Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 960 https://internationalpubls.com Generalised Four Step Adams-Moulton Second Derivative Method for the Solution Stiff Ordinary Differential Equations Buba M. T Hambagda, Kaze Atsi, Babatunde Aina Mathematics Department, Federal University Gashua, Yobe state, Nigeria bmthambagda@gmail.com. kzeeat@yahoo.com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: A uniform order ten second derivative block method is presented for solving stiff ordinary differential equations. The proposed method was derived by using the approach of collocation and interpolation of four step Adams-Moulton method. Four individual schemes that made up the block method are obtained at step number, k = 4. The stability properties of the new method have been ascertained and it has shown to be consistent, zero–stable and A–stable. The solutions of two problems have been computed and compared with the corresponding exact and other existing solutions. Solutions are presented on graphs and the associated absolute errors are compared with some existing solution-errors in tables. Keywords: Second Derivative, Adams-Moulton Method, Collocation and Interpolation, Block Methods, Stiff Ordinary Differential Equations, Stability Properties, Absolute Error. 1.0 Introduction Differential equations are mathematical models of problems emanating from almost every field of study, where measurement can be taken. They are of different types and exhibit different phenomenon. Stiff ordinary differential equations are special class of problems of differential equation which are mostly found in engineering sciences and other areas of studies, [1]. So many researchers are involved in finding solution to the initial valued problems of stiff ordinary differential equations of the form: 𝑦′ = 𝑓(π‘₯, 𝑦), 𝑦(π‘₯ 0) = 𝑦0, π‘₯ ∈ [π‘Ž, 𝑏], 𝑦 ∈ ℝ (1.1) as can be seen in [1] and [2]. As it has been stated in [2], about the availability of the analytical solution of (1), there is a need for a suitable numerical method to solve (1.1). Dauda et. al., [3], presented a second derivative block hybrid method suitable for the continuous integration of stiff ordinary differential equations and was stated to compete favourably with some strong stability stiff integrators. In [4], Rufai et. al., derive a new continuous hybrid block method using Chebyshev polynomials of the first kind as basis function and the new proposed method for the solution of initial value problems of systems of stiff ordinary differential equations. It can also be seen in Skwame