Solutions of Nonlinear Integro-Differential Equations Using a Hybrid Adomian Decomposition Method with Modified Bernstein Polynomials

Mohammad Almousa; Suhaila Saidat; Ahmad Al-Hammouri; Sultan Alsaadi; Ghada Banihani

doi:10.5391/IJFIS.2024.24.3.271

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International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(3): 271-279

Published online September 25, 2024

https://doi.org/10.5391/IJFIS.2024.24.3.271

Solutions of Nonlinear Integro-Differential Equations Using a Hybrid Adomian Decomposition Method with Modified Bernstein Polynomials

Mohammad Almousa¹, Suhaila Saidat¹, Ahmad Al-Hammouri¹, Sultan Alsaadi², and Ghada Banihani¹

¹Department of Mathematics, College of Science and Technology, Irbid National University, Irbid, Jordan
²Mathematics Education Program, Faculty and Arts, Sohar University, Sohar, Oman

Correspondence to :
Mohammad Almousa (mohammadalmousa12@yahoo.com)

Received: December 17, 2023; Revised: August 8, 2024; Accepted: September 9, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Abstract
1. Introduction
2. Preliminaries
3. Description of the Method
4. Convergence of the Method
5. Examples
6. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

In various scientific disciplines, including mathematics, physics, chemistry, biology, and engineering, numerous challenges are modeled using linear and nonlinear integro-differential equations. Researchers have developed analytical methods to address these issues and to find effective solutions. This study explored the effectiveness of an Adomian decomposition method based on modified Bernstein polynomials to solve nonlinear first-order integro-differential equations. This hybrid method does not require any diminution presumptions or linearization to solve these types of equations, and the arrangement methodology is extremely straightforward with little emphasis on prompting a highly exact solution. This produced an extremely effective strategy among the alternative strategies. The performance of the proposed method was confirmed by comparing the exact and approximate solutions using examples. A comparison of the results shown in numerical tables demonstrates the practical applicability of this method. The computations were performed using the Maple software.

Keywords: Adomian decomposition method (ADM), Bernstein polynomials, Modified Bernstein polynomials, Integro-differential equations, Approximate solutions

1. Introduction

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Abstract
1. Introduction
2. Preliminaries
3. Description of the Method
4. Convergence of the Method
5. Examples
6. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

Recently, modified ADM has been successfully applied to linear and nonlinear problems in various fields, for example, differential equations [1, 2], multi-dimensional time fractional model of the Navier-Stokes equation [3], Van der Pol equation [4], fractional partial differential equations [5], nonlinear integral equations [6], three-dimensional Fredholm integral equations [7], heat and wave equations [8], Newell-Whitehead-Segel Equation [9], time-delay integral equations [10] and others as in [11–25]. Modified Bernstein polynomials were combined with the Adomian decomposition method (ADM) to solve the integro-differential equations. This study focuses on nonlinear integro-differentials of the type.

(1)dudx=f(x)+∫0xK(t,u(t),u′(t))dt,

where K(t, u(t), u’(t)) is the kernel function and f (x) is called the source-term. It should be noted that many methods and techniques have been developed for solving these types of equations. Examples include the variational iteration [26], Adomian decomposition and Tau Methods [27], homotopy analysis [28], ADM [29], Laplace transform-optimal homotopy asymptotic [30], Mahgoub transform [31], optimal homotopy asymptotic method [32], homotopy perturbation method [33], wavelet-Galerkin method [34], Sumudu and Elzaki integral transforms [35].

In this study, an ADM based on modified Bernstein polynomials was applied to solve nonlinear integro-differential equations. Our aim was to modify the ADM to provide approximate solutions for nonlinear first-order integro-differential equations. This hybrid method finds a solution without discretization or restrictive assumptions and avoids round-off errors. The fundamental advantage of this method is that it can be used specifically for all types of linear and nonlinear differential, integral, and integro-differential equations. First, we briefly present the definitions, properties, and notation of Bernstein polynomials. Herein, we describe our new idea. Furthermore, the convergence and maximum absolute errors of the method were presented. Herein, we discuss some examples of this phenomenon. Additionally, we performed a comparative study with other methods to test the accuracy of the proposed method.

2. Preliminaries

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Abstract
1. Introduction
2. Preliminaries
3. Description of the Method
4. Convergence of the Method
5. Examples
6. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

In the following section, we briefly present the basic definitions, most important properties, and notation of the Bernstein polynomials. Bernstein first used these polynomials in 1912 [36, 37].

Definition 1. A linear combination Bernstein based polynomial

(2)Bn(x)=∑i=0nBi,n(x)βi

is called the Bernstein polynomials, where x ∈ [0, 1], β_i are the Bernstein coefficients, and Bi,n(x)=n!i!(n-i)xi(1-x)n-i.

Definition 2. The nth Bernstein polynomial for f (x) can be written as

(3)Bn(f)=∑i=0nBi,n(x)f(in).

We are now giving the most important properties of these polynomials.

