On an approximate solution of a boundary value problem for a nonlinear integro-differential equation

Abstract The aim of this work is to discuss the solvability of a boundary value problem for a nonlinear integro-differential equation. First, we derive an equivalent nonlinear Fredholm integral equation (NFIE) to this problem. Second, we prove the existence of a solution to the NFIE using the Krasnosel’skii fixed point theorem under verifying some sufficient conditions. Third, we solve the NFIE numerically and study the convergence rate via methods based upon applying the modified Adomian decomposition method and Liao’s homotopy analysis method. As applications, some examples are illustrated to support our work. The results in this work refer to both methods are efficient and converge rapidly, but the homotopy analysis method may converge faster when we succeed in choosing the optimal homotopy control parameter.


Introduction
Many problems arising in applied physics, biology, chemistry and engineering can be described using mathematical models that depend on utilizing integral and differential operators with imposed conditions. For example, the so-called Chandrasekhar H-function that appears in the radiative transfer processes, is defined in terms of integral equation. The evolution of biological populations is characterized by employing delayed integro-differential equations of the Volterra type. Continuous medium-nuclear reactors are analysed using models that employ systems of integro-differential equations. Also, there are the singular integral equations that occur during the process of formulating mixed boundary value problems in mathematical physics, especially in solid mechanics and elasticity. Using the Green function approach, we can transform many partial differential equations to equivalent integral equations. So, trying to find solutions for these equations attracts many researchers. Because we cannot determine the exact solutions for most of these equations, "many numerical" or "semi-analytic" techniques are developed to overcome this gap. From these methods, there are the Adomian decomposition method (ADM), homotopy perturbation method (HPM), variational iteration method (VIM), homotopy analysis method (HAM) and many other methods, see Abdou, Soliman, and Abdel-Aty (2020), Hamoud and Ghadle (2018), He (2020aHe ( , 2020b, Mirzaee and Alipour (2019), Rezabeyk, Abbasbandy, and Shivanian (2020), Saeedi, Tari, and Babolian (2020) among others. These techniques can be applied to find approximate solutions for a large class of linear and nonlinear integral equations and many functional equations as well. Also, in some special cases, when the series solution converges to a known function we can get a closed form-solution using these methods. For example, Wazwaz (2010) confirmed that the VIM is very reliable in solving first-and second-kind integral equations of the Volterra type and most calculations can be significantly reduced. Alhendi, Shammakh, and Al-Badrani (2017) found that the VIM and HPM are very effective when applying them to solve quadratic fractional integro-differential equations. Elborai, Abdou, and Youssef (2013) studied the mean square convergence of the series solution for a stochastic integro-differential equation and estimated the truncation error by the ADM. Kurt and Tasbozan (2019) utilized the HAM to solve the modified Burgers equation. Singh, Kumar, Baleanu, and Rathore (2018) used the Sumudu transform along with the HAM to find approximate solutions to some fractional equations of the Drinfeld-Sokolov-Wilson type. Hetmaniok, Słota, Trawi nski, and Wituła (2014) explained the applicability of the HAM in solving nonlinear integral equations of second kind. Hamoud, Ghadle, and Atshan (2019) applied the MADM (modified Adomian's decomposition method) to find an approximate solution for a class of fractional nonlinear integro-differential equation of the Caputo-Volterra-Fredholm type. Issa, Hamoud, and Ghadle (2021) used the MADM, VIM and HPM to solve a fuzzy integro-differential equation of the Volterra type numerically and compared the results, see Adomian (1994), Alidema andGeorgieva (2018), Bakodah, Al-Mazmumy, andAlmuhalbedi (2019), Liao (2012), Maitama and Zhao (2019) and Singh, Nelakanti, and Kumar (2014) for more applications regarding these elegant methods and the references therein.
The current article discusses the solvability of a two-point boundary value problem for a nonlinear integro-differential equation in the form (1.1) subject to the boundary conditions where p ! 0 is a finite non-negative integer. The parameters fl, kg are two non zero real parameters. The functions A 1 , A 2 , f and the kernel K are known functions satisfying certain conditions to be assigned in the next section while x7 !wðxÞ is the sought function to be determined in the space C 2 ð½a, b, RÞ, (see Def. (2.1)). The present form of Eq. (1.1) is not discussed before to the best of our knowledge. This work is organized as follows. In the section "Outcomes for existence and uniqueness", we derive a corresponding nonlinear Fredholm integral equation (NFIE) for Equation (1.1) under verifying condition (1.2). Then, we apply the Krasnosel'skii fixed point theorem to prove the existence of solutions for this NFIE under satisfying some sufficient conditions. The uniqueness of this solution is investigated as well. The convergence of solution and truncation error is studied in the "Analysis of MADM" section. The homotopy analysis technique is applied to problem (1.1) under verifying condition (1.2) in the "Analysis of HAM" section. Next, we present some applications in the section "Numerical and analytical outcomes". The section "Conclusion" is devoted to the discussion of results.

