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Adomian decomposition method for solution of fourteenth order boundary value problems

  • Aasma Khalid , Muhammad Nawaz Naeem , Neelam Jamal , Sameh Askar and Hijaz Ahmad EMAIL logo
Published/Copyright: March 27, 2023
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Open Physics
From the journal Open Physics Volume 21 Issue 1

Abstract

Differential equations (DEs) performed a vital role in the implementation of almost all the mechanical, physical, or biological processes. Higher order DEs had always been challenging to solve for the researchers so numerous numerical techniques were developed to attain the vital numerical approximations of such types of problems. In this work, highly advanced numerical techniques are established for the approximation of the fourteenth (14th)-order boundary value problems using Adomian decomposition method. The mathematical outcomes of the equations are attained in the form of convergent series that have effortlessly assessable components having step size h = 10. Some numerical examples are also deliberated to demonstrate the capability and application of the established procedure.

Keywords: nonlinear; linear; Adomian decomposition method; absolute errors; fourteenth order

1 Introduction

During the last decade, numerous analytical and approximate strategies had been evolved to resolve the linear and nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) [1,2,3,4,5,6,7,8,9,10]. Amongst them is the Adomian decomposition technique. The Adomian decomposition method (ADM) is a very operative method for explaining comprehensive classes of ODEs and PDEs, with significant usages in diverse topics of everyday life sciences. The ADM desires fewer efforts in contrast with the already established procedures. This technique drops substantial amount of computations. The decomposition process of Adomian is attained easily not including linearization of the discussed problem by employing the decomposition technique somewhat than the usual procedures meant for the precise solutions.

In the recent decades, the expansion of the high-pitched digital computer and enlarged attention towards the non-linear phenomena have directed to an exhaustive analysis of the mathematical explanation of ODEs and PDEs. Higher order boundary value problems (BVPs) appear in the investigation of fluid dynamic forces, hydro-dynamic, hydro-magnetic steadiness, space science, induction automobiles, engineering, and implemented quantum mechanics. Such type of higher order BVPs have been assessed by means of their scientific importance in various implemented sciences. It is as of now not so smooth to decide the scientific response for such classification of BVPs and examination in this track might be expected in its underlying stages.

The literature regarding mathematical clarifications of fourteenth (14th)-order BVPs is exceptionally uncommon. The ADM is a numerical technique that decomposes a differential equation (DE) into simpler parts and then solves each part individually. This technique is especially useful for nonlinear DEs which are difficult or impossible to solve using traditional analytical methods. The method involves the use of a nonlinear operator, called the Adomian operator, which is applied to the differential equation. The Adomian operator is then decomposed into a series of terms, each of which is a simpler differential equation. These simpler equations are solved iteratively, using recursive formulas to obtain the solution to the original differential equation. One of the main advantages of the ADM is that it does not require any linearization or small parameter assumptions, making it applicable to a wide range of problems. In addition, the method can handle systems of differential equations and can be used to find approximate solutions for problems that do not have exact solutions. The ADM offers the solution in a speedily convergent sequences with effortlessly assessable elements. The key benefit of the technique is that it can be exhausted straight to explain all kinds of differential equations with boundary conditions. Additional benefit of the technique is that it diminishes the computational work in an evident means, whereas sustaining better precision of the mathematical result. The ADM has been successfully applied to many fields, including physics, engineering, finance, biology, and economics. It has been used to solve problems in heat transfer, fluid mechanics, elasticity, population dynamics, and many other areas. The method has also been used to study nonlinear phenomena such as chaos and bifurcation.

Various angles and accentuations were exhibited aiming the ADM analyses in literature. Al-Jawary combined novel iterative techniques to tackle Cauchy problems [11]. With this iterative approach, the solution is produced in a series form with easily calculable components that converge to the exact solution. Modifications of nonlinear PDEs was used by Al-Mazmumy and Al-Malki [12] by solving using ADM. These well-organized modifications gave a simple prevailing implement for gaining the solutions without a need for huge size of calculations. Ali et al. [13] adapted the procedure for 12th-order BVPs by Optimal homotopy asymptotic. This procedure had been utilized to interpret the actions of nonlinear automatic vibrations of power-driven device. Bhalekar and Gejji [14] discovered the convergence of a new iterative. The new iterative was an effective strategy to settle nonlinear conditions. He found conditions for the convergence of DJM and some modified form of Adomian decomposition method. More identicalness of ADM hadset up. The solution of the considered problem, i.e., 14th-order DEs with the numerical approach, 4th-order Runge–Kutta technique was defined by Chapra and Canale [15]. Adomian polynomials was used by Elsaid [16] for iterative strategy for a series of solution of nonlinear conditions. The motion of a beam on a nanowire was explored using a novel fractional-order Lagrangian by Erturk et al. [17]. In beam theory, boundary and continuity conditions along with the behavior equations for the entire whereabouts are consequent explicitly, by means of five unknown quantities: horizontal and vertical deflections of the advanced and subordinate skins and the shear strain in the core. Frostig et al. [18] described such equations a 14th-order DE in forms of the unknown quantities. Frostig and Thomsen [19] have explained that the outward symmetrical circular sandwich plate, that have nonlinear equations, may be established in the form a set of 14th-order ODEs.

ADM was utilized by Hassan and Erturk [20] and Hassan and Zhu [21] for singular 2nd-order ODEs and various kinds of linear and nonlinear higher order BVPs. Differential transformation method (DTM) along with ADM is expended to solve the 4th-order BVPs. Hayani [22,23] explained the usefulness of ADM for 10th- and 12th-order BVPs. ADM and DTM was applied on boundary value problems in ref. [24]. Hymavathi and Kumar [25] reviewed the solution of 12th-order BVPs. This method was a prevailing process to transform solutions of linear and nonlinear ordinary equations. Approximation solution of this problem was calculated and was rapidly convergent. Hajipour et al. [26] investigated an accurate discretization method for the solution of multi-dimensional highly nonlinear Bratu-type problems.

Numerical solution of the nonlinear diffusion equation with convection period with initial condition was studied by Jebari et al. [27] using the ADM. The solution was considered in the process of a convergent power series using simply calculable mechanisms. A new and general fractional formulation is presented by Jajarmi et al. [28] to investigate the complex behaviors of a capacitor microphone dynamical system. Jajarmi and Baleanu [29] developed an efficient numerical method for solving a class of nonlinear fractional BVPs. Lamnii et al. [30] applied the ADM to understand the 2nd-order differential condition with initial conditions. Marasi and Nikbakht [31] associated the ADM to get the arrangements of some eigenvalue problems of 2nd- and 4th-orders and exhibited the convergent of the arrangement. The ADM was a prevailing method to study the estimated root of a non-linear equation as an infinite series that typically converges to the precise root. Nhawu and Mushanyn [32] used the ADM as a prevailing method to study the projected explanation of a non-linear equation as a countless series which typically joins to the exact solution. It was exposed that the series solutions converge to the solution for each problem. This method was proposed to solve eigenvalue problems approximately. ADM was used by Olga and Zdenek [33] to explain the singular initial value problems. They settled the 2nd-order differential condition utilizing Adomian strategy. That was a great contribution in the field of numerical analysis to solve such problems by concern method.