et. al., [5], Kumleng et. al., [6], Ajie et. al., [7] and Kumleng et. al., [8], how new methods are presented for the solution of problems of the form (1.1). Interpolation and collocation approach has been adopted in many literatures for the construction of block method of solving differential equations. Kumleng et. al., [8], adopted the approach and methods Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 961 https://internationalpubls.com of uniform orders 10 and 11 were constructed. In a similar approach, [1] generated a block method of order 7. It can also be seen in [9], where a new block method of order 4 has been formed. Other methods in which this approach is adopted are [10], [11], [12]. Kayode et at., [14], presened a continuous two step trigonometrically-fitted second order method for the solution of linear and nonlinear initial value oscillatory problems. The coefficients of the developed approaches are determined by the approximate solution’s frequency and step size, a discrete trigonometrically -fitted second order ordinary differential equation was recovered as a by-product. , The stability and other properties qualities are described to demonstrate the method’s usefulness and efficiency. And real-world oscillatory problems of ordinary differential equations implemented to the suitability of the new method. We present a second derivative block method for the solution (1.1). The new block method is derived from a four step Adams-Moulton method. 2.0 Derivation of the Method Harnessing the concept of interpolation and collocation technique, we consider the following method for our new method: οΏ½Μ…οΏ½(π‘₯) = 𝛼3(π‘₯)𝑦𝑛+3 + β„Žβˆ‘π›½π‘—(π‘₯)𝑓𝑛+𝑗 4 𝑗=0 (π‘₯)) + β„Ž2βˆ‘π›Ύπ‘—(π‘₯)𝑔𝑛+𝑗 4 𝑗=0 (𝟐. 𝟏) where (2.1) is the approximation of a continuously differentiable solution of (1.1) 𝛼3(π‘₯), 𝛽𝑗(π‘₯) and 𝛾𝑗(π‘₯) are the continuous coefficients β„Ž is the step size of the method with a step number, π‘˜ = 4 𝑦𝑛+𝑗 is the approximation to the theoretical solution of (1.1) at π‘₯𝑛+𝑗 and, the first and second derivative of 𝑦𝑛+𝑗 are 𝑓𝑛+𝑗 ≑ 𝑓(π‘₯𝑛+𝑗𝑦𝑛+𝑗) and 𝑔𝑛+𝑗 ≑ 𝑔(π‘₯𝑛+𝑗 , 𝑦𝑛+𝑗, 𝑓𝑛+𝑗), respectively. Adopting the technique in [8] and other work, we obtain a matrix defined by 𝐷, as follows: 𝐷 = ( 𝐷1 𝐷3 𝐷2 𝐷4 ) where, 𝐷1 = ( 1 π‘₯𝑛 + 3β„Ž (π‘₯𝑛 + 3β„Ž) 2 (π‘₯𝑛 + 3β„Ž) 3 (π‘₯𝑛 + 3β„Ž) 4 (π‘₯𝑛 + 3β„Ž) 5 0 1 2π‘₯𝑛 3π‘₯𝑛 2 4π‘₯𝑛 3 5π‘₯𝑛 4 0 1 2(π‘₯𝑛 + β„Ž) 3(π‘₯𝑛 + β„Ž) 2 4(π‘₯𝑛 + β„Ž) 3 5(π‘₯𝑛 + β„Ž) 4 0 1 2(π‘₯𝑛 + 2β„Ž) 3(π‘₯𝑛 + 2β„Ž) 2 4(π‘₯𝑛 + 2β„Ž) 3 5(π‘₯𝑛 + 2β„Ž) 4 0 1 2(π‘₯𝑛 + 3β„Ž) 3(π‘₯𝑛 + 3β„Ž) 2 4(π‘₯𝑛 + 3β„Ž) 3 5(π‘₯𝑛 + 