Outcomes for existence and uniqueness
In order to prove Thm. (2.1) we suppose the following postulates.
(i) the functions A 1 and A 2 are elements in the space Cð½a, b, RÞ: (ii) the known free function f belongs to the space C 2 ð½a, b; RÞ: (iii) the known kernel ðx, yÞ7 !KðxÀyÞ is continuous in x 8y 2 ½a, b with values in R and: where: & It is easy to see that (2.12) Substituting Equations (2.10) (2.11) and (2.12) in Equation (1.1) gives The converse can be done easily and thereby it is omitted. The proof is completed.
w Remark 2.1. It is worth mentioning that Thm. (2.1) is valid whether the kernel KðxÀyÞ is continuous or has a singularity of the weak type at the straight line y ¼ x.
Definition 2.1. By a solution for the boundary value problem Equation (1.1), we mean proving the existence of a function w 2 C 2 ð½a, b; RÞ satisfying Equation (1.1), and the boundary conditions (1.2).
Let the constant l 2 fa 2 Z : 0 a pg: We define the following positive real constants. We consider the following assumption.

Proof.
Let juðxÞj rg: The radius r is a finite positive solution for the equation jkj P p l¼2 c Ã ðlÞr l þ ðb þ jkjc Ã ð1Þ À jljÞr þ jjFjj 1 ¼ 0: Let u 1 , u 2 be any two functions in the set X r . Define the following two operators.
Sðx, y; 1ÞM 2 ðy, tÞdy ! u 1 ðtÞdt: (2.14) Applying the Cauchy inequality and then, simplifying the right-hand side yield Using condition (iii), passing the supremum over x 2 ½a, b and then, utilizing the value of c Ã ð1Þ give Using similar arguments as we used above implies (2.16) Using Equations (2.15) and (2.16) give c Ã ðlÞr l ¼ r: (2.17) Therefore, Tðu 1 Þ þ Wðu 2 Þ 2 X r , 8u 1 , u 2 2 X r : Now, suppose x 1 <x 2 be two elements in ½a, b: The functions F, H 1 and H 2 are continuous in x from applying conditions ðiÞÀðiiiÞ and therefore, we have dy ! 0 as x 2 ! x 1 : (2.18) Also, we have So, Tu 1 and Wu 2 are elements in the space Cð½a, b, RÞ: Consequently, the operator T þ W is a self-operator on X r . Let u, u Ã be any two functions in the set X r . So, Þj uÀu Ã jj 1 : (2.20) Therefore, the operator T is a contraction operator on X r from applying condition (iv). Consider the sequence ðu n Þ n2N with u n 2 X r such that u n ! u, when n ! 1: It is clear that u 2 X r and sup x2½a, b u n ðxÞ j j r, 8n 2 N: Applying the Arzela convergence theorem implies lim n!1 jðWu n ÞðxÞ À ðWuÞðxÞj jkj jlj lim where e(l) is a finite positive constant depends on l. Therefore, the operator W is a sequentially continuous operator on X r and hence it is continuous on X r . It is clear from Equation (2.16) that 8 Wu2WX r sup x2½a, b ðWuÞðxÞ j j jkj jlj P p l¼2 c Ã ðlÞr l and hence the set WX r is uniformly bounded. Consider the sequence ðWu n Þ n2N with Wu n 2 WX r : Using similar steps as we followed in Equation (2.19) implies ðWu n Þðx 2 Þ À ðWu n Þðx 1 Þ j j <, 8n 2 N when jx 2 Àx 1 j<d: Therefore, there exists a sub-sequence ðWu n k Þ k2N which converges uniformly in WX r from applying the Arzela-Ascoli theorem and consequently the set WX r is compact. As a result, the operator W is completely continuous. Now all conditions of the Krasnosel'skii theorem are satisfied and therefore, the operator T þ W has at least one fixed point in the set X r which is a solution for the NFIE (2.1). The proof is completed.
w In what follows we suppose that: (2.21) Using Equations (2.20) and (2.21) lead to So, the operator T þ W is contraction on X r from utilizing condition (v) and consequently, the NFIE (2.1) posses a unique continuous solution in X r from applying the Banach fixed point theorem. The proof is completed. w For the next theorem, let S Ã ðx, y; 1Þ :¼ Sðx, y; 1Þ j p¼1 , (2.23) possesses a unique continuous solution in X r Ã : Proof. The proof is similar to the arguments that we have used above. So, it is omitted.