ADM was established by Singh and Kumar [34], which explained the higher order BVPs. The nonlinear system of fractional DEs which appear in a model of HIV infection of CD4+T cells were established by Sefidgar et al. [35] and Laplace Adomian method was used for solving this system. A system of linear and nonlinear integral algebraic equations (IAEs) of Hessenberg type was presented by Shiri [36]. Convergence analysis of the discontinuous collocation methods was investigated for the large class of IAEs based on the new definitions. The generalization of the DTM was established by Shiri [37] to solve the integro-differential equation.

Wazwaz [38] employed ADM to provide an explanation for the 5th-order BVPs. In ref. [39] the variation iterative method was used for solving linear and nonlinear ODEs and logical models with steady constants. Unique Lagrange multiplier was utilized such type of ODE. The ordinary differential conditions with variable constants show up in several parts of functional sciences. The ADM, the homotopy-perturbation, and variationally iteration method, were samples of recently developed methods. The remaining of this article is sorted out as follows. The development of ADM is presented in Section 2. In Section 3, the development of application of ADM on 14th-order BVP is introduced. Results and discussions are given in Section 4. Likewise, a few problems are reasoned right now to reveal the effectiveness of the ADM. The exactness of this method for detailed investigation is equated with the precise solution and conveyed through tables. At last, the concluding comments are given in Section 5.

2 ADM

Consider ODE

(1) Lw + Rw + Nw = g ( t ) ,

R is the linear differential operator, L is called the operator that is the highest order derivatives, and N is the nonlinear differential operator.

(2) Lw = g ( t ) − Rw − Nw .

Taking L −1 on both sides of the above equation, we get

L − 1 Lw ( t ) = L − 1 g ( t ) − L − 1 Rw − L − 1 Nw ,

w ( t ) = L − 1 g ( t ) − L − 1 Rw − L − 1 Nw ,

(3) w ( t ) = f ( t ) − L − 1 Rw − L − 1 Nw ,

where f(t) denotes the function made by integrating g(t). The unknown function may be inscribed in the form of infinite series.

(4) w ( t ) = ∑ n = 0 ∞ w n ( t ) .

The nonlinear span was stated in the form of an infinite series of the Adomian polynomials and is inscribed in the following form:

(5) Nw = ∑ n = 0 ∞ A n .

where A n denotes the Adomian polynomials.

A n = 1 n ! d n d λ n N ∑ n = 0 ∞ λ j w j λ = 0 .

Substituting (3) and (4) in Eq. (5), we obtain the following equation:

∑ n = 0 ∞ w n = f ( t ) − L − 1 ∑ n = 0 ∞ R w n − L − 1 ∑ n = 0 ∞ A n .

Observing the above equations, we have the components w 0 , w 1 , w 2 , … in recursive relation. The ADM classifies the zeroth element that we have from boundary conditions at x = 0.

w 0 = f ( t ) ,

w n + 1 = L − 1 R w n + L 1 A n .

We compute A n for nonlinear operator

A 0 = F ( w 0 ) ,

A 1 = w 1 F ( w 0 ) ,

A 2 = w 2 F ( 1 ) + w 1 2 2 ! F ( 2 ) ( w 0 ) .

We can calculate the component w 0 , w 1 , w 2 … by Adomian method and the solution will be

w = lim x → ∞ ϕ n .

We describe n-term estimation to the root of “w” as follows:

ϕ n = ∑ j = 0 n − 1 w j .

2.1 Adomian polynomials

ADM u(t) is the series solution given by the sum of components

u ( t ) = ∑ n = 0 ∞ u n ( t ) .

Nonlinear operator N is decomposed by

N ( y ) = ∑ n = 0 ∞ A n .

A n is called the Adomian polynomials and are formed for each and every nonlinearity in such a way that A 0 has dependance only on y 0, A 1 has dependance only on y 0 and y 1, A 2 has dependance on y 0, y 1, y 2, etc. Different approaches are used for different functions.

A 0 = f ( y 0 ) , A 1 = f ( 1 ) ( y 0 ) y 1 , A 2 = y 2 f ( 1 ) ( y 0 ) + y 1 2 2 ! f ( 2 ) ( y 0 ) , . . . A n = A n − 1 ( y 0 , y 1 , … , y n − 1 ) .

3 Application of ADM on 14th-order BVP

Here we will consider the 14th-order BVP of the form given below:

(6) u ( 14 ) = f ( x ) + g ( y ) , 0 < x < p ,

with the boundary conditions

u ( 0 ) = b 0 , u ( 1 ) ( 0 ) = b 1 , u ( 2 ) ( 0 ) = b 2 , u ( 3 ) ( 0 ) = b 3 ,

u ( 4 ) ( 0 ) = b 4 , u ( 5 ) ( 0 ) = b 5 , u ( 6 ) ( 0 ) = b 6 , u ( p ) = c 0 ,

(7) u ( 1 ) ( p ) = c 1 , u ( 2 ) ( p ) = c 2 , u ( 3 ) ( p ) = c 3 , u ( 4 ) ( p ) = c 4 , u ( 5 ) ( p ) = c 5 , u ( 6 ) ( p ) c 6 ,

where the function g(y) is a linear, nonlinear, and continuous function on the given interval [0, p] and f(x) is a basis term function.

b i , i = 0, 1,…,6. c i , i = 0, 1,…,6.

Now, we can inscribe Eq. (6) in the operator form as follows:

(8) Lu = f ( x ) + g ( y ) ,

where L is the differential operator

(9) L = d 14 d x 14 ( . ) .

Therefore, L −1 is the 14th integral operator

(10) L − 1 ( . ) = ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ( . ) d t d t d t d t d t d t d t d t d t d t d t d t d t d t .

Applying L −1, we have

u ( t ) = A + Bt + C t 2 2 ! + D t 3 3 ! + E t 4 4 ! + F t 5 5 ! + G t 6 6 ! + a t 7 7 ! + b t 8 8 ! + c t 9 9 ! + d t 10 10 ! + e t 11 11 ! + f t 12 12 ! + g t 13 12 ! + L − 1 [ f ( x ) + g ( y ) ] .

Applying the boundary condition at x = 0, we get

u 0 ( t ) = A + Bt + C t 2 2 ! + D t 3 3 ! + E t 4 4 ! + F t 5 5 ! + G t 6 6 ! + a t 7 7 ! + b t 8 8 ! + c t 9 9 ! + d t 10 10 ! + e t 11 11 ! + f t 12 12 ! + g t 13 13 ! + L − 1 [ f ( x ) ] ,

where constants a, b, c, d, e, f, and g can be determined by using boundary conditions at x = p.

The ADM directs towards the solution u(t) by decomposition series of components

(11) u ( t ) = ∑ n = 0 ∞ u n ( t ) .

The nonlinear term g(y) is taken in the form of infinite series of the Adomian polynomials and can be written in a form as follows

(12) g ( y ) = ∑ n = 0 ∞ A n ,

where u n (t) will be determined by recursive relation and An are called Adomian polynomials, substituting Eqs. (9) and (10) in Eq. (8) we get the following relation:

(13) ∑ n = 0 ∞ u n = A + Bt + C t 2 2 ! + D t 3 3 ! + E t 4 4 ! + F t 5 5 ! + G t 6 6 ! + a t 7 7 ! + b t 8 8 ! + c t 9 9 ! + d t 10 10 ! + e t 11 11 ! + e t 12 12 ! + f t 13 13 ! + L − 1 [ f ( x ) ] + L − 1 [ ∑ n = 0 ∞ A n ] .

u n (t) recurrence relation will be used

(14) u 0 ( t ) = A + Bt + C t 2 2 ! + D t 3 3 ! + E t 4 4 ! + F t 5 5 ! + G t 6 6 ! + a t 7 7 ! + b t 8 8 ! + c t 9 9 ! + d t 10 10 ! + e t 11 11 ! + e t 12 12 ! + f t 13 13 ! + L − 1 [ f ( x ) ] ,

(15) u k + 1 = L − 1 ∑ n = 0 ∞ A K , k .

Here boundary condition at x = p is applied to find the coefficients a, b, c, d, e, and f. To apply the above discussed method, two numerical examples are restrained in Section 4.

4 Results and discussion

The above-defined methods are applied on two examples where one example is linear 14th-order BVP and the other is nonlinear 14th-order BVP and the results accomplished are appropriately exact up to nine-decimal places as displayed in tables that shows the authenticity of the built-up process.

4.1 Problem 1

Consider the following 14th-order DE:

(16) u ( 14 ) ( t ) = e − t u ( t ) ,

with the given boundary conditions

u ( 0 ) = 1 , u ( 1 ) ( 0 ) = 1 , u ( 2 ) ( 0 ) = 1 , u ( 3 ) ( 0 ) = 1 , u ( 4 ) ( 0 ) = 1 ,

u ( 5 ) ( 0 ) = 1 , u ( 6 ) ( 0 ) = 1 , u ( 1 ) = e , u ( 1 ) ( 1 ) = e , u ( 2 ) ( 1 ) = e ,

(17) u ( 3 ) ( 1 ) = e , u ( 4 ) ( 1 ) = e , u ( 5 ) ( 1 ) = e , u ( 6 ) ( 1 ) = e ,

having the exact solution

L ( u ) = e − t u ( t ) ,

where we have the differential operator “L” from Eq. (9).

u ( t ) = A + Bt + C t 2 2 ! + D t 3 3 ! + E t 4 4 ! + F t 5 5 ! + G t 6 6 ! + a t 7 7 ! + b t 8 8 ! + c t 9 9 ! + d t 10 10 ! + e t 11 11 ! + f t 12 12 ! + g t 13 13 ! + L − 1 [ e − t u ] ,

u ( 2 ) ( t ) = C + Dt + E t 2 2 ! + F t 3 3 ! + G t 4 4 ! + a t 5 5 ! + b t 6 6 ! + c t 7 7 ! + d t 8 8 ! + e t 9 9 ! + f t 10 10 ! + g t 11 11 ! ,

u ( 3 ) ( t ) = D + Et + F t 2 2 ! + G t 3 3 ! + a t 4 4 ! + b t 5 5 ! + c t 6 6 ! + d t 7 7 ! + e t 8 8 ! + f t 9 9 ! + g t 10 10 ! ,

u ( 4 ) ( t ) = E + Ft + G t 2 2 ! + a t 3 3 ! + b t 4 4 ! + c t 5 5 ! + d t 6 6 ! + e t 7 7 ! + f t 8 8 ! + g t 9 9 ! ,

u ( 5 ) ( t ) = F + Gt + a t 2 2 ! + b t 3 3 ! + c t 4 4 ! + d t 5 5 ! + e t 6 6 ! + f t 7 7 ! + g t 8 8 ! ,

u ( 6 ) ( t ) = G + at + b t 2 2 ! + c t 3 3 ! + d t 4 4 ! + e t 5 5 ! + f t 6 6 ! + g t 7 7 ! ,

u ( 7 ) ( t ) = a + bt + c t 2 2 ! + d t 3 3 ! + e t 4 4 ! + f t 5 5 ! + g t 6 6 ! .

using the given boundary conditions in Eq. (17), we get

(18) u ( t ) = 1 + t + t 2 2 ! + t 3 3 ! + t 4 4 ! + t 5 5 ! + t 6 6 ! + a t 7 7 ! + b t 8 8 ! + c t 9 9 ! + d t 10 10 ! + e t 11 11 ! + f t 12 12 ! + g t 13 13 ! + L − 1 [ e − t u ] .

applying the decomposition method on Eq. (18), we have the following expression:

∑ n = 0 ∞ u n ( t ) = 1 + t + t 2 2 ! + t 3 3 ! + t 4 4 ! + t 5 5 ! + t 6 6 ! + a t 7 7 ! + b t 8 8 ! + c t 9 9 ! + d t 10 10 ! + e t 11 11 ! + f t 12 12 ! + g t 13 13 ! + L − 1 ∑ n = 0 ∞ e − t u k ,

u k = 1 + t + t 2 2 ! + t 3 3 ! + t 4 4 ! + t 5 5 ! + t 6 6 ! + a t 7 7 ! + b t 8 8 ! + c t 9 9 ! + d t 10 10 ! + e t 11 11 ! + f t 12 12 ! + g t 13 13 ! .

Now, we have the recursive relation from the above equation

u k + 1 = L − 1 ∑ n = 0 ∞ e − t u k ,

u 1 = L − 1 ∑ n = 0 ∞ e − t ( 1 + t + t 2 2 ! + t 3 3 ! + t 4 4 ! + t 5 5 ! + t 6 6 ! + a t 7 7 ! + b t 8 8 ! + c t 9 9 ! + d t 10 10 ! + e t 11 11 ! + f t 12 12 ! + g t 13 13 ! ,

u 1 = L − 1 ∑ n = 0 ∞ e − t u 0 ,

u 1 = L − 1 ∑ n = 0 ∞ e − t 1 + t + t 2 2 ! + t 3 3 ! + t 4 4 ! + t 5 5 ! + t 6 6 ! + a t 7 7 ! + b t 8 8 ! + c t 9 9 ! + d t 10 10 ! + e t 11 11 ! + f t 12 12 ! + g t 13 13 ! ,

where we already know that

e − t = 1 − t + t 2 2 ! − t 3 3 ! + t 4 4 ! − … .