3β„Ž) 4 0 1 2(π‘₯𝑛 + 4β„Ž) 3(π‘₯𝑛 + 4β„Ž) 2 4(π‘₯𝑛 + 4β„Ž) 3 5(π‘₯𝑛 + 4β„Ž) 4) 𝐷2 = ( 0 0 2 6π‘₯𝑛 12π‘₯𝑛 2 20π‘₯𝑛 3 0 0 2 6(π‘₯𝑛 + β„Ž) 12(π‘₯𝑛 + β„Ž) 2 20(π‘₯𝑛 + β„Ž) 3 0 0 2 6(π‘₯𝑛 + 2β„Ž) 12(π‘₯𝑛 + 2β„Ž) 2 20(π‘₯𝑛 + 2β„Ž) 3 0 0 2 6(π‘₯𝑛 + 3β„Ž) 12(π‘₯𝑛 + 3β„Ž) 2 20(π‘₯𝑛 + 3β„Ž) 3 0 0 2 6(π‘₯𝑛 + 4β„Ž) 12(π‘₯𝑛 + 4β„Ž) 2 20(π‘₯𝑛 + 4β„Ž) 3) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 962 https://internationalpubls.com 𝐷3 = ( (π‘₯𝑛 + 3β„Ž) 6 (π‘₯𝑛 + 3β„Ž) 7 (π‘₯𝑛 + 3β„Ž) 8 (π‘₯𝑛 + 3β„Ž) 9 (π‘₯𝑛 + 3β„Ž) 10 6π‘₯𝑛 5 7π‘₯𝑛 6 8π‘₯𝑛 7 9π‘₯𝑛 8 10π‘₯𝑛 9 6(π‘₯𝑛 + β„Ž) 5 7(π‘₯𝑛 + β„Ž) 6 8(π‘₯𝑛 + β„Ž) 7 9(π‘₯𝑛 + β„Ž) 8 10(π‘₯𝑛 + β„Ž) 9 6(π‘₯𝑛 + 2β„Ž) 5 7(π‘₯𝑛 + 2β„Ž) 6 8(π‘₯𝑛 + 2β„Ž) 7 9(π‘₯𝑛 + 2β„Ž) 8 10(π‘₯𝑛 + 2β„Ž) 9 6(π‘₯𝑛 + 3β„Ž) 5 7(π‘₯𝑛 + 3β„Ž) 6 8(π‘₯𝑛 + 3β„Ž) 7 9(π‘₯𝑛 + 3β„Ž) 8 10(π‘₯𝑛 + 3β„Ž) 9 6(π‘₯𝑛 + 4β„Ž) 5 7(π‘₯𝑛 + 4β„Ž) 6 8(π‘₯𝑛 + 4β„Ž) 7 9(π‘₯𝑛 + 4β„Ž) 8 10(π‘₯𝑛 + 4β„Ž) 9) 𝐷4 = ( 30π‘₯𝑛 4 42π‘₯𝑛 5 56π‘₯𝑛 6 72π‘₯𝑛 7 90π‘₯𝑛 8 30(π‘₯𝑛 + β„Ž) 4 42(π‘₯𝑛 + β„Ž) 5 56(π‘₯𝑛 + β„Ž) 6 72(π‘₯𝑛 + β„Ž) 7 90(π‘₯𝑛 + β„Ž) 8 30(π‘₯𝑛 + 2β„Ž) 4 42(π‘₯𝑛 + 2β„Ž) 5 56(π‘₯𝑛 + 2β„Ž) 6 72(π‘₯𝑛 + 2β„Ž) 7 90(π‘₯𝑛 + 2β„Ž) 8 30(π‘₯𝑛 + 3β„Ž) 4 42(π‘₯𝑛 + 3β„Ž) 5 56(π‘₯𝑛 + 3β„Ž) 6 72(π‘₯𝑛 + 3β„Ž) 7 90(π‘₯𝑛 + 3β„Ž) 8 30(π‘₯𝑛 + 4β„Ž) 4 42(π‘₯𝑛 + 4β„Ž) 5 56(π‘₯𝑛 + 4β„Ž) 6 72(π‘₯𝑛 + 4β„Ž) 7 90(π‘₯𝑛 + 4β„Ž) 8) From the inverse of the D matrix, and by putting π‘₯ βˆ’ π‘₯𝑛 = πœ‡, the values of the continuous coefficients in (2.1) are obtained and presented as follows: 𝛼3 = 1 β„Žπ›½0 = βˆ’ 1 4354560β„Ž9 (58509β„Ž7 βˆ’ 102771β„Ž6πœ‡ βˆ’ 122274β„Ž5πœ‡2 + 457350β„Ž4πœ‡3 βˆ’ 446265β„Ž3πœ‡4 + 198135β„Ž2πœ‡5 βˆ’ 40810β„Žπœ‡6 + 3150πœ‡7)(3β„Ž βˆ’ πœ‡)3 β„Žπ›½1 = 1 272160β„Ž9 (1260πœ‡7 βˆ’ 14420πœ‡6β„Ž + 57015πœ‡5β„Ž2 βˆ’ 88065πœ‡4β„Ž3 + 38910πœ‡3β„Ž4 + 5526πœ‡2β„Ž5 + 8289πœ‡β„Ž6 + 8289β„Ž7)(3β„Ž βˆ’ πœ‡)3 β„Žπ›½2 = βˆ’ 3 560β„Ž8 (216β„Ž6 + 216β„Ž5πœ‡ + 144β„Ž4πœ‡2 βˆ’ 480β„Ž3πœ‡3 + 810β„Ž2πœ‡4 βˆ’ 315β„Žπœ‡5 + 35πœ‡6)(β„Ž βˆ’ πœ‡)3 β„Žπ›½3 = βˆ’ 1 272160β„Ž9 (57591β„Ž9 + 19197β„Ž8πœ‡ + 6399β„Ž7πœ‡2 βˆ’ 320427β„Ž6πœ‡3 + 807111β„Ž5πœ‡4 βˆ’ 850515β„Ž4πœ‡5 + 466335β„Ž3πœ‡6 βˆ’ 138075β„Ž2πœ‡7 + 20860β„Žπœ‡8 βˆ’ 1260πœ‡9)(3β„Ž βˆ’ πœ‡) β„Žπ›½4 = βˆ’ 1 4354560β„Ž9 (3699β„Ž7 + 3699β„Ž6πœ‡ + 2466β„Ž5πœ‡2 βˆ’ 57990β„Ž4πœ‡3 + 115065β„Ž3πœ‡4 βˆ’ 87255β„Ž2πœ‡5 + 28490β„Žπœ‡6 βˆ’ 3150πœ‡7)(3β„Ž βˆ’ πœ‡)3 β„Ž2𝛾0 = βˆ’ 1 725760β„Ž8 (1017β„Ž7 + 1017β„Ž6πœ‡ βˆ’ 12762β„Ž5πœ‡2 + 24270β„Ž4πœ‡3 βˆ’ 20205β„Ž3πœ‡4 + 8379β„Ž2πœ‡5 βˆ’ 1666β„Žπ‘§6 + 126𝑧7)(3β„Ž βˆ’ πœ‡)3 β„Ž2𝛾1 = 1 90720β„Ž8 (837β„Ž7 + 837β„Ž6πœ‡ + 558β„Ž5πœ‡2 βˆ’ 17610β„Ž4πœ‡3 + 24795β„Ž3πœ‡4 βˆ’ 13293β„Ž2πœ‡5 + 3052β„Žπœ‡6 