Projection-iterated method for solving numerically the nonlinear mixed integral equation in position and time

In this work, we consider a nonlinear Volterra-Hammerstein integral equation of the second kind (V-HIESK). The existence of a unique solution to the integral equation, under certain conditions, is guaranteed. The projection method was used, as the best numerical method. This method is better than the previous methods in the possibility of obtaining the lowest relative error. The method of obtaining the best approximate solution, through the projection method, depends on presenting three consecutive algorithms and depends mainly on the method of iterative projection (P-IM). In each algorithm, we get the approximate solution and the corresponding relative error. Moreover, we demonstrated that the estimated error of P-IM in the first algorithm is better than that of the successive approximation method (SAM). Some numerical results were calculated, and the error estimate was computed, in each case. Finally, the numerical results have been compared between this method and previous research, and it is clear through the same examples that the relative error of the first algorithm is better than the comparative error in the other methods.


Introduction
Consider, the nonlinear mixed integral equation of type V-HIESK, Here, the IE (1) is considered in time for the Volterra integral term in the class C[0, T ], T < 1, and has a continuous kernel f (t, ), and in position, with continuous kernel k(x, y), for the Hammerstein integral term in the space L 2 [Ω]; Ω is the domain of integration. The given two functions (t, x, (x, t)) and g(x, t) are defined in the space L 2 [Ω] × C[0, T ]. The constant defines the kind of (1), for the second kind ≠ 0, while for the first kind = 0; the constant , has many physical meaning. The unknown function (x, t) will be discussed and obtained in the space The technique discussing of solution depends upon presenting three different algorithms. These algorithms based on the projection-iteration method to obtain the approximate solution and the corresponding error, in each algorithm.
The presence of integral equations in various sciences and their fields of applications led to different techniques of solutions. For example, in the contact problem (Abdou, et al. [1], Popv [2]), fluid mechanics (Haggag and Dosqiyas k(x, y) ( , y, (y, )) dyd , * Sharifah E. Al Hazmi, sehazmi@uqu.edu.sa | 1 Department of Mathematics, Al-Qunfudah University College, Umm Al-Qura University, Mecca, Saudi Arabia. 1 3 [3], Constanda and Perez [4]), displacement problems in mechanics, and the theory of elasticity (Aleksandrovsk and Covalence [5], Abdou and Asseri [6], Zisis et al. [7]), in laser theory (Gao et al. [8]), in thermoplastic plate, in theormoelasticity (Abdou et al. [9,10]). In Thermal-shock problem (Zenkour [11]), and in Dual-Phase-Lag Model on Thermoelastic (Abbas and Zenkour [12]). For this progress in the various basic sciences, we find that the integral equations, whether analytical or numerical methods, played an important role in solving these problems. One of these important methods that helped to solve some problems in different sciences is the method of orthogonal polynomials; see Abdou, et al. [13][14][15]. Diego and Lima,in [15] used collocation method to discuss the convergence of a class of weakly singular integral equation. Hafez and Youssri, in [16] discussed the numerical solution of twodimensional integral equations of linear Volterra-Fredholm types, after applying collocation method via the Legendre-Chebyshev polynomials. Nemati, et al. in [17], used the Legendre polynomials method to obtain, numerically the solution of a class of two-dimensional nonlinear Volterra integral equation of the second kind with continuous kernel. Almasieh and Meleh used, in [18] Hybrid function method to solve a nonlinear integral equation of the Fredholm type. Brezinski and Zalglia, in [19] applied the extrapolation method for obtaining the numerical solution of a nonlinear Fredholm integral equation. Katani, in [20] used the quadrature method to discuss the solution of Fredholm integral equations of the second kind. Al-Bugami, in [21] applied Trapezoidal and Simpson methods to obtain the numerical solution of the integral equation in the two-dimensional problem in a surface crack layer. Bakhshayesh, in [22] used the Galerkin method to discuss the approximate solution of Volterra integral equations with discontinuous kernel. Elzaki and Alamri, in [23] used the homotopy perturbation method and the Adomian decomposition method to obtain the numerical solution of some kinds of nonlinear integral equations. Almousa and Ismail, in [24] used the same two numerical methods to discuss the numerical solution of Volterra-Fredholm integral equation in two-dimensional. Also, Basseem and Alalyani, in [25] used the Adomain decomposition method and homotropy perturbation method for solving quadratic integral equation with a continuous kernel. Abdou, et al., in [26] used the Chebyshev polynomial method to solve numerically, quadratic integral equation with discontinuous continuous.
The projection method is to create three algorithms to discuss how to solve the nonlinear mixed integral equation. So that we study the existence of a single solution for each algorithm, and then compare the relative error in each case with the other most used methods.
In the remainder of this research, the necessary and guaranteeing conditions for the existence of a unique solution to the integral equation in the space L 2 [Ω] × C[0, T ], will be established. In section three, the projection -iteration method will be used by considering three algorithms to discuss, numerically the solution of the integral Eq. (1). The existence of a unique solution, in each algorithm, will be proved in two cases. (1) When Lipschitz condition is satisfied in the space of integration. (2) When Lipschitz condition is not verified in the whole space. Some numerical results will be calculated and the error estimate, in each case, will be computed. Moreover, we will deduce that the estimate error here is the best error corresponding to many other methods.

