90000 × 10−12 0.3 0.28347034959096070 0.28347034959097905 1.83742 × 10−14 1.10000 × 10−12 0.4 0.35803792743390490 0.35803792743392630 2.14273 × 10−14 7.00000 × 10−12 0.5 0.41218031767503205 0.41218031767503670 4.66294 × 10−15 1.00000 × 10−11 0.6 0.43730851209372210 0.43730851209368804 3.40838 × 10−14 1.70000 × 10−12 0.7 0.42288806856880007 0.42288806856871670 8.33777 × 10−14 8.80000 × 10−11 0.8 0.35608654855879485 0.35608654855866470 1.30174 × 10−13 9.42000 × 10−11 0.9 0.22136428000412547 0.22136428000395672 1.68754 × 10−13 1.36800 × 10−10 1.0 0.00000000000000000 1.83070 × 10−13 1.83070 × 10−13 0.00000 × 10−00 Table 2. Comparison of Maximum absolute error in HBM with other existing methods for Problem 1 References Maximum Absolute Error Current method-HBM 1.8307 × 10−13 [30] 1.3700 × 10−10 [31] 5.3000 × 10−07 [32] 1.8400 × 10−06 [33] 2.3700 × 10−07 [34] 8.1200 × 10−04 [35] 1.6400 × 10−02 [36] 2.6400 × 10−07 [37] 8.2900 × 10−09 Table 3. Numerical results of HBM and Absolute error for Problem 2 taking N = 100 or h = 0.01 x y-Exact solution y-Computed solution AE in HBM 0.01 −0.009998833350833248 −0.009998833350833078 1.70003 × 10−16 0.02 −0.019990667226655746 −0.019990667226655066 6.80012 × 10−16 0.03 −0.029968504252313413 −0.029968504252311883 1.53003 × 10−15 0.04 −0.039925351251935550 −0.039925351251932820 2.72699 × 10−15 0.05 −0.049854221347501636 −0.049854221347497375 4.26048 × 10−15 0.06 −0.059748136056118590 −0.059748136056112450 6.14092 × 10−15 0.07 −0.069600127385578860 −0.069600127385570460 8.39606 × 10−15 0.08 −0.079403239927770000 −0.079403239927759020 1.09773 × 10−14 0.09 −0.089150532949507160 −0.089150532949493310 1.38500 × 10−14 0.10 −0.098835082480359870 −0.098835082480342770 1.70974 × 10−14 Table 4. Numerical results of HBM and Absolute error for Problem 2 taking N = 10 or h = 0.1 x y-Exact solution y-Computed solution AE in HBM 0.1 −0.098835082480359870 −0.09883508248034277 1.70974 × 10−14 0.2 −0.190722557563258760 −0.19072255756318998 6.87783 × 10−14 0.3 −0.268923388061819000 −0.26892338806166494 1.54043 × 10−13 0.4 −0.327111407539266430 −0.32711140753900390 2.62512 × 10−13 0.5 −0.359569153953152255 −0.35956915395277234 3.79918 × 10−13 0.6 −0.361371182972822670 −0.36137118297233617 4.86500 × 10−13 0.7 −0.328551020491222400 −0.32855102049065060 5.71820 × 10−13 0.8 −0.258248192723828200 −0.25824819272318880 6.39433 × 10−13 0.9 −0.148832112829221850 −0.14883211282853925 6.82593 × 10−13 1.0 0.0000000000000000000 6.98759 × 10−13 6.98759 × 10−13 In Table 1, The theoretical solution, approximate solution, absolute error in HBM and the absolute error in other existing method were presented. On the other hand, Table 2 shows the comparison of Maximum absolute error in HBM against nu- 6 Familua et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 706 7 Table 5. Comparison of Maximum absolute error in HBM with other existing methods for Problem 2 References Maximum Absolute Error Current method-HBM 6.98759 × 10−13 [34] 8.55940 × 10−05 [35] 8.88390 × 10−03 [36] 2.15720 × 10−08 Table 6. Numerical results of HBM and Absolute error Problem 3 with using (N = 10) or (h = 0.1) x y-Exact solution y-Computed solution AE in HBM 0.1 0.09531017980432493 0.09531017980433536 1.04222 × 10−14 0.2 0.18232155679395460 0.18232155679398550 3.09197 × 10−14 0.3 0.26236426446749106 0.26236426446754463 5.35683 × 10−14 0.4 0.33647223662121290 0.33647223662128860 7.57172 × 10−14 0.5 0.40546510810816440 0.40546510810826103 9.66449 × 10−14 0.6 0.47000362924573563 0.47000362924585110 1.15463 × 10−13 0.7 0.53062825106217040 0.53062825106230150 1.31117 × 10−13 0.8 0.58778666490211910 0.58778666490226180 1.42775 × 10−13 0.9 0.64185388617239470 0.64185388617254560 