βˆ’ 252πœ‡7)(3β„Ž βˆ’ πœ‡)3 β„Ž2𝛾2 = 1 320β„Ž8 (3β„Ž βˆ’ πœ‡)3(3β„Ž4 + 12β„Ž3πœ‡ + 29β„Ž2πœ‡2 βˆ’ 16β„Žπ‘§3 + 2πœ‡4)(β„Ž βˆ’ πœ‡)3 β„Ž2𝛾3 = 1 90720β„Ž8 (1647β„Ž8 + 1098β„Ž7πœ‡ + 549β„Ž6πœ‡2 βˆ’ 17676β„Ž5πœ‡3 + 39675β„Ž4πœ‡4 βˆ’ 35874β„Ž3πœ‡5 + 15729β„Ž2πœ‡6 βˆ’ 3248β„Žπœ‡7 + 252πœ‡8)(3β„Ž βˆ’ πœ‡)2 β„Ž2𝛾4 = 1 725760β„Ž8 (135β„Ž7 + 135β„Ž6πœ‡ + 90β„Ž5πœ‡2 βˆ’ 2190β„Ž4πœ‡3 + 4365β„Ž3πœ‡4 βˆ’ 3339β„Ž2πœ‡5 + 1106β„Žπœ‡6 βˆ’ 126πœ‡7)(3β„Ž βˆ’ πœ‡)3 } (2.2) To obtain the continuous form of the new method, we substitute the values of the continuous coefficients into (2.1). This is presented in the following form: 𝑦(π‘₯) = 𝑦𝑛+3 + (βˆ’ 1 4354560β„Ž9 (58509β„Ž7 βˆ’ 102771β„Ž6πœ‡ βˆ’ 122274β„Ž5πœ‡2 + 457350β„Ž4πœ‡3 βˆ’ 446265β„Ž3πœ‡4 + 198135β„Ž2πœ‡5 βˆ’ 40810β„Žπœ‡6 + 3150πœ‡7)(3β„Ž βˆ’ πœ‡)3) 𝑓𝑛 +( 1 272160β„Ž9 (1260πœ‡7 βˆ’ 14420πœ‡6β„Ž + 57015πœ‡5β„Ž2 βˆ’ 88065πœ‡4β„Ž3 + 38910πœ‡3β„Ž4 + 5526πœ‡2β„Ž5 + 8289πœ‡β„Ž6 + 8289β„Ž7)(3β„Ž βˆ’ πœ‡)3) 𝑓𝑛+1 +(βˆ’ 3 560β„Ž8 (216β„Ž6 + 216β„Ž5πœ‡ + 144β„Ž4πœ‡2 βˆ’ 480β„Ž3πœ‡3 + 810β„Ž2πœ‡4 βˆ’ 315β„Žπœ‡5 + 35πœ‡6)(β„Ž βˆ’ πœ‡)3) 𝑓𝑛+2 +(βˆ’ 1 272160β„Ž9 (57591β„Ž9 + 19197β„Ž8πœ‡ + 6399β„Ž7πœ‡2 βˆ’ 320427β„Ž6πœ‡3 + 807111β„Ž5πœ‡4 βˆ’ 850515β„Ž4πœ‡5 + 466335β„Ž3πœ‡6 βˆ’ 138075β„Ž2πœ‡7 + 20860β„Žπœ‡8 βˆ’ 1260πœ‡9)(3β„Ž βˆ’ πœ‡)) 𝑓𝑛+3 +(βˆ’ 1 4354560β„Ž9 (3699β„Ž7 + 3699β„Ž6πœ‡ + 2466β„Ž5πœ‡2 βˆ’ 57990β„Ž4πœ‡3 + 115065β„Ž3πœ‡4 βˆ’ 87255β„Ž2πœ‡5 + 28490β„Žπœ‡6 βˆ’ 3150πœ‡7)(3β„Ž βˆ’ πœ‡)3) 𝑓𝑛+4 +(βˆ’ 1 725760β„Ž8 (1017β„Ž7 + 1017β„Ž6πœ‡ βˆ’ 12762β„Ž5πœ‡2 + 24270β„Ž4πœ‡3 βˆ’ 20205β„Ž3πœ‡4 + 8379β„Ž2πœ‡5 βˆ’ 1666β„Žπ‘§6 + 126𝑧7)(3β„Ž βˆ’ πœ‡)3)𝑔𝑛 +( 1 90720β„Ž8 (837β„Ž7 + 837β„Ž6πœ‡ + 558β„Ž5πœ‡2 βˆ’ 17610β„Ž4πœ‡3 + 24795β„Ž3πœ‡4 βˆ’ 13293β„Ž2πœ‡5 + 3052β„Žπœ‡6 βˆ’ 252πœ‡7)(3β„Ž βˆ’ πœ‡)3)𝑔𝑛+1 +( 1 320β„Ž8 (3β„Ž βˆ’ πœ‡)3(3β„Ž4 + 12β„Ž3πœ‡ + 29β„Ž2πœ‡2 βˆ’ 16β„Žπ‘§3 + 2πœ‡4)(β„Ž βˆ’ πœ‡)3)𝑔𝑛+2 +( 1 90720β„Ž8 (1647β„Ž8 + 1098β„Ž7πœ‡ + 549β„Ž6πœ‡2 βˆ’ 17676β„Ž5πœ‡3 + 39675β„Ž4πœ‡4 βˆ’ 35874β„Ž3πœ‡5 + 15729β„Ž2πœ‡6 βˆ’ 3248β„Žπœ‡7 + 252πœ‡8)(3β„Ž βˆ’ πœ‡)2) 𝑔𝑛+3 +( 1 725760β„Ž8 (135β„Ž7 + 135β„Ž6πœ‡ + 90β„Ž5πœ‡2 βˆ’ 2190β„Ž4πœ‡3 + 4365β„Ž3πœ‡4 βˆ’ 3339β„Ž2πœ‡5 + 1106β„Žπœ‡6 βˆ’ 126πœ‡7)(3β„Ž βˆ’ πœ‡)3)𝑔𝑛+4 (2.3) The continuous scheme (2.3) is interpolated at some points for πœ‡ = 0, β„Ž, 2β„Ž and 4β„Ž, thereby, yielding 4 discrete methods. These can be presented as: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 963 https://internationalpubls.com [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] [ 𝑦𝑛+1 𝑦𝑛+2 𝑦𝑛+3 𝑦𝑛+4 ] = [ 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 ] [ π‘¦π‘›βˆ’3 π‘¦π‘›βˆ’2 