1. Non negativity.
2. Symmetry, so B_i,n(x) = B_n_–_i,n(1 – x).
3. Linearly, so B_n (α f ∓ βg) = αB_n ( f ) ∓ βB_n (g).
4. Each polynomial has only one maximum at x=in.
5. ∑i=0nBi,n(x)=Bn(1,x)=1.
6. For mathematical convenience, we write B_i,n(x) = 0 if i < 0 or i > n.
7. The derivative of the n-th degree was

(4)ddxBi,n(x)=n(Bi-1,n-1(x)-Bi,n-1(x)).
8. It can be written as

(5)Bi,n(x)=((1-x)Bi,n-1(x)+xBi-1,n-1(x)), 0≤i≤n.

Remark 1. The 2k-th order derivative f⁽²^k⁾ given by

(6)Bnf(x)=f(x)+∑a=22k-1f(a)(x)a!naTn,a(x)+O(1nk),

where

(7)Tn,a(x)=∑k(k-nx)a(nk)xk(1-x)n-k.

3. Description of the Method

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Abstract
1. Introduction
2. Preliminaries
3. Description of the Method
4. Convergence of the Method
5. Examples
6. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

This section describes the application of the modified ADM with a Bernstein polynomial for solving the nonlinear first-order integro-differential equations. We will rewrite Eq. (1) as follows:

(8)ℒu=f(x)+∫0xN(u)dt,

where ℒu=dudx and N(u) is a nonlinear term.

To solve this problem, we take ℒ⁻¹ such that ℒ-1(.)=∫0x.dx to both sides

(9)u(x)-u(0)=g(x)+ℒ-1∫0xN(u)dt,

where g(x)=∫0xf(t)dx.

The Adomian polynomial N(u) is as follows:

(10)N(u)=∫n=0∞An,

where An=1n!dndγn[N(∑i=0∞γiui)],n=0,1,2,….

Using Eqs. (3), (4), and (6), we obtain the following modified Bernstein series:

(11)f(x)=∑i=0n(ni)xi(1-x)n-ig(in)-∑a=22k-1(dadxa)Bi,n(x)a!naTn,a(x).

The solution u(x) is given by

(12)u(x)=∫j=0∞uj(x).

Substituting Eqs. (10), (11), and (12) into Eq. (9), we have

(13)∑j=0∞uj(x)= u(0)+∑i=0n(ni)xi(1-x)n-ig(in) -∑a=22k-1(dadxa)Bi,n(x)a!naTn,a(x)+ℒ-1∫0xAndt.

Therefore, the solutions of Eq. (1) are given by

(14)u0(x)= u(0)+ℒ-1∑i=0n(ni)xi(1-x)n-ig(in)-∑a=22k-1(dadxa)Bi,n(x)a!naTn,a(x),u1(x)= ℒ-1∫0xA0dt,u2(x)= ℒ-1∫0xA1dt,⋮ un+1(x)=ℒ-1∫0xAndt, n=0,1,2,….

4. Convergence of the Method

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Abstract
1. Introduction
2. Preliminaries
3. Description of the Method
4. Convergence of the Method
5. Examples
6. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

This section presents the convergence of the proposed method for solving nonlinear integro-differential equations. Furthermore, the maximum absolute error is determined.

Theorem 1. The series un(x)=∑j=0∞uj(x) of Eq. (1) converges if α ∈ [0, 1) and ||u_n₊₁(x)|| = α ||u_n(x)|| such that ||u₀(x)|| < ∞.

Proof. Let S _n = u₁(x) + u₂(x) + ... + u_n(x). For n ≤ m, we are going to prove that S _n is a Cauchy sequence.

‖Sm-Sn‖=‖∫i=0mui(x)-∫i=0nui(x)‖=∫i=n+1mui(x).

We have the following sequence

‖Sm-Sn‖ =‖(Sn+1-Sn)+(Sn+2-Sn+1)+(Sn+3-Sn+2)+⋯+(Sm-Sm-1)‖ ≤‖(Sn+1-Sn)‖+‖(Sn+2-Sn+1)‖+‖(Sn+3-Sn+2)‖+⋯+‖(Sm-Sm-1)‖ ≤αn+1‖(u0)‖+αn+2‖(u0)‖+αn+3‖(u0)‖+⋯+αm‖(u0)‖ ≤(αn+1+αn+2+αn+3+⋯+αm)‖(u0)‖ ≤αn+1(1+α+α2+α3⋯+αm-n-1) ‖(u0)‖ ≤αn+1(1-αm-n1-α) ‖(u0)‖.

Since α ∈ [0, 1) then 1 – α^m^–ⁿ < 1.

This in turn gives

‖Sm-Sn‖≤(αn+11-α) ‖(u0)‖.

But we know that ||u₀(x)|| < ∞, yields

‖Sm-Sn‖→0 as n→0.

Therefore, Theorem 1 is proven.

Theorem 2. The maximum absolute error of un(x)=∑j=0∞uj(x) is

max∀x∈J|u(x)-∑i=0nui(x)|≤(αn+11-α)max∀x∈J‖(u0)‖.