Then, we have

(19) e − t u 0 = 1 − t + t 2 2 ! − t 3 3 ! + t 4 4 ! − . . . 1 + t + t 2 2 ! + t 3 3 ! + t 4 4 ! + t 5 5 ! + t 6 6 ! + a t 7 7 ! + b t 8 8 ! + c t 9 9 ! + d t 10 10 ! + e t 11 11 ! + f t 12 12 ! + g t 13 13 ! = 1 − 1 5 , 040 a t 7 t + 1 10 , 080 a t 7 t 2 − 1 30 , 240 a t 7 t 3 + 1 1 , 20 , 960 a t 7 t 4 − 1 604 , 800 a t 7 t 5 − 1 40 , 320 b t 8 t + 1 80 , 640 b t 8 t 2 − 1 241 , 920 b t 8 t 3 + 1 967 , 680 b t 8 t 4 − 1 4 , 838 , 400 b t 8 t 5 − 1 362 , 880 c t 9 t + 1 725 , 760 c t 9 t 2 − 1 2 , 177 , 280 c t 9 t 3 + 1 8 , 709 , 120 c t 9 t 4 − 1 43 , 545 , 600 c t 9 t 5 − 1 3 , 628 , 800 d t 10 t + 1 7 , 257 , 600 d t 10 t 2 − 1 21 , 772 , 800 d t 10 t 3 + 1 87 , 091 , 200 d t 10 t 4 − 1 43 , 54 , 56 , 000 d t 10 t 5 − 1 39 , 916 , 800 e t 11 t + 1 79 , 833 , 600 e t 11 t 2 − 1 239 , 500 , 800 e t 11 t 3 + 1 958 , 003 , 200 e t 11 t 4 − 1 4 , 790 , 016 , 000 e t 11 t 5 − 1 479 , 001 , 600 f t 12 t + 1 958 , 003 , 200 f t 12 t 2 − 1 2 , 874 , 009 , 600 f t 12 t 3 + 1 11 , 496 , 038 , 400 f t 12 t 4 − 1 57 , 480 , 192 , 000 f t 12 t 5 − 1 6 , 227 , 020 , 800 g t 13 t + 1 1 , 245 , 041 , 600 g t 13 t 2 + 1 37 , 362 , 124 , 800 g t 13 t 3 + 1 149 , 448 , 499 , 200 g t 13 t 4 − 1 747 , 242 , 496 , 000 g t 13 t 5 − 1 720 t 7 1 2 , 880 t 8 − 1 4 , 320 t 9 − 1 86 , 400 t 10 − 1 86 , 400 t 11 + 1 6 , 227 , 020 , 800 g t 13 1 479 , 001 , 600 f t 12 + 1 39 , 916 , 800 e t 11 + 1 362 , 880 c t 9 + 1 40 , 320 b t 8 + 1 5 , 040 a t 7 − 1 720 t 6 .

Now, applying L −1 operator on Eq. (19), we have

(20) L − 1 1 − 1 5 , 040 a t 7 t + 1 10 , 080 a t 7 t 2 − 1 30 , 240 a t 7 t 3 + 1 120 , 960 a t 7 t 4 − 1 604 , 800 a t 7 t 5 − 1 40 , 320 b t 8 t + 1 80 , 640 b t 8 t 2 − 1 241 , 920 b t 8 t 3 + 1 967 , 680 b t 8 t 4 − 1 4 , 838 , 400 b t 8 t 5 − 1 362 , 880 c t 9 t + 1 725 , 760 c t 9 t 2 − 1 2 , 177 , 280 c t 9 t 3 + 1 8 , 709 , 120 c t 9 t 4 − 1 43 , 545 , 600 c t 9 t 5   − 1 3 , 628 , 800 d t 10 t + 1 7 , 257 , 600 d t 10 t 2 − 1 21 , 772 , 800 d t 10 t 3 + 1 87 , 091 , 200 d t 10 t 4 − 1 435 , 456 , 000 d t 10 t 5 − 1 39 , 916 , 800 e t 11 t + 1 79 , 833 , 600 e t 11 t 2 − 1 239 , 500 , 800 e t 11 t 3 + 1 958 , 003 , 200 e t 11 t 4 − 1 4 , 790 , 016 , 000 e t 11 t 5 − 1 479 , 001 , 600 f t 12 t + 1 958 , 003 , 200 f t 12 t 2 − 1 2 , 874 , 009 , 600 f t 12 t 3 + 1 11 , 496 , 038 , 400 f t 12 t 4 − 1 57 , 480 , 192 , 000 f t 12 t 5 − 1 6 , 227 , 020 , 800 g t 13 t + 1 1 , 245 , 041 , 600 g t 13 t 2 1 37 , 362 , 124 , 800 g t 13 t 3 + 1 149 , 448 , 499 , 200 g t 13 t 4 − 1 747 , 242 , 496 , 000 g t 13 t 5 − 1 720 t 7 1 2 , 880 t 8 − 1 4 , 320 t 9 − 1 86 , 400 t 10 − 1 86 , 400 t 11 + 1 6 , 227 , 020 , 800 g t 13 1 479 , 001 , 600 f t 12 + 1 39 , 916 , 800 e t 11 +   1 362 , 880 c t 9 + 1 40 , 320 b t 8 + 1 5 , 040 a t 7 − 1 720 t .

Now, on applying the inverse differential operator L − 1 from Eq. (10), Eq. (20) takes the form

(21) ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ( . ) d t d t d t d t d t d t d t d t d t d t d t d t d t d t − 1 33,574,047,712,837,632,000,000 t 25 − 1 14,772,580,993,648,558,080,000 t 24 − 1 307,762,104,034,344,960,000 t 23 − 1 80,285,766,269,829,120,000 t 22 − 1 7,298,706,024,529,920,000 t 21 − 1 2,432,902,008,176,640,000 t 20 + 1 19 1 32,267,963,476,869,120,000 a t 7 − 1 258,143,707,814,952,960,000 b t 8 − 1 2,323,293,370,334,576,640,000 c t 9 − 1 23,232,933,703,345,766,400,000 d t 10 − 1 255,562,270,736,803,430,400,000 e t 11 − 1 3,066,747,248,841,641,164,800,000 f t 12 − 1 39,867,714,234,941,335,142,400,000 g t 13 t 19 + 1 18 1 1,792,664,637,603,840,000 a t 7 + 1 14,341,317,100,830,720,000 b t 8 + 1 129,071,853,907,476,480,000 c t 9 + 1 1,290,718,539,074,764,800,000 d t 10 + 1 14,197,903,929,822,412,800,000 e t 11 + + 1 170,374,847,157,868,953,600,000 f t 12 + 1 2,214,873,013,052,296,396,800,000 g t 13 t 18 + 1 17 1 105,450,861,035,520,000 a t 7 − 1 843,606,888,284,160,000 b t 8 − 1 7,592,461,994,557,440,000 c t 9 − 1 75,924,619,945,574,400,000 d t 10 − 1 835,170,819,401,318,400,000 e t 11 − 1 10,022,049,832,815,820,800,000 f t 12 − 1 130,286,647,826,605,670,400,000 g t 13 t 17 + 1 16 1 6,590,678,814,720,000 a t 7 + 1 52,725,430,517,760,000 b t 8 + 1 474,528,874,659,840,000 c t 9 + 1 4745,288,746,598,400,000 d t 10 − 1 52,198,176,212,582,400,000 e t 11 + 1 626,378,114,550,988,800,000 f t 12 + 1 8,142,915,489,162,854,400,000 g t 13 t 16 + 1 15 − 1 439,378,587,648,000 a t 7 − 1 3,515,028,701,184,000 b t 8 − 1 31,635,258,310,656,000 c t 9 − 1 316,352,583,106,560,000 d t 10 − 1 3,479,878,414,172,160,000 e t 11 − 1 41,758,540,970,065,920,000 f t 12 − 1 542,861,032,610,856,960,000 g t 13 t 5 + 1 14 1 6,227,020,800 + 1 38,775,788,043,632,640,000 g t 13 + 1 2,982,752,926,433,280,000 f t 12 + 1 248,562,743,869,440,000 e t 12 + 1 22,596,613,079,040,000 d t 10 + 1 2,259,661,307,904,000 c t 9 + 1 251,073,478,656,000 b t 8 + 1 31,384,184,832,000 a t 7 t 14 .