The MADM for the NFIE
This section is devoted to using the MADM (Wazwaz, 1999)  ( 3.2) where Adomain's polynomial, A m , m ! 0, is evaluated using the equation below.

Numerical and analytical outcomes
(5.7) where Table 1 presents the absolute errors between the exact solution and the first four approximate solutions using the MADM. We observe that the approximate solutions, that are obtained using the MADM, converge very fast to the exact solution, see Figure 1.
(5.10)  The values of h that ensure the convergence of the approximate solution to the exact (closed form) solution are evaluated from the line segments that are nearly parallel to the hÀ axis in the hÀ curves in Figure 2. Minimizing the squared of residual yields the optimal value of h, see Figure 3. For example, minimizing the squared residual that is based on utilizing B 1 ðxÞ, B 2 ðxÞ, B 3 ðxÞ, B 4 ðxÞ, where x 2½0,1, gives h ' À1:035526,À1:012718,À1:004832,À1:010456, see Figure 3. Using h ¼ À1 gives the same results that we obtained when we utilized the MADM in Table 1. So, from the results that are obtained in Tables 1 and 2, we can notice that the two methods are very efficient in solving Ex. (5.1) and converge very fast to the exact solution under satisfying conditions of Thm. (2.3). But the HAM may converge slightly faster than the MADM when we use the optimal value of h n corresponding to each B n ðxÞ,n ¼ 1,2,3,:::: The exact (closed form) solution of Equation (5.13) is   where Table 3 shows the absolute errors between the exact solution and the first four approximate solutions using the MADM. We can observe that the approximate solutions, that are obtained using the MADM, converge very fast to the exact solution, see Figure 4.
(2.) Approximate solution using the HAM The zero-order deformation is defined, from using Equation (4.6), as below.      The values of h that ensure the convergence of the approximate solution to the exact (closed form) solution are evaluated from the line segments that are nearly parallel to the hÀ axis in the hÀ curves in Figure 5. Minimizing the squared of residual yields the optimal value of h, see Figure 6. For example, minimizing the squared residual that is based on utilizing B 1 ðxÞ, B 2 ðxÞ, B 3 ðxÞ, B 4 ðxÞ, where x 2 ½0, 1, gives h ' À0:995435, À0:996601, À0:993321, À0:993270, see Figure 6. Using h ¼ À1 gives the same results that we obtained when we utilized the MADM in Table 3. So, from the results that are obtained in Tables 3 and  4, we can notice that the two methods are very efficient in solving Ex. (5.2) and converge very fast to the exact solution under satisfying conditions of Thm.
(2.3). But the HAM may converge slightly faster than the MADM when we use the optimal value of h n corresponding to each B n ðxÞ, n ¼ 1, 2, 3, ::::

Conclusion
In this work, we have studied a boundary value problem for a nonlinear integro-differential equation.
An equivalent NFIE has been derived for the proposed problem, then the Krasnosel'skii fixed point has been applied to investigate the existence of continuous solutions. Moreover, the sufficient conditions which guarantee the uniqueness of the solution are proved. We have determined an approximate solution to the NFIE using the MADM. The convergence and error estimates of this approximate solution are studied as well. After that, the homotopy analysis technique is applied to get another approximate solution for the NFIE. We compared the accuracy and convergence rate of the approximate solution using the two techniques. It turns out for us that both methods are efficient and converge very rapidly, but the HAM may converge slightly faster when we succeed in choosing the optimal homotopy control parameter. We may study the fractal version that is corresponding to Equation (1.1) for future suggested work (He, 2020a(He, , 2020b.