The necessary conditions for existing a unique solution
To guarantee the existence of a unique solution of V-HIE we consider the following conditions: The given function g(x, t) with its partial derivatives are continuous in the space L 2 [Ω] × C[0, T ] and its norm, for a constant G, is defined as: 4. The known continuous function (t, x, (x, t)), for the two finite constants and q 1 , satisfies the following conditions: Theorem 1 (see Abdou et al. [14]): In view of the above conditions, the V-HIESK of (1) has a unique solution (x, t) in the space L 2 (Ω) × C[0, T ] under the condition: Proof: To prove the theorem, we write Eq. (1) in the integral operator form where Then, we follow Hence, we have The previous inequality (4) shows that, the operator K maps the ball S into itself, where In addition for the continuity, and for two different Hence, under the condition | | > | |qTCM, the integral operator K (x, t) is a contracting mapping. Using fixed point theorem the solution of Eq. (1) is unique.

Projection-iteration method
To use the projection-iteration method for discussing the numerical solution of the V-HIE (1), rewrite (1) in the following operator effect: where, Then, present various variants of the projectioniteration method of Eq. (2), in the space L 2 (Ω) × C[0, T ] depending upon the following property: Definition 1. Let H be an inner product space and f n is a Cauchy sequence in H. Such a sequence has the property that for every > 0 , we can find an L( ) such that, The inner product H is said to be a Hilbert space if every Cauchy sequence converges to an element in H.
Property 1 (See Zhuang et al. [27]). For the projection operator P , there exists an orthogonal projection operator Q satisfies the relation Now, to discuss the approximate solution of Eq. (7) we construct three different kinds of a logarithm based on projection-iteration method.

The first algorithm
Construct the sequence of the numerical solution n in the form, where and Then, we have (8) Γ( ) = ( , y, (y, )),

Original Article
J.Umm Al-Qura Univ. Appll. Sci. (2023) 9:107-114 | https://doi.org/10.1007/s43994-023-00025-w 1 3 The formula (12) represents the first algorithm via the sequence of the solution n . Theorem 2. If the two operators P Γ, QΓ with the two constants D 1 , D 2 respectively, are satisfying Lipschitz condition in the space L 2 (Ω) × C[0, T ] , with the condition, Then, the sequence of a unique solution n of (12) converges to a unique solution { } of (2) and the estimate error satisfies the following inequality Proof: To prove the theorem, assume the sequence of solutions in the form = lim j→∞ { j }.
Case (1). If the Lipschitz condition is verified. Let n and n−1 are two arbitrary distinct partial sums in the sequence of solution { n }.
Hence, from (12) we have Using the relation (10), and applying the Lipschitz condition for P Γ and Q Γ respectively, we obtain Applying the successive approximation method and using the mathematical induction to get.
Finally, in general we can have the following inequality The inequality (16) shows that n represents Cauchy sequence. Since L 2 (Ω) × C[0, T ] is a complete space, then there exists a function such that Case (2). If the Lipschitz condition is not verified in the whole Banach space. Therefore, we go to prove that ‖ ‖ − n ‖ ‖ → 0, as n → ∞, in a certain ball of the space.  (14) is valid, then for r ′ > 0 we have the inequality Under the condition (17), the sequence of solution n of (7) in the ball H(‖ ‖ ≤ r � ) converges to the exact solution of Eq. (7) in that ball for every 0 ∈ H(‖ ‖ ≤ r � ) . In this case, the estimate error holds.
Proof: Consider the norm of Eq. (12), and then with the aid of (5), we have Using (17) in the above inequality (18), we get ‖ ‖ n ‖ ‖ ≤ r. Since the above inequality is valid for r ′ , therefore n ∈ H(‖ ‖ ≤ r � ) for every 0 ∈ H(‖ ‖ ≤ r � ) . Then Eq. (12) has a unique solution in that ball and the condition (18) is valid.