Applying boundary condition at x = 1

e = 2 . 7181 + 1 7 ! a + 1 8 ! b + 1 9 ! c + 1 10 ! d + 1 11 ! e + 1 12 ! f + 1 13 ! g ,

1.81828 × 1 0 − 4 = 1 7 ! a + 1 8 ! b + 1 9 ! c + 1 10 ! d + 1 11 ! e + 1 12 ! f + 1 13 ! g ,

1.81828 × 1 0 − 4 = 1 6 ! a + 1 7 ! b + 1 8 ! c + 1 9 ! d + 1 10 ! e + 1 11 ! f + 1 12 ! g ,

1.81828 × 1 0 − 4 = 1 5 ! a + 1 6 ! b + 1 7 ! c + 1 8 ! d + 1 9 ! e + 1 10 ! f + 1 11 ! g ,

1.81828 × 1 0 − 4 = 1 4 ! a + 1 5 ! b + 1 6 ! c + 1 7 ! d + 1 8 ! e + 1 9 ! f + 1 10 ! g ,

1.81828 × 1 0 − 4 = 1 3 ! a + 1 4 ! b + 1 5 ! c + 1 6 ! d + 1 7 ! e + 1 8 ! f + 1 9 ! g ,

1.81828 × 1 0 − 4 = 1 2 ! a + 1 3 ! b + 1 4 ! c + 1 5 ! d + 1 6 ! e + 1 7 ! f + 1 8 ! g ,

1.81828 × 1 0 − 4 = a + 1 2 ! b + 1 3 ! c + 1 4 ! d + 1 5 ! e + 1 6 ! f + 1 7 ! g .

The values of the constants a, b, c, d, e, f, and g are as follows:

a = 1 . 00000000422 , b = 0 . 99999985263 , c = 0 . 8956666623452 , d = 0 . 8464592356

e = 0 . 7956203689 , f = 0 . 7785632013 , g = 0 . 72794632112

By considering two components

u ( t ) = u 0 ( t ) + u 1 ( t ) ,

and substituting the values, finally the series can be written as follows:

(22) u ( t ) = 1 + t + t 2 2 ! + t 3 3 ! + t 4 4 ! + t 5 5 ! + t 6 6 ! + a t 7 7 ! + b t 8 8 ! + c t 9 9 ! + d t 10 10 ! + e t 11 11 ! + f t 12 12 ! + g t 13 13 ! − 1 33,574,047,712,837,632,000,000 t 25 − 1 14,772,580,993,648,558,080,000 t 24 − 1 307,762,104,034,344,960,000 t 23 − 1 80,285,766,269,829,120,000 t 22 − 1 7,298,706,024,529,920,000 t 21 − 1 2,432,902,008,176,640,000 t 20 + 1 19 ( 1 32,267,963,476,869,120,000 a t 7 − 1 258,143,707,814,952,960,000 b t 8 − 1 2,323,293,370,334,576,640,000 c t 9 − 1 23,232,933,703,345,766,400,000 d t 10 − 1 255,562,270,736,803,430,400,000 e t 11 − 1 3,066,747,248,841,641,164,800,000 f t 12 − 1 39,867,714,234,941,335,142,400,000 g t 13 ) t 19 + 1 18 ( 1 1,792,664,637,603,840,000 a t 7 + 1 14,341,317,100,830,720,000 b t 8 + 1 129,071,853,907,476,480,000 c t 9 + 1 1,290,718,539,074,764,800,000 d t 10 + 1 14,197,903,929,822,412,800,000 e t 11 + 1 170,374,847,157,868,953,600,000 f t 12 + 1 2,214,873,013,052,296,396,800,000 g t 13 ) t 18 + 1 17 ( 1 105,450,861,035,520,000 a t 7 − 1 843,606,888,284,160,000 b t 8 − 1 7,592,461,994,557,440,000 c t 9 − 1 75,924,619,945,574,400,000 d t 10 − 1 835,170,819,401,318,400,000 e t 11 − 1 10,022,049,832,815,820,800,000 f t 12 − 1 130,286,647,826,605,670,400,000 g t 13 ) t 17 + 1 16 1 6,590,678,814,720,000 a t 7 + 1 5,272,543,0517,760,000 b t 8 + 1 474,528,874,659,840,000 c t 9 + 1 4,745,288,746,598,400,000 d t 10 + 1 52,198,176,212,582,400,000 e t 11 + 1 626,378,114,550,988,800,000 f t 12 + 1 8,142,915,489,162,854,400,000 g t 13 t 16 + 1 15 − 1 439,378,587,648,000 a t 7 − 1 3,515,028,701,184,000 b t 8 − 1 31,635,258,310,656,000 c t 9 − 1 316,352,583,106,560,000 d t 10 − 1 3,479,878,414,172,160,000 e t 11 − 1 41,758,540,970,065,920,000 f t 12 − 1 542,861,032,610,856,960,000 g t 13 t 5 + 1 14 1 6,227,020,800 + 1 38,775,788,043,632,640,000 g t 13 + 1 2,982,752,926,433,280,000 f t 12 + 1 248,562,743,869,440,000 e t 12 + 1 22,596,613,079,040,000 d t 10 + 1 2,259,661,307,904,000 c t 9 + 1 251,073,478,656,000 b t 8 + 1 31,384,184,832,000 a t 7 t 14 .

The Eq. (22) takes the form

(23) u ( t ) = 1 + t + t 2 2 ! + t 3 3 ! + t 4 4 ! + t 5 5 ! + t 6 6 ! + 1 . 984126942 × 1 0 − 04 t 7 + 2 . 480158365 × 1 0 − 05 t 8 + 2 . 468217213 × 1 0 − 06 t 9 + 1 . 993196772 × 1 0 − 08 t 11 + 1 . 625387475 × 1 0 − 09 t 12 + 4 . 4847911959 × 1 0 − 11 t 13 + [ 1 . 14707456 × 1 0 − 11 + 1 . 340944141 × 1 0 − 21 t 13 + 1 . 064440623 × 1 0 − 20 t 12 + 2 . 675683023 × 1 0 − 18 t 10 + 17312298 + 2 . 831229174 × 1 0 − 17 t 9 + 2 . 844926565 × 1 0 − 16 t 8 + 2 . 275941596 × 1 0 − 15 t 7 ] t 14 .

Now, we have compared the solution using ADM with exact solution. The outcome is specified in Table 1.

Table 1

Algebraic assessment

T Precise solution ADM solution Absolute error
0.0 1.00000 1.00000 0.0000
0.1 1.1051709181 1.1051642 1.0 × 10−1
0.2 1.2214027582 1.22139875 4.0 × 10−6
0.3 1.349858807 1.34985804 7.6 × 10−7
0.4 1.4918246976 1.491811322 1.3 × 10−4
0.5 1.64872127 1.648716379 4.8 × 10−6
0.6 1.8221188004 1.82211835 4.5 × 10−7
0.7 2.0137527075 2.013741769 1.0 × 10−5
0.8 2.225540928 2.225440482 1.0 × 10−4
0.9 2.4596031112 2.459610745 7.6 × 10−6
1.0 2.7182818285 2.52704044 1.9 × 10−1

Exact solution Cubic non-polynomial solution Cubic polynomial solution Cubic non-polynomial absolute error Cubic polynomial absolute error Cubic non-polynomial relative error Cubic polynomial reletive error
0.2 1.2214027581 1.2214020778 1.2214020778 6.80 × 10−7 3.71 × 10−4 5.57 × 10−7 3.04 × 10−4
0.4 1.4918246976 1.4918236132 1.4918236132 1.08 × 10−6 5.92 × 10−4 7.24 × 10−7 3.97 × 10−4
0.6 1.8221188003 1.8221176295 1.8221176295 1.17 × 10−6 6.35 × 10−4 6.42 × 10−7 3.48 × 10−4
0.8 2.2255409284 2.2255400739 2.2255400739 8.54 × 10−7 4.59 × 10−4 3.84 × 10−7 2.06 × 10−4
The results obtained for Problem 4.1 are also compared with that in the previous study [40] in the table below at h = 0.5.

4.2 Problem 2

Consider the following 14th-order DE:

(24) u ( 14 ) ( t ) = e t u 2 ( t ) ,

with the given boundary conditions

(25) u ( 0 ) = 1 , u ( 1 ) ( 0 ) = 1 , u ( 2 ) ( 0 ) = 1 , u ( 3 ) ( 0 ) = 1 , u ( 4 ) ( 0 ) = 1 , u ( 5 ) ( 0 ) = 1 , u ( 6 ) ( 0 ) = 1 , u ( 1 ) = e , u ( 1 ) ( 1 ) = e , u ( 2 ) ( 1 ) = e , u ( 3 ) ( 1 ) = e , u ( 4 ) ( 1 ) = e , u ( 5 ) ( 1 ) = e , u ( 6 ) ( 1 ) = e ,

having the exact solution

L ( u ) = e t u 2 ( t ) ,

where we have the differential operator “L” from Eq. (9)

(26) = a + bt + c t 2 2 ! + d t 3 3 ! + e t 4 4 ! + f t 5 5 ! + g t 6 6 ! + a t 7 7 ! + b t 8 8 ! + c t 9 9 ! + d t 10 10 ! + e t 11 11 ! + f t 12 12 ! + g t 13 13 ! + L − 1 [ e − t u ] .

Using the boundary condition in Eq. (26), we get the form

(27) u ( t ) = 1 + t + t 2 2 ! + t 3 3 ! + t 4 4 ! + t 5 5 ! + t 6 6 ! + a t 7 7 ! + b t 8 8 ! + c t 9 9 ! + d t 10 10 ! + e t 11 11 ! + f t 12 12 ! + g t 13 13 ! + L − 1 [ e − t u ] .

Now, applying the decomposition method on Eq. (27), we have

∑ n = 0 ∞ u n ( t ) = 1 + t + t 2 2 ! + t 3 3 ! + t 4 4 ! + t 5 5 ! + t 6 6 ! + a t 7 7 ! + b t 8 8 ! + c t 9 9 ! + d t 10 10 ! + e t 11 11 ! + f t 12 12 ! + g t 13 13 ! + L − 1 ∑ n = 0 ∞ e − t u k ,

(28) u k = 1 + t + t 2 2 ! + t 3 3 ! + t 4 4 ! + t 5 5 ! + t 6 6 ! + a t 7 7 ! + b t 8 8 ! + c t 9 9 ! + d t 10 10 ! + e t 11 11 ! + f t 12 12 ! + g t 13 13 ! .

The recursive relation of the above Eq. (28) is

u k + 1 = L − 1 ∑ n = 0 ∞ e t u 2 k ,

u 1 = L − 1 ∑ n = 0 ∞ e t ( 1 + t + t 2 2 ! + t 3 3 ! + t 4 4 ! + t 5 5 ! + t 6 6 ! + a t 7 7 ! + b t 8 8 ! + c t 9 9 ! + d t 10 10 ! + e t 11 11 ! + f t 12 12 ! + g t 13 13 ! ) 2 ,

u 1 = L − 1 ∑ n = 0 ∞ e t u 2 0 ,

u 1 = L − 1 ∑ n = 0 ∞ e t ( 1 + t + t 2 2 ! + t 3 3 ! + t 4 4 ! + t 5 5 ! + t 6 6 ! + a t 7 7 ! + b t 8 8 ! + c t 9 9 ! + d t 10 10 ! + e t 11 11 ! + f t 12 12 ! + g t 13 13 ! ) 2 .

we know that

e t = 1 + t + t 2 2 ! + t 3 3 ! + t 4 4 ! + … ,

(29) e t u 0 2 = 1 + 3 t + 1 518,400 t 13 + 1 207,360 t 14 + 1 3,110,400 t 15 + 1 4,828,336,128,000 b t 8 e t 11 t 3 + 1 73,156,608,000 b t 8 d t 10 t + 1 1,207,084,032,000 a t 7 f t 12 t + 1 18,289,152,000 a t 7 d t 10 t 2 + 1 3,950,456,832,000 c t 9 d t 10 t 3 + 1 6,778,983,923,712,000 c t 9 g t 13 t 3 + 1 609,638,400 a t 7 b t 8 t 3 + 1 5,214,603,018,240,000 d t 10 f t 1 2 t 3 + 1 57,940,033,536,000 b t 8 f t 12 t 3 + 1 658,409,472,000 c t 9 d t 10 t + 1 1,828,915,200 a t 7 c t 9 t 2 + 1 43,455,025,152,000 c t 9 e t 11 t 3 + 1 67,789,839,237,120,000 d t 10 g t 13 t 3 + 1 146,313,216,000 b t 8 d t 10 t 2 + 1 1,609,445,376,000 b t 8 e t 11 t 2 + 1 914,457,600 a t 7 c t 9 t + 1 11,298,306,539,520,000 d t 10 g t 13 t + 1 1,609,445,376,000 b t 8 e t 11 t 2 + 1 914,457,600 a t 7 c t 9 t 1 100,590,336,000 a t 7 e t 11 t + 1 7,242,504,192,000 c t 9 e t 11 t + 1 125,536,739,328,000 b t 8 g t 13 t + 1 11,298,306,539,520,000 d t 10 g t 13 t + 1 54,867,456,000 a t 7 d t 10 t 3 + 1 19,120,211,066,880,000 e t 11 f t 12 t 2 + 1 43,893,964,800 b t 8 c t 9 t 3 + 1 869,100,503,040,000 d t 10 f t 12 t + 1 173,820,100,608,000 c t 9 f t 12 t 2 + 1 1,129,830,653,952,000 c t 9 g t 13 t + 1 86,910,050,304,000 c t 9 f t 12 t + 1 1,316,818,944,000 c t 9 d t 10 t 2 + 1 144,850,083,840,000 d t 10 e t 11 t 2 + 1 201,180,672,000 a t 7 e t 11 t 2 + 1 22,596,613,079,040,000 d t 10 g t 13 t 2 + 1 9,656,672,256,000 b t 8 f t 12 t + 1 72,425,041,920,000 d t 10 e t 11 t + 1 94,152,554,496,000 a t 7 g t 13 t 3 + 1 9,560,105,533,440,000 e t 11 f t 12 t 1 521,460,301,824,000 c t 9 f t 12 t 3 + 1 57,360,633,200,640,000 e t 11 f t 12 t 3 + 1 124,281,371,934,720,000 e t 11 g t 13 t + 1 248 , 562 , 743 , 869 , 440 , 000 e t 11 g t 13 x 2 + 1 101 , 606 , 400 a t 7 b t 8 t + 1 745 , 688 , 231 , 608 , 320 , 000 e t 11 g t 13 t 3 + 1 1 , 491 , 376 , 463 , 216 , 640 , 000 f t 12 g t 13 t + 1 804 , 722 , 688 , 000 b t 8 e t 11 t + 1 2 , 982 , 752 , 926 , 433 , 280 , 000 f t 12 g t 13 t 2 + 1 8 , 948 , 258 , 779 , 299 , 840 , 000 f t 12 g t 13 t 3 + 1 14 , 485 , 008 , 384 , 000 c t 9 e t 11 t 2 + 1 251 , 073 , 478 , 656 , 000 b t 8 g t 13 t 2 + + 1 14 , 485 , 008 , 384 , 000 c t 9 e t 11 t 2 + 1 251 , 073 , 478 , 656 , 000 b t 8 g t 13 t 2 + 1 804 , 722 , 688 , 000 b t 8 t 11 t + 1 10 , 080 t b t 8 + 1 90 , 720 t c t 9 + 1 907 , 200 t d t 10 + 1 9 , 979 , 200 t e t 11 + 1 119 , 750 , 400 t f t 12 + 1 1556755200 t g t 1 3 + 1 14,370,048,000 t 5 g t 13 + 1 43,200 t 6 a t 7 + 1 345,600 t 6 b t 8 + 1 3,110,400 t 6 c t 9 + 1 31,104,000 t 6 d t 10 + 1 239,500,800 f t 12 + 1 19,958,400 e t 11 + 1 1,814,400 d t 10 + 1 181,440 c t 9 + 1 20,160 b t 8 + 1 2,520 a t 7 41 45 t 6 + 29 15 t 5 + 10 3 t 4 + 9 2 t 3 + 9 2 t 2 .

Now applying the inverse differential operator L − 1 from Eq. (10), Eq. (29) takes the form

(30) L − 1 ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ∫ 0 t ( . ) d t d t d t d t d t d t d t d t d t d t d t d t d t d t 1 7,006,757,783,548,723,200,000 c t 9 + 1 70,067,577,835,487,232,000,000 d t 10 + 1 770,743,356,190,359,552,000,000 e t 11 + 1 145,670,494,319,977,955,328,000,000 f t 12 + 1 1,893,716,426,159,713,419,264,000,000 g t 13 ) t 22 + 1 14 ( 1 1,428,743,424,166,223,413,248,000,000 f t 2 4 + 1 241,457,638,684,091,756,838,912,000,000 g t 26 + 1 158,176,291,553,280,000 a t 14 1 101,23,282,659,409,920,000 b t 16 + 1 819,985,895,412,203,520,000 c t 18 1 81,998,589,541,220,352,000,000 d t 20 + 1 23 ( 1 90,107,481,784,320,000 + 1 269760,174,666,625,843,200,000 b t 8 + 1 2,427,841,571,999,632,588,800,000 c t 9 + 1 24,278,415,719,996,325,888,000,000 d t 10 + 1 267,062,572,919,959,584,768,000,000 e t 11 + 1 3,204,750,875,039,515,017,216,000,000 + 1 41,661,761,375,513,695,223,808,000,000 g t 13 + 1 33,720,021,833,328,230,400,000 a t 7 ) t 23 + 1 22 ( 23 243,290,200,817,664,000 + 1 1,532,728,265,151,283,200,000 a t 7 + 1 12,261,826,121,210,265,600,000 b t 8 + 1 110,356,435,090,892,390,400,000 c t 9 + 1 1,103,564,350,908,923,904,000,000 d t 10 + 1 12,139,207,859,998,162,944,000,000 e t 11 + 1 145,670,494,319,977,955,328,000,000 f t 12 + 1 1,893,716,426,159,713,419,264,000,000 g t 13 ) t 22 + 1 22 ( 43 57,926,238,289,920,000 + 1 97,316,080,327,065,600,000 a t 7 + 1 778,528,642,616,524,800,000 b t 8 + 1 7,006,757,783,548,723,200,000 c t 9 + 1 70,067,577,835,487,232,000,000 d t 10 + 1 770,743,356,190,359,552,000,000 e t 11 + 1 145,670,494,319,977,955,328,000,000 f t 12 + 1 1,893,716,426,159,713,419,264,000,000 g t 13 ) t 22 + 1 14 ( 1 1,428,743,424,166,223,413,248,000,000 f t 2 4 + 1 241,457,638,684,091,756,838,912,000,000 g t 26 + 1 158,176,291,553,280,000 a t 14 + 1 10,123,282,659,409,920,000 b t 16 + 1 819,985,895,412,203,520,000 c t 18 1 81,998,589,541,220,352,000,000 d t 20 + 1 9,921,829,334,487,662,592,000,000 e t 22 + 1 632,705,166,213,120,000 a t 7 b t 8 + 1 5,694,346,495,918,080,000 a x 7 c x 9 + 1 56,943,464,959,180,800,000 a t 7 d t 1 0 + 1 626,378,114,550,988,800,000 a t 7 e t 1 1 + 1 7,516,537,374,611,865,600,000 a t 7 f t 1 2 + 1 97,714,985,869,954,252,800,000 a t 7 g t 1 3 + 1 450,992,242,476,711,936,000,000 d t 1 0 e t 1 1 + 1 5,411,906,909,720,543,232,000,000 d t 1 0 f t 1 2 + 1 124,281,371,934,720,000 e t 1 1 + 1 11,298,306,539,520,000 d t 1 0 + 1 1,129,830,653,952,000 c t 9 1 125,536,739,328,000 b t 8 + 1 15,692,092,416,000 a t 7 ) t 14 .

Applying boundary condition at x = 1

e = 2 . 7181 + 1 7 ! a + 1 8 ! b + 1 9 ! c + 1 10 ! d + 1 11 ! e + 1 12 ! f + 1 13 ! g ,

1 . 81828 × 1 0 − 4 = 1 7 ! a + 1 8 ! b + 1 9 ! c + 1 10 ! d + 1 11 ! e + 1 12 ! f + 1 13 ! g ,

1 . 81828 × 1 0 − 4 = 1 6 ! a + 1 7 ! b + 1 8 ! c + 1 9 ! d + 1 10 ! e + 1 11 ! f + 1 12 ! g ,

1 . 81828 × 1 0 − 4 = 1 5 ! a + 1 6 ! b + 1 7 ! c + 1 8 ! d + 1 9 ! e + 1 10 ! f + 1 11 ! g ,

1 . 81828 × 1 0 − 4 = 1 4 ! a + 1 5 ! b + 1 6 ! c + 1 7 ! d + 1 8 ! e + 1 9 ! f + 1 10 ! g ,

1 . 81828 × 1 0 − 4 = 1 3 ! a + 1 4 ! b + 1 5 ! c + 1 6 ! d + 1 7 ! e + 1 8 ! f + 1 9 ! g ,

1 . 81828 × 1 0 − 4 = 1 2 ! a + 1 3 ! b + 1 4 ! c + 1 5 ! d + 1 6 ! e + 1 7 ! f + 1 8 ! g ,

1 . 81828 × 1 0 − 4 = a + 1 2 ! b + 1 3 ! c + 1 4 ! d + 1 5 ! e + 1 6 ! f + 1 7 ! g .

To find out the constant a, b, c, d, e, f, and g, let us consider two components

u ( t ) = u 0 ( t ) + u 1 ( t ) ,

and substituting the values finally, the series can be written as follows:

u ( t ) = 1 + t + t 2 2 ! + t 3 3 ! + t 4 4 ! + t 5 5 ! + t 6 6 ! + 1 . 00000000422 7 ! t 7 + 0 . 99999985263 8 ! t 8 + 0 . 8956666623452 9 ! t 9 1 21,030,783,487,321,492,684,800,000 t 9 + 1 725,199,430,597,292,851,200,000 t 8 + 1 39,413,012,532,461,568,000,000 t 7 + 1 2,710,947,951,968,256,000,000 x 6 + 1 219,438,220,345,344,000,000 t 5 + 43 868,975,352,567,562,240,000 t 4 + 1 23 ( 1 90,107,481,784,320,000 + 0.99999985263 269,760,174,666,625,843,200,000 t 8 + 0.8956666623452 2,427,841,571,999,632,588,800,000 t 9 + 0.8645923564 24,278,415,719,996,325,888,000,000 t 10 + 0.7956203689 267,062,572,919,959,584,768,000,000 t 11 + 1 3,204,750,875,039,515,017,216,000,000 + 0.72794632112 41,661,761,375,513,695,223,808,000,000 t 13 + 1.00000000422 33,720,021,833,328,230,400,000 t 7 ) t 3 + 1 22 ( 23 243,290,200,817,664,000 + 1.00000000422 1,532,728,265,151,283,200,000 t 7 + 0.99999985263 12,261,826,121,210,265,600,000 t 8 + 0.8956666623452 110,356,435,090,892,390,400,000 t 9 + 0.8464592356 1,103,564,350,908,923,904,000,000 t 10 + 0.7956203689 12,139,207,859,998,162,944,000,000 t 11 + 0.7785632012 145,670,494,319,977,955,328,000,000 t 12 + 0.72794632112 1,893,716,426,159,713,419,264,000,000 t 13 ) t 2 + 1 22 ( 43 57,926,238,289,920,000 + 1.00000000422 97,316,080,327,065,600,000 t 7 + 0.99999985263 778,528,642,616,524,800,000 t 8 + 0.8956666623452 7,006,757,783,548,723,200,000 t 9 + 0.8464592356 70,067,577,835,487,232,000,000 t 10 + 0.7956203689 770,743,356,190,359,552,000,000 t 11 + 0.7785632013 145,670,494,319,977,955,328,000,000 t 12 + 0.72794632112 1,893,716,426,159,713,419,264,000,000 t 13 ) t 2 +

1 14 ( 0.7785632013 1,428,743,424,166,223,413,248,000,000 t 2 + 0.72794632112 241,457,638,684,091,756,838,912,000,000 t 6 + 1.00000000422 158,176,291,553,280,000 t 4 + 0.99999985263 10,123,282,659,409,920,000 t 6 + 0.8956666623452 819,985,895,412,203,520,000 t 8 + 0.84592356 81,998,589,541,220,352,000,000 t 10 + 0.7956203689 9,921,829,334,487,662,592,000,000 t 6 + 0.999999856 632,705,166,213,120,000 t 7 t 8 + 0.895666665 5,694,346,495,918,080,000 t 7 t 9 + 0.846459239 56,943,464,959,180,800,000 t 7 t 10 + 0.795620372 626,378,114,550,988,800,000 t 7 t 11 + 0.778563204 7,516,537,374,611,865,600,000 t 7 t 12 + 0.727946324 97,714,985,869,954,252,800,000 t 7 t 13 + 0.778563204 7,516,537,374,611,865,600,000 t 7 t 12 + 0.727946324 97,714,985,869,954,252,800,000 t 7 t 13 + 0.673460209 450,992,242,476,711,936,000,000 t 10 t 11 + 0.659022012 5,411,906,909,720,543,232,000,000 t 10 t 11 + 0.616176886 70,354,789,826,367,062,016,000,000 t 10 t 13 + 0 . 619440741 59,530,976,006,925,975,552,000,000 t 11 t 12 0 . 57916892 773,902,688,090,037,682,176,000,000 t 12 t 11 0 . 566752218 9,286,832,257,080,452,186,112,000,000 t 13 t 10 + 0 . 72794632112 19,387,894,021,816,320,000 t 13 + 0 . 7785632013 1,491,376,463,216,640,000 t 12 + 0 . 7956203689 124,281,371,934,720,000 t 11 + 0 . 8464592356 11,298,306,539,520,000 t 10 + 0 . 8956666623452 1,129,830,653,952,000 t 9 0 . 99999985263 125,536,739,328,000 t 8 + 1 . 00000000422 15,692,092,416,000 t 7 ) t 14 + 0 . 846459235 10 ! t 10 + 0 . 778563 11 ! t 11 + 0 . 7785632 12 ! t 12 .

Now we have compared our solution with precise solution. The outcome is specified in Table 2.

Table 2

Algebraic assessment

T Precise solution ADM solution Absolute error
0.0 1.00000 1.00000 0.00000
0.1 1.10517091707565 1.105170418 4.44 × 10−7
0.2 1.22140275816017 1.22149856463 9.5006 × 10−5
0.3 1.34985880757600 1.349856463 2.144 × 10−6
0.4 1.49182469764127 1.491824156 4.41 × 10−7
0.5 1.64872127070013 1.648715018 6.152 × 10−6
0.6 1.82211880029051 1.82211200 6.8 × 10−6
0.7 2.01375270747048 2.013734018 1.6089 × 10−5
0.8 2.22554092840247 2.222414468 2.0222 × 10−9
0.9 2.45960311115695 2.459496163 1.067485 × 10−4
1.0 2.71828182845905 2.718055456 2.26272444 × 10−4

5 Conclusion

ADM is an authentic technique to solve an extensive class of problems normally in a fast-convergent series solution. This method is numerically convenient and it is flexible to apply on wide variety of classifications of linear and nonlinear ODEs and PDEs, with substantial uses in various areas of daily life sciences. ADM, because of its numerous uses, has gotten countless researchers' consideration and has been effectively useful to apply on several problems of DEs, integral equations, and differential-integral equations. The key benefit of ADM is that it can be used promptly by devoiding any assumptions or alteration formula and the estimated numerical solution attained by ADM might be presented in terms of a speedily convergent power series with beneficially assessible terms.

ADM has been used to solve 14th-order ODE with boundary conditions. ADM is considered in extensive application, a modest scheming procedure, and a speedy convergence rate with no estimated conditions. The high accuracy calculation of the equation can be attained even with exact solutions. The technique is applied on two examples and the results accomplished are appropriately exact up to eight-decimal places as illustrated in the tables, indicating the authenticity of the built-up process. This work can be extended further to higher order BVPs like 15th-order and 16th-order for linear and nonlinear cases.

Acknowledgments

The authors are thankful for Research Supporting Project number (RSP2023R167), King Saud University, Riyadh, Saudi Arabia.

  1. Funding information: This Project is funded by King Saud University, Riyadh, Saudi Arabia.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2022-07-05
Revised: 2023-02-07
Accepted: 2023-02-20
Published Online: 2023-03-27

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2022-0236
Keywords for this article
nonlinear; linear; Adomian decomposition method; absolute errors; fourteenth order
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Articles in the same Issue

  1. Regular Articles
  2. Dynamic properties of the attachment oscillator arising in the nanophysics
  3. Parametric simulation of stagnation point flow of motile microorganism hybrid nanofluid across a circular cylinder with sinusoidal radius
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