1.50879 × 10−13 1.0 0.69314718055994530 0.69314718056009890 1.53655 × 10−13 Table 7. Comparison of the numerical errors in HBM and other existing methods for Problem 3 with N = 10 x AE in HBM AE in [37] AE in [38] 0.1 1.04222 × 10−14 2.22000 × 10−06 2.947641 × 10−10 0.2 3.09197 × 10−14 4.67000 × 10−06 2.084566 × 10−11 0.3 5.35683 × 10−14 1.890000 × 10−06 4.147355 × 10−11 0.4 7.57172 × 10−14 3.560000 × 10−06 2.208859 × 10−12 0.5 9.66449 × 10−14 7.430000 × 10−07 2.777215 × 10−11 0.6 1.15463 × 10−13 1.220000 × 10−06 2.647594 × 10−11 0.7 1.31117 × 10−13 2.780000 × 10−06 1.190750 × 10−11 0.8 1.42775 × 10−13 3.74000 × 10−06 1.816331 × 10−12 0.9 1.50879 × 10−13 4.11000 × 10−06 7.666364 × 10−10 1.0 1.53655 × 10−13 0.00000 × 10−00 0.00000 × 10−00 merous methods proposed by various authors in the literatures. It is very clear that the efficiency and accuracy of HBM is es- tablished comprehensively. 5.2. Test problem 2 Consider the third-order boundary value problem solved by Abdullah et al. [34]. y′′′ − y + (7 − x2)cosx + (x2 − 6x − 1)sinx, y(0) = 0, y′(0) = −1, y′(1) = 2sin1 (24) with the analytical solution given as y(x) = (x2 − 1)sin(x) (25) Here, the interpretation of Tables 3, 4 and 5 are made. The computational results of problem 2 using the proposed method named HBM with h = 0.01 is shown, while in Table 2, we presents the computational results of problem 2 using HBM with h = 0.1. It could be observed that as the value of h decreases, the accuracies also increases whereas as the h in- creases, the accuracies of the method decreases as demonstrated in Tables 3 and 4. furthermore, we computed the maximum ab- solute error of the HBM for problem 2 and compared with other numerical methods in the cited literature. This is illustrated in the Table 5. The HBM gives a minimal error when compared with other existing method. 5.3. Test problem 3 Lastly, a non-linear third-order problem studied by Akram et al. [37] and, Hossain et al. [38] is taking into consideration. y′′′ + 2e3y = −4(1 + x)−3, y(0) = 0, y′(0) = 1, y′(1) = In(2) (26) with the exact solution. y(x) = In(1 + x) (27) Finally, we considered a non-linear third-order bvp in other to determine the strength and advantage of the HBM. In Table 6, 7 Familua et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 706 8 Figure 1. Comparison of Absolute error in HBM with Absolute error in Akram et al. [37] Figure 2. Comparison of Absolute error in HBM with Absolute error in Hossain et al. [38] The computational results and absolute error on HBM is pre- sented while Table 7 presented the comparison of absolute er- ror of the non-linear problem using HBM and compared with other similar methods in the cited work. In addition, the com- parison of errors in curves for the non-linear problem were also presented in Figure 1 and 2. without any iota of doubt, we have been able to demonstrate the convergence, efficiency and accu- racy of the new method namely HBM over other techniques. 6. Conclusion This study has successfully presented a construction of new numerical method (HBM), analyse and implemented for solv- ing numerous type of third-order BVPs directly without utiliz- ing the conventional method such as shooting method, reduc- tion to first-order system of equations, finite difference method. In the derivation, the method make use of power series basic function due to its great stability and convergence attributes. The multi-collocation points was introduced for the purpose of improving the order of the work, which also enhances good ac- curacy, the points of interpolation and collocation were also chosen strategically in order to obtain a desired results. 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