π‘¦π‘›βˆ’1 𝑦𝑛 ] +β„Ž ( [ 89371 272160 103 630 38341 272160 59681 4354560 6616 8505 208 315 1576 8505 1153 68040 921 1120 81 70 711 1120 411 17920 8192 8505 416 315 8192 8505 3202 8505 ] [ 𝑓𝑛+1 𝑓𝑛+2 𝑓𝑛+3 𝑓𝑛+4 ] + [ 0 0 0 1539551 4354560 0 0 0 24463 68040 0 0 0 6501 17920 0 0 0 3202 8505 ] [ 𝑓𝑛+3 𝑓𝑛+2 𝑓𝑛+1 𝑓𝑛 ] ) +β„Ž2 ( [ βˆ’ 31207 90720 βˆ’ 81 320 βˆ’ 1243 18144 βˆ’ 2237 725760 βˆ’ 152 567 βˆ’ 2 5 βˆ’ 248 2835 βˆ’ 43 11340 βˆ’ 279 1120 βˆ’ 81 320 βˆ’ 183 1120 βˆ’ 9 1792 βˆ’ 512 2835 0 512 2835 βˆ’ 116 2835 ] [ 𝑔𝑛+1 𝑔𝑛+2 𝑔𝑛+3 𝑔𝑛+4 ] + [ 0 0 0 26051 725760 0 0 0 421 11340 0 0 0 339 8960 0 0 0 116 2835 ] [ π‘”π‘›βˆ’3 π‘”π‘›βˆ’2 π‘”π‘›βˆ’1 𝑔𝑛 ] ) (2.4) 3.0 Analysis of the Block Methods A summary on the order, error constant and the convergence of the proposed block method are given using the technique presented in [1]. The block method (2.4) is represented by a matrix finite difference equation in block form as: 𝐴(0)π‘Œπ‘š = 𝐴(1)π‘¦π‘š + β„Ž{𝐡 (0)πΉπ‘š + 𝐡 (1)π‘“π‘š} + β„Ž 2{𝐢(1)π‘”π‘š + 𝐢 (0)πΊπ‘š} (3.1) where, π‘Œπ‘š = [𝑦𝑛+1 𝑦𝑛+2 𝑦𝑛+3 𝑦𝑛+4]𝑇, π‘¦π‘š = [π‘¦π‘›βˆ’3 π‘¦π‘›βˆ’2 π‘¦π‘›βˆ’1 𝑦𝑛]𝑇, πΉπ‘š = [𝑓𝑛+1 𝑓𝑛+2 𝑓𝑛+3 𝑓𝑛+4]𝑇, π‘“π‘š = [π‘“π‘›βˆ’3 π‘“π‘›βˆ’2 π‘“π‘›βˆ’1 𝑓𝑛]𝑇, π‘”π‘š = [π‘”π‘›βˆ’3 π‘”π‘›βˆ’2 π‘”π‘›βˆ’1 𝑔𝑛]𝑇 and πΊπ‘š = [𝑔𝑛+1 𝑔𝑛+2 𝑔𝑛+3 𝑔𝑛+4]𝑇 and the matrices 𝐴(1), 𝐴(0), 𝐡(1), 𝐡(0), 𝐢(0)and 𝐢(1) are 4 by 4 matrices. 3.1 Zero-stability As in (12), the roots of the first characteristics polynomial, 𝜌(πœ†), defined by: 𝜌(πœ†) = |πœ†π΄(0) βˆ’ 𝐴(1)| satisfies |π‘Ÿ| ≀ 1, are, |π‘Ÿ| = 0, 0, 0, or 1. Therefore, the new block method is zero-stable. 3.2 Order and Error Constants The new block method is of uniform order 𝑝 = [10 10 10 10]𝑇, with an error constant of 𝐢𝑝+1 = [ 551 314344800 4 1964655 1 431200 8 1964655 ] 𝑇 3.3 Consistency By definition, the order is 𝑝 β‰₯ 1 implies that the new block method is consistence, see [1]. 3.4 Absolute stability region The absolute stability region of the new block method is obtained using [13]. The method, (2.4) is reformulated as a general linear method and is plotted using the Matlab program as revealed in the figure 1 to be A-stable. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 964 https://internationalpubls.com Figure 1: Region of Absolute stability of the new block method 3.5 Convergence According to [1], the new block method is convergent, since it’s consistent and zero-stable. 4.0 Numerical experiments To demonstrate the performance of the new block method, we consider two linear stiff ordinary differential equations of the form (1.1). We present the calculated absolute errors in tables and the solution curves are shown in figures. Example 1: This problem is taken from [1]: on the range, 0 ≀ π‘₯ ≀ 1 and a step size, β„Ž = 0.1: 𝑦′ = βˆ’1000𝑦 + 999π‘’βˆ’π‘₯, 𝑦(0) = 1 With exact solution: 𝑦(π‘₯) = π‘’βˆ’π‘₯. Example 2: This problem is taken from [2]: 𝑦′(π‘₯) = [ 21 19 βˆ’21 19 βˆ’21 20 40 βˆ’40 40 ] [ 𝑦1(π‘₯) 𝑦2(π‘₯) 𝑦3(π‘₯) ] , 𝑦(0) = [ 1 0 βˆ’1 ] the exact solutions for 0 ≀ π‘₯ ≀ 10, are: 𝑦(π‘₯) = [ π‘’βˆ’π‘₯ βˆ’π‘’βˆ’π‘₯ ] Result: Table 1: Errors in New block method and in [1] with Exact solution: Example 1 x Error in [1] Error in New block method 0 0 0 0.100000000000000 1.88737914186277e-15 0 0.200000000000000 2.10942374678780e-15 0 0.300000000000000 8.88178419700125e-16 1.11022302462516e-16 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 965 https://internationalpubls.com 0.400000000000000 1.55431223447522e-15 1.11022302462516e-16 0.500000000000000 9.99200722162641e-16 1.11022302462516e-16 0.600000000000000 1.22124532708767e-15 0 0.700000000000000 1.49880108324396e-15 0 0.800000000000000 9.99200722162641e-16 0 0.900000000000000 4.99600361081320e-16 0 1 8.32667268468867e-16 1.11022302462516e-16 Figure 2: Showing the New block method with Exact solutions: Example 1 Table 2: Comparison of error in the new block method with the exact solution in Example 2 x Error of y1 Error of y2 Error of y3 0 0 0 0 0.1000 0 0.0260987027814653 0.00206310803809157 0.2000 0.0122373815672981 0.000719747107311319 0.00107577664370589 0.3000 0.000387854389990783 3.24631645365336e-06 5.40015326179073e-05 0.4000 6.54313444264920e-06 9.83441426322118e-07 9.66591002920572e-07 0.5000 9.51042133906510e-07 3.83220316313437e-08 3.02159582896780e-08 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 966 https://internationalpubls.com 0.6000 3.64403553165804e-08 4.77281492106130e-10 2.72539843546488e-09 0.7000 4.43050762743980e-10 2.67659783226293e-11 8.80135840652426e-11 0.8000 2.69109318162819e-11 1.81354931072519e-12 5.81808897278642e-13 0.9000 1.84649517898094e-12 6.90142387682613e-14 8.58072112004154e-14 1 3.18495230189342e-14 1.68198788230711e-14 4.72864743371380e-15 Figure 3: Showing the New block method with Exact solutions: Example 2 5. DISCUSSION OF RESULTS The confirmation of the region of absolute stability is presented in figure 1. The region containing the entire left-half plane, reveals that the new block method is A-stable, thereby ensuring unconditional stability for all step sizes. The region of absolute stability is larger than that of the fourth-order Runge- Kutta (RK4) method, making it more stable for a