The second algorithm
The second algorithm can be constructed in the form where With the aid of Eq. (10), rewrite Eq. (14) to take the form where the sequence { n } represents the unique solution of (20) and 0 ∈ L 2 (Ω) × C[0, T ].

Theorem 4 Under condition
the sequence n of equation ( 20) converges to the unique solution of Eq. (2) and the estimate error in this case, is given by the relation, Proof: By following the same way of Theorem 2, and when the Lipschitz condition is holding, we can easily proof the theorem.
When the Lipschitz condition is not holding in all space, we state the following: Proof: The proof is completely determined after using Eq. (10) and the property 3. Case (1). When the Lipschitz condition for P Γ and Q Γ respectively is hold. Applying Cauchy-Schwarz inequality, then using the relation (5), and the Lipschitz condition for P Γ and Q Γ respectively, to have (23) ‖g‖ + � � ‖FK ‖ V (r) ≤ � �r.
In general, for two solutions n+p , n ∈ { n } n=∞ n=0 , after applying the successive approximation and the mathematical induction we obtain Hence, the sequence n of the inequality (28) represents Cauchy sequence in the complete space L 2 (Ω) × C[0, T ] then there exists a function such that ‖ ‖ − n ‖ ‖ → 0, (n → ∞). . Then, is the unique solution of Eq. (7) in L 2 (Ω) × C[0, T ].

Remark:
The last theorem can be proved in a certain ball of H when Lipschitz condition is not hold in the whole Hilbert space.

Numerical results
In this section, we study some problems that were solved by some researchers using some different numerical methods. Then we compare the relative error between the method used previously and the method that we dealt with in this research.

Example 1:
Consider the integral equation, see [28] From Table 1, it is noticed that the results of the first algorithm were compared with the approximate solution of example (1) of Kumar's [28], who solved the Hammerstein integral equation using the discrete collocation |x − y| 1.05 2 (y)dy, ( (x) = x 2 , |x| ≤ 1) Table 1 The approximate solution of Hammerstein integral equation, with three estimate error method. It was found from the numerical results using the first algorithm that the projection method gives less relative error and the stability of the solution is better than the collocation method.Example 2: Consider the Hammerstein integral equation, see Abou El-Seoud, et al. [29] In example (2), the integral equation was solved numerically by Abu Al-Saud, et al. [29] after developing the successive approximation method. In this research, we also solved the same example by the projection-iteration method. By comparing the relative error from the results of the first algorithm, we conclude that the projection method is better than the successive approximation method as demonstrated in Table 2.
In all previous research, there was no explicit study, whether analytical or numerical, to reveal and clarify the effect of time on the integral equations. Abdou, et al., [30][31][32][33][34] were the first to study the effect of time on the integral equations. In this research, we will conduct a numerical study that shows the effect of increasing time on numerical results using the projection method.
Example 3: Consider the integral equation and the results are displayed in Table 3.  Table 4.  Table 1, is better than the error of the collocation method (see Kumer [28[). In addition, the relative error of the first algorithm, in Table 2, is better than the successive approximation method (see Abu Al-Saud, et al. [29]). (3-ii) For the second algorithm we construct the sequence n that converges to the unique solution of Eq. (1) and to the sequence of the projection-iteration method. (3-iii) For the third algorithm we construct the sequence n of unique solution in the space L 2 [Ω] × C[0, T ] converges to the unique solution of the projection-iteration method. (4) In Table 3. We considered the nonlinear mixed integral equation of the second kind when the exact solution is (x;t) = x 2 t 2 and time t = 0.8. As the value of x increases the error increases. Also, the error of the first algorithm is bigger than the error of the second and third algorithm. (5) In Table 4. We considered the same mixed integral equation of Table 3. with its exact solution (x;t) = x 2 t 2 but at time t = 0.1. We have the same behavior of example three. Moreover, we deduce    Table 3 and Table 4, as the time increases the error increases. (6) From the result of example (1) and example (2), we deduce that the Projection-iterated method is the best method to estimate the error.
Funding The author would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code: (22UQU4282396DSR21).
Data availability Data is available from the corresponding author.

Conflicts of interest
The author declares that she has no conflicts of interest to report regarding the present study.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.