Proof. If m → ∞ then S _m → u(x), yields

‖u(x)-Sn‖≤(αn+11-α) ‖(u0)‖.

Therefore, we have

max∀x∈J|u(x)-∫i=0nui(x)|≤(αn+11-α)max∀x∈J‖(u0)‖.

Therefore, Theorem 2 is proven.

5. Examples

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Abstract
1. Introduction
2. Preliminaries
3. Description of the Method
4. Convergence of the Method
5. Examples
6. Conclusion
Conflict of Interest
Figures
Tables
References
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The main objective here is to solve some examples of nonlinear first-order integro-differential equations using the new modified method to demonstrate its accuracy.

Example 1. Consider the nonlinear integro-differential equation as follows:

(15)dudx=-1+∫0xu2(t)dt, x∈[0,1],

with boundary condition u(0) = 0 and exact solution u(x) = −x.

Introducing Eq. (10) into Eq. (15) and taking ℒ⁻¹ to both sides, yields

(16)u(x)-u(0)=ℒ-1(-1+∫0x∑n=0∞An(t)dt).

Combining Eq. (12) and Eq. (16), we have

(17)∑j=0∞uj(x)=ℒ-1(-1+∫0x∑n=0∞An(t)dt).

As suggested in Eqs. (11)–(14), when m = 6 and k = 2, the following solutions were obtained.

u0(x) =ℒ-1∑i=06(6i)xi(1-x)6-if(i6)-∑a=23(dadxa)Bi,6(x)a!6aT6,a(x) =-x(1-x)5-5x2(1-x)4-10x3(1-x)3-10x4(1-x)2-5x5(1-x)-x6=-x,u1(x) =∫0x∫0xA0dtdx=∫0x∫0xu02(t)dtdx=112x4,u2(x) =∫0x∫0xA1dtdx=∫0x∫0x2u0(t)u1(t)dtdx=-1252x7,u3(x) =∫0x∫0xA2dtdx=∫0x∫0x2u0(t)u2(t)+u12(t)dtdx=16048x10,u4(x) =∫0x∫0xA3dtdx=∫0x∫0x2u0(t)u3(t)+2u(t)u2(t)dtdx=-1157248x13,

and so on.

The solution of Eq. (15) becomes

(18)um(x) =∫j=0mui=u0(x)+u1(x)+u2(x)+… =-x+112x4-1252x7+16048x10-1157248x13+…

This converges to the exact solution u(x) = −x. Table 1 shows a comparison between the first fourth-order approximate solutions u₄(x) of Example 1 and the solutions in [29, 33, 34]. Figure 1 presents a numerical comparison of the proposed solution, u₄(x) and the exact solution. Figure 2 shows the absolute errors for Example 1.

Example 2. Consider the nonlinear integro-differential equation as follows:

(19)dudx=1+∫0xe-tu2(t)dt,

with boundary condition u(0) = 1 and exact solution u(x) = e^x.

Using Eq. (10) in Eq. (19) and ℒ⁻¹ on both sides, we obtain

∑j=0∞uj(x)=u(0)+ℒ-1(1+∫0x∑n=0∞An(t)dt).

As suggested in Eqs. (11)–(14) when m = 6 and k = 2, yields

u0(x)= u(0)+ℒ-1∑i=06(6i)xi(1-x)6-if(i6) -∑a=23(dadxa)Bi,6(x)a!6aT6,a(x)=1+x(1-x)5+5x2(1-x)4+10x3(1-x)3 +10x4(1-x)2+5x5(1-x)+x6=1+x,u1(x)= ∫0x∫0xA0dtdx=∫0x∫0xe-tu02(t)dtdx=-11+x2e-x+6xe-x+11e-x+5x,u2(x)= ∫0x∫0xA1dtdx=∫0x∫0x2e-tu0(t)u1(t)dtdx=-1394+394x+14e-x+5e-2xx2+714e-2xx +834e-2x+28xe-x+12x3e-2x+10x2e-x,u3(x)= ∫0x∫0xA2dtdx=1972(15267xe3x+43254x2e2x+7290x3ex +216x4-676073+17496xe2x+49572x2ex +3060x3-1458e2x+103275xex+17010x2 +26487ex+43212x+42578)e-3x,u4(x)= ∫0x∫0xA3dtdx=1248832(5536734xe4x+32077824x2e3x +12970368x3e2x+1105920x4ex+22464x5 -28049407e4x-38330368xe3x+53374464x2e2x +12128256x3ex+409824x4+14800896e3x +47138112xe2x+49489920x2ex+3090096x3 -46453824e2x+81899520xex+11923584x2 +41003008ex+23455582x+18699327)e-4x,

and so on. Therefore, the solution of Eq. (19) is expressed as

(20)um(x) =∑j=0mui =u0(x)+u1(x)+u2(x)+… =1+x-11+x2e-x+6xe-x+11e-x+5x+….

This converges to the exact solution, u(x) = e^x. Table 2 and Figures 3 and 4 present a numerical comparison between our solution, u₄(x) and the exact solution for Example 2.

Example 3. Consider the following nonlinear integro-differential equation:

(21)dudx=32ex-12e-3x+∫0xex-tu3(t)dt,u(0)=1.

Applying the Adomian polynomial and ℒ⁻¹ to both sides,

we have

∑j=0∞uj(x)=u(0)+ℒ-1(32ex-12e-3x+∫0x∑n=0∞An(t)dt).

This formula gives the following series solution:

(22)um(x) =∑j=0mui=u0(x)+u1(x)+u2(x)+… =1+x+x22+x33!+x44!+x55!+….

This converges to the exact solution, u(x) = e^x. Table 3 and Figure 5 present a numerical comparison between the proposed solution and the exact solution for Example 3.

6. Conclusion

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Abstract
1. Introduction
2. Preliminaries
3. Description of the Method
4. Convergence of the Method
5. Examples
6. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

We constructed an ADM based on modified Bernstein polynomials to solve the nonlinear first-order integro-differential equations. The performance of this method was validated by comparing the exact and approximate solutions for some examples. The results confirmed that this hybrid method can compete with other efficient methods for solving these types of equations. This method did not require any diminutive presumptions to solve the nonlinear integro-differential equations and produced an extremely effective strategy among the alternate strategies. This hybrid method is suitable for solving nonlinear problems.

Conflict of Interest

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Abstract
1. Introduction
2. Preliminaries
3. Description of the Method
4. Convergence of the Method
5. Examples
6. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

No potential conflict of interest relevant to this article was reported.

Figures

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Abstract
1. Introduction
2. Preliminaries
3. Description of the Method
4. Convergence of the Method
5. Examples
6. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

Fig. 1.

Numerical comparison between our solution u₄(x) and the exact solution for Example 1.

Fig. 2.

Absolute error for Example 1.

Fig. 3.

Numerical comparison between our solution u₄(x) and exact solution for Example 2.

Fig. 4.

Absolute error for Example 2.

Fig. 5.

Numerical comparison between our solution and the exact solution for Example 3.

Tables

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Abstract
1. Introduction
2. Preliminaries
3. Description of the Method
4. Convergence of the Method
5. Examples
6. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

Table. 1.

Table 1. Comparison between our first fourth-order approximate solutions u₄(x) of Example 1 and the solutions in other studies.

x	Exact	u₄(x)	ADM [29]	WGM [33]	HPM [34]
0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
0.0312	− 0.0312	−0.03119992103	− 0.0311999	−0.0312	−0.03120
0.0625	− 0.0625	−0.06249872844	− 0.0624987	−0.0625	−0.06250
0.0938	− 0.0938	−0.09379354920	− 0.0937935	−0.0937	−0. 09380
0.1250	− 0.1250	−0.3978731510	− 0.1249800	−0.1250	−0. 12498
0.1562	− 0.1562	−0.1561504020	− 0.1561500	−0.1562	−0. 15615
0.1875	− 0.1875	−0.1873970355	− 0.1873970	−0.1874	−0.18740
0.2188	− 0.2188	−0.2186091064	− 0.2186090	− 0.2186	−0.21861
0.2500	− 0.2500	−0.2496747212	− 0.2496750	− 0.2497	−0.24968
0.2812	− 0.2812	−0.2806795005	− 0.2806800	− 0.2807	−0.28068
0.3125	− 0.3125	−0.3117064248	− 0.3117060	− 0.3117	−0.31171

Table. 2.

Table 2. Numerical comparison between our solution u₄(x) and the exact solution for Example 2.

x	Exact	u₄(x)	Absolute error
0	1.000000000	1.000000000	0.00000000000
0.1	1.105170918	1.105170947	2.748263881 e⁻⁸
0.2	1.221402758	1.221402795	3.470977888 e⁻⁸
0.3	1.349858808	1.349858866	5.617735570 e⁻⁸
0.4	1.491824698	1.491824726	2.548139800 e⁻⁸
0.5	1.648721271	1.648721069	2.030066670 e⁻⁷
0.6	1.822118800	1.822117695	1.105495050 e⁻⁶
0.7	2.013752707	2.013747690	5.019781840 e⁻⁶
0.8	2.225540928	2.225522980	1.795120140 e⁻⁵
0.9	2.459603111	2.459549162	5.394816930 e⁻⁵
1.0	2.718281828	2.718138102	1.437281820 e⁻⁴

Table. 3.

Table 3. Numerical comparison between our solution and the exact solution for Example 3.

x	Exact	u_App(x)	Absolute error
0	1.000000000	1.000000000	0.00000000000
0.1	1.105170918	1.105170917	1.33329967 e⁻⁹
0.2	1.221402758	1.221402667	9.1333663 e⁻⁸
0.3	1.349858808	1.349857750	1.05800000 e⁻⁶
0.4	1.491824698	1.491818667	6.03132967 e⁻⁶
0.5	1.648721271	1.648697917	2.33543363 e⁻⁵
0.6	1.822118800	1.822048000	7.08000000 e⁻⁵
0.7	2.013752707	2.013571417	1.81290327 e⁻⁴
0.8	2.225540928	2.225130667	4.10261323 e⁻⁴
0.9	2.459603111	2.458758250	8.44861000 e⁻⁴
1.0	2.718281828	2.716666667	1.615161297 e⁻³

References

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Abstract
1. Introduction
2. Preliminaries
3. Description of the Method
4. Convergence of the Method
5. Examples
6. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

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Biographies

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Abstract
1. Introduction
2. Preliminaries
3. Description of the Method
4. Convergence of the Method
5. Examples
6. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

Mohammad Almousa received his Ph.D. degree in mathematics from the Universiti Sains Malaysia, Malaysia. His research interests are applied mathematics, numerical analysis, and differential equations.

Suhaila Saidat received her Ph.D. in Applied Mathematics from the Universiti Malaysia Perlis, Malaysia. Her research interests include applied mathematics and optimization.

Ahmad Al-Hammouri received his Ph.D. degree in Mathematics from Near East University, Cyprus. His research interests include mathematics, stability and numerical analyses, and differential equations.

Sultan Alsaadi received his Ph.D. degree in Mathematics from Near East University, Cyprus. His research interests include stability analysis, mathematical modeling, and mathematical biology.

Ghada Banihani received her Master’s degree in mathematics from the Jordan University of Science and Technology. Her research interests lie in mathematics.

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(3): 271-279

Published online September 25, 2024 https://doi.org/10.5391/IJFIS.2024.24.3.271

Solutions of Nonlinear Integro-Differential Equations Using a Hybrid Adomian Decomposition Method with Modified Bernstein Polynomials

Mohammad Almousa¹, Suhaila Saidat¹, Ahmad Al-Hammouri¹, Sultan Alsaadi², and Ghada Banihani¹

¹Department of Mathematics, College of Science and Technology, Irbid National University, Irbid, Jordan
²Mathematics Education Program, Faculty and Arts, Sohar University, Sohar, Oman

Correspondence to:Mohammad Almousa (mohammadalmousa12@yahoo.com)

Received: December 17, 2023; Revised: August 8, 2024; Accepted: September 9, 2024

Abstract

Keywords: Adomian decomposition method (ADM), Bernstein polynomials, Modified Bernstein polynomials, Integro-differential equations, Approximate solutions

1. Introduction

(1)

\frac{d u}{d x} = f (x) + \int_{0}^{x} K (t, u (t), u^{'} (t)) d t,

2. Preliminaries

In the following section, we briefly present the basic definitions, most important properties, and notation of the Bernstein polynomials. Bernstein first used these polynomials in 1912 [36, 37].

Definition 1. A linear combination Bernstein based polynomial

(2)

B_{n} (x) = \sum_{i = 0}^{n} B_{i, n} (x) β_{i}

is called the Bernstein polynomials, where x ∈ [0, 1], β_i are the Bernstein coefficients, and $B_{i, n} (x) = \frac{n!}{i! (n - i)} x^{i} {(1 - x)}^{n - i}$ .

Definition 2. The nth Bernstein polynomial for f (x) can be written as

(3)

B_{n} (f) = \sum_{i = 0}^{n} B_{i, n} (x) f (\frac{i}{n}) .

We are now giving the most important properties of these polynomials.

1. Non negativity.
2. Symmetry, so B_i,n(x) = B_n_–_i,n(1 – x).
3. Linearly, so B_n (α f ∓ βg) = αB_n ( f ) ∓ βB_n (g).
4. Each polynomial has only one maximum at $x = \frac{i}{n}$ .
5. $\sum_{i = 0}^{n} B_{i, n} (x) = B_{n} (1, x) = 1$ .
6. For mathematical convenience, we write B_i,n(x) = 0 if i < 0 or i > n.
7. The derivative of the n-th degree was

(4) $\frac{d}{d x} B_{i, n} (x) = n (B_{i - 1, n - 1} (x) - B_{i, n - 1} (x)) .$
8. It can be written as

(5) $B_{i, n} (x) = ((1 - x) B_{i, n - 1} (x) + x B_{i - 1, n - 1} (x)), 0 \leq i \leq n .$

Remark 1. The 2k-th order derivative f⁽²^k⁾ given by

(6)

B_{n}^{f} (x) = f (x) + \sum_{a = 2}^{2 k - 1} \frac{f^{(a)} (x)}{a! n^{a}} T_{n, a} (x) + O (\frac{1}{n^{k}}),

where

(7)

T_{n, a} (x) = \sum_{k} {(k - n x)}^{a} (\begin{matrix} n \\ k \end{matrix}) x^{k} {(1 - x)}^{n - k} .

3. Description of the Method

This section describes the application of the modified ADM with a Bernstein polynomial for solving the nonlinear first-order integro-differential equations. We will rewrite Eq. (1) as follows:

(8)

ℒ u = f (x) + \int_{0}^{x} N (u) d t,

where $ℒ u = \frac{d u}{d x}$ and N(u) is a nonlinear term.

To solve this problem, we take ℒ⁻¹ such that $ℒ^{- 1} (.) = \int_{0}^{x} . d x$ to both sides

(9)

u (x) - u (0) = g (x) + ℒ^{- 1} \int_{0}^{x} N (u) d t,

where $g (x) = \int_{0}^{x} f (t) d x$ .

The Adomian polynomial N(u) is as follows:

(10)

N (u) = \int_{n = 0}^{\infty} A_{n},

where $A_{n} = \frac{1}{n!} \frac{d^{n}}{d γ^{n}} [N (\sum_{i = 0}^{\infty} γ^{i} u_{i})], n = 0, 1, 2, \dots .$

Using Eqs. (3), (4), and (6), we obtain the following modified Bernstein series:

(11)

f (x) = \sum_{i = 0}^{n} (\begin{matrix} n \\ i \end{matrix}) x^{i} {(1 - x)}^{n - i} g (\frac{i}{n}) - \sum_{a = 2}^{2 k - 1} \frac{(\frac{d^{a}}{d x^{a}}) B_{i, n} (x)}{a! n^{a}} T_{n, a} (x) .

The solution u(x) is given by

(12)

u (x) = \int_{j = 0}^{\infty} u_{j} (x) .

Substituting Eqs. (10), (11), and (12) into Eq. (9), we have

(13)

\begin{array}{l} \sum_{j = 0}^{\infty} u_{j} (x) = & u (0) + \sum_{i = 0}^{n} (\begin{matrix} n \\ i \end{matrix}) x^{i} {(1 - x)}^{n - i} g (\frac{i}{n}) \\ - \sum_{a = 2}^{2 k - 1} \frac{(\frac{d^{a}}{d x^{a}}) B_{i, n} (x)}{a! n^{a}} T_{n, a} (x) + ℒ^{- 1} \int_{0}^{x} A_{n} d t . \end{array}

Therefore, the solutions of Eq. (1) are given by

(14)

\begin{array}{l} u_{0} (x) = & u (0) + ℒ^{- 1} \sum_{i = 0}^{n} (\begin{matrix} n \\ i \end{matrix}) x^{i} {(1 - x)}^{n - i} g (\frac{i}{n}) - \sum_{a = 2}^{2 k - 1} \frac{(\frac{d^{a}}{d x^{a}}) B_{i, n} (x)}{a! n^{a}} T_{n, a} (x), \\ u_{1} (x) = & ℒ^{- 1} \int_{0}^{x} A_{0} d t, \\ u_{2} (x) = & ℒ^{- 1} \int_{0}^{x} A_{1} d t, \\ ⋮ \\ u_{n + 1} (x) & = ℒ^{- 1} \int_{0}^{x} A_{n} d t, n = 0, 1, 2, \dots . \end{array}

4. Convergence of the Method

This section presents the convergence of the proposed method for solving nonlinear integro-differential equations. Furthermore, the maximum absolute error is determined.

Theorem 1. The series $u_{n} (x) = \sum_{j = 0}^{\infty} u_{j} (x)$ of Eq. (1) converges if α ∈ [0, 1) and ||u_n₊₁(x)|| = α ||u_n(x)|| such that ||u₀(x)|| < ∞.

Proof. Let S _n = u₁(x) + u₂(x) + ... + u_n(x). For n ≤ m, we are going to prove that S _n is a Cauchy sequence.

‖ S_{m} - S_{n} ‖ = ‖ \int_{i = 0}^{m} u_{i} (x) - \int_{i = 0}^{n} u_{i} (x) ‖ = \int_{i = n + 1}^{m} u_{i} (x) .

We have the following sequence

\begin{array}{l} ‖ S_{m} - S_{n} ‖ & = ‖ (S_{n + 1} - S_{n}) + (S_{n + 2} - S_{n + 1}) + (S_{n + 3} - S_{n + 2}) + \dots + (S_{m} - S_{m - 1}) ‖ \\ \leq ‖ (S_{n + 1} - S_{n}) ‖ + ‖ (S_{n + 2} - S_{n + 1}) ‖ + ‖ (S_{n + 3} - S_{n + 2}) ‖ + \dots + ‖ (S_{m} - S_{m - 1}) ‖ \\ \leq α^{n + 1} ‖ (u_{0}) ‖ + α^{n + 2} ‖ (u_{0}) ‖ + α^{n + 3} ‖ (u_{0}) ‖ + \dots + α^{m} ‖ (u_{0}) ‖ \\ \leq (α^{n + 1} + α^{n + 2} + α^{n + 3} + \dots + α^{m}) ‖ (u_{0}) ‖ \\ \leq α^{n + 1} (1 + α + α^{2} + α^{3} \dots + α^{m - n - 1}) ‖ (u_{0}) ‖ \\ \leq α^{n + 1} (\frac{1 - α^{m - n}}{1 - α}) ‖ (u_{0}) ‖ . \end{array}

Since α ∈ [0, 1) then 1 – α^m^–ⁿ < 1.

This in turn gives

‖ S_{m} - S_{n} ‖ \leq (\frac{α^{n + 1}}{1 - α}) ‖ (u_{0}) ‖ .

But we know that ||u₀(x)|| < ∞, yields

‖ S_{m} - S_{n} ‖ \to 0 as n \to 0.

Therefore, Theorem 1 is proven.

Theorem 2. The maximum absolute error of $u_{n} (x) = \sum_{j = 0}^{\infty} u_{j} (x)$ is

max_{\forall x \in J} | u (x) - \sum_{i = 0}^{n} u_{i} (x) | \leq (\frac{α^{n + 1}}{1 - α}) max_{\forall x \in J} ‖ (u_{0}) ‖ .

Proof. If m → ∞ then S _m → u(x), yields

‖ u (x) - S_{n} ‖ \leq (\frac{α^{n + 1}}{1 - α}) ‖ (u_{0}) ‖ .

Therefore, we have

max_{\forall x \in J} | u (x) - \int_{i = 0}^{n} u_{i} (x) | \leq (\frac{α^{n + 1}}{1 - α}) max_{\forall x \in J} ‖ (u_{0}) ‖ .

Therefore, Theorem 2 is proven.

5. Examples

The main objective here is to solve some examples of nonlinear first-order integro-differential equations using the new modified method to demonstrate its accuracy.

Example 1. Consider the nonlinear integro-differential equation as follows:

(15)

\frac{d u}{d x} = - 1 + \int_{0}^{x} u^{2} (t) d t, x \in [0, 1],

with boundary condition u(0) = 0 and exact solution u(x) = −x.

Introducing Eq. (10) into Eq. (15) and taking ℒ⁻¹ to both sides, yields

(16)

u (x) - u (0) = ℒ^{- 1} (- 1 + \int_{0}^{x} \sum_{n = 0}^{\infty} A_{n} (t) d t) .

Combining Eq. (12) and Eq. (16), we have

(17)

\sum_{j = 0}^{\infty} u_{j} (x) = ℒ^{- 1} (- 1 + \int_{0}^{x} \sum_{n = 0}^{\infty} A_{n} (t) d t) .

As suggested in Eqs. (11)–(14), when m = 6 and k = 2, the following solutions were obtained.

\begin{array}{l} u_{0} (x) & = ℒ^{- 1} \sum_{i = 0}^{6} (\begin{matrix} 6 \\ i \end{matrix}) x^{i} {(1 - x)}^{6 - i} f (\frac{i}{6}) - \sum_{a = 2}^{3} \frac{(\frac{d^{a}}{d x^{a}}) B_{i, 6} (x)}{a! 6^{a}} T_{6, a} (x) \\ = - x {(1 - x)}^{5} - 5 x^{2} {(1 - x)}^{4} - 10 x^{3} {(1 - x)}^{3} - 10 x^{4} {(1 - x)}^{2} - 5 x^{5} (1 - x) - x^{6} = - x, \\ u_{1} (x) & = \int_{0}^{x} \int_{0}^{x} A_{0} d t d x = \int_{0}^{x} \int_{0}^{x} {u_{0}}^{2} (t) d t d x = \frac{1}{12} x^{4}, \\ u_{2} (x) & = \int_{0}^{x} \int_{0}^{x} A_{1} d t d x = \int_{0}^{x} \int_{0}^{x} 2 u_{0} (t) u_{1} (t) d t d x = - \frac{1}{252} x^{7}, \\ u_{3} (x) & = \int_{0}^{x} \int_{0}^{x} A_{2} d t d x = \int_{0}^{x} \int_{0}^{x} 2 u_{0} (t) u_{2} (t) + {u_{1}}^{2} (t) d t d x = \frac{1}{6048} x^{10}, \\ u_{4} (x) & = \int_{0}^{x} \int_{0}^{x} A_{3} d t d x = \int_{0}^{x} \int_{0}^{x} 2 u_{0} (t) u_{3} (t) + 2 u (t) u_{2} (t) d t d x = - \frac{1}{157248} x^{13}, \end{array}

and so on.

The solution of Eq. (15) becomes

(18)

\begin{array}{l} u_{m} (x) & = \int_{j = 0}^{m} u_{i} = u_{0} (x) + u_{1} (x) + u_{2} (x) + \dots \\ = - x + \frac{1}{12} x^{4} - \frac{1}{252} x^{7} + \frac{1}{6048} x^{10} - \frac{1}{157248} x^{13} + \dots \end{array}

Example 2. Consider the nonlinear integro-differential equation as follows:

(19)

\frac{d u}{d x} = 1 + \int_{0}^{x} e^{- t} u^{2} (t) d t,

with boundary condition u(0) = 1 and exact solution u(x) = e^x.

Using Eq. (10) in Eq. (19) and ℒ⁻¹ on both sides, we obtain

\sum_{j = 0}^{\infty} u_{j} (x) = u (0) + ℒ^{- 1} (1 + \int_{0}^{x} \sum_{n = 0}^{\infty} A_{n} (t) d t) .

As suggested in Eqs. (11)–(14) when m = 6 and k = 2, yields

\begin{array}{l} u_{0} (x) = & u (0) + ℒ^{- 1} \sum_{i = 0}^{6} (\begin{matrix} 6 \\ i \end{matrix}) x^{i} {(1 - x)}^{6 - i} f (\frac{i}{6}) \\ - \sum_{a = 2}^{3} \frac{(\frac{d^{a}}{d x^{a}}) B_{i, 6} (x)}{a! 6^{a}} T_{6, a} (x) \\ = & 1 + x {(1 - x)}^{5} + 5 x^{2} {(1 - x)}^{4} + 10 x^{3} {(1 - x)}^{3} \\ + 10 x^{4} {(1 - x)}^{2} + 5 x^{5} (1 - x) + x^{6} \\ = & 1 + x, \\ u_{1} (x) = & \int_{0}^{x} \int_{0}^{x} A_{0} d t d x \\ = & \int_{0}^{x} \int_{0}^{x} e^{- t} u_{0}^{2} (t) d t d x \\ = & - 11 + x^{2} e^{- x} + 6 x e^{- x} + 11 e^{- x} + 5 x, \\ u_{2} (x) = & \int_{0}^{x} \int_{0}^{x} A_{1} d t d x \\ = & \int_{0}^{x} \int_{0}^{x} 2 e^{- t} u_{0} (t) u_{1} (t) d t d x \\ = & \frac{- 139}{4} + \frac{39}{4} x + 14 e^{- x} + 5 e^{- 2 x} x^{2} + \frac{71}{4} e^{- 2 x} x \\ + \frac{83}{4} e^{- 2 x} + 28 x e^{- x} + \frac{1}{2} x^{3} e^{- 2 x} + 10 x^{2} e^{- x}, \\ u_{3} (x) = & \int_{0}^{x} \int_{0}^{x} A_{2} d t d x \\ = & \frac{1}{972} (15267 x e^{3 x} + 43254 x^{2} e^{2 x} + 7290 x^{3} e^{x} \\ + 216 x^{4} - 676073 + 17496 x e^{2 x} + 49572 x^{2} e^{x} \\ + 3060 x^{3} - 1458 e^{2 x} + 103275 x e^{x} + 17010 x^{2} \\ + 26487 e^{x} + 43212 x + 42578) e^{- 3 x}, \\ u_{4} (x) = & \int_{0}^{x} \int_{0}^{x} A_{3} d t d x \\ = & \frac{1}{248832} (5536734 {x e}^{4 x} + 32077824 x^{2} e^{3 x} \\ + 12970368 x^{3} e^{2 x} + 1105920 x^{4} e^{x} + 22464 x^{5} \\ - 28049407 e^{4 x} - 38330368 x e^{3 x} + 53374464 x^{2} e^{2 x} \\ + 12128256 x^{3} e^{x} + 409824 x^{4} + 14800896 e^{3 x} \\ + 47138112 x e^{2 x} + 49489920 x^{2} e^{x} + 3090096 x^{3} \\ - 46453824 e^{2 x} + 81899520 x e^{x} + 11923584 x^{2} \\ + 41003008 e^{x} + 23455582 x + 18699327) e^{- 4 x}, \end{array}

and so on. Therefore, the solution of Eq. (19) is expressed as

(20)

\begin{array}{l} u_{m} (x) & = \sum_{j = 0}^{m} u_{i} \\ = u_{0} (x) + u_{1} (x) + u_{2} (x) + \dots \\ = 1 + x - 11 + x^{2} e^{- x} + 6 x e^{- x} + 11 e^{- x} + 5 x + \dots . \end{array}

This converges to the exact solution, u(x) = e^x. Table 2 and Figures 3 and 4 present a numerical comparison between our solution, u₄(x) and the exact solution for Example 2.

Example 3. Consider the following nonlinear integro-differential equation:

(21)

\frac{d u}{d x} = \frac{3}{2} e^{x} - \frac{1}{2} e^{- 3 x} + \int_{0}^{x} e^{x - t} u^{3} (t) d t, u (0) = 1.

Applying the Adomian polynomial and ℒ⁻¹ to both sides,

we have

\sum_{j = 0}^{\infty} u_{j} (x) = u (0) + ℒ^{- 1} (\frac{3}{2} e^{x} - \frac{1}{2} e^{- 3 x} + \int_{0}^{x} \sum_{n = 0}^{\infty} A_{n} (t) d t) .

This formula gives the following series solution:

(22)

\begin{array}{l} u_{m} (x) & = \sum_{j = 0}^{m} u_{i} = u_{0} (x) + u_{1} (x) + u_{2} (x) + \dots \\ = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + \frac{x^{5}}{5!} + \dots . \end{array}

This converges to the exact solution, u(x) = e^x. Table 3 and Figure 5 present a numerical comparison between the proposed solution and the exact solution for Example 3.

Fig 1.

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