wider range of problems. Table 1 shows the comparisons between the error in the proposed methods with the presented in [1]. With the same step number and step size, the new method has slightly more error stability compared to the error in [1]. The numerical and exact solutions for example 1 are presented in figure 2, with the numerical solution competing favorably with the exact solution. The representation of the absolute errors in the new method with the exact solution, in table 2, shows a healthy competition with the exact solution. Similar favoritism is observed in figure 3 of this work. Furthermore, convergence of the new method ensures a reduction in inherent error, in comparison with other existing methods. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 967 https://internationalpubls.com 6. CONCLUSION The generalized four step Adam Moulton Method has been presented in this research. Following the technique adopted in [1], four discrete methods were constructed. Stability analysis demonstrates that the method satisfies essential properties such as consistency, zero-stability; and the stability region is plotted using MATLAB, visually confirming A-stability. Furthermore, numerical results obtained in examples 1and 2 confirm the method’s ability to handle stiff ODEs efficiently. Tables and graphs are presented to provide a clear comparison between the new method and existing approaches. The paper does not analyze the computational cost of the proposed method. Including execution time, function evaluations, and memory usage. 7. DECLARATIONS Availability of data and material: All data generated or analyzed during this study are included in this published article and its supplementary references. Conflicts of interest/Competing interests: All authors declare that they have no conflicts of interest. Funding: This work was supported through TETFund. Authors’ contributions: We understand that the Corresponding Author is the sole contact for the Editorial process (including Editorial Manager and direct communications with the office). He is responsible for communicating with the other authors about progress, submissions of revisions and final approval of proofs. We confirm that we have provided a current, correct email address which is accessible by the Corresponding Author. All authors read and approved the final manuscript. Acknowledgments We are grateful to God for inspiration to carry out this research. TETFund has helped us in conducting our study and completing the work. 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