The combined reproducing kernel method and Taylor series to solve nonlinear Abel’s integral equations with weakly singular kernel

Abstract: The reproducing kernel method and Taylor series to determine a solution for nonlinear Abel’s integral equations are combined. In this technique, we first convert it to a nonlinear differential equation by using Taylor series. The approximate solution in the form of series in the reproducing kernel space is presented. The advantages of this method are as follows: First, it is possible to pick any point in the interval of integration and as well the approximate solution. Second, numerical results compared with the existing method show that fewer nodes are required to obtain numerical solutions. Furthermore, the present method is a reliable method to solve nonlinear Abel’s integral equations with weakly singular kernel. Some numerical examples are given in two different spaces.


PUBLIC INTEREST STATEMENT
In mathematical modeling of real-life problems, we need to deal with functional equations, e.g. partial differential equations, integral and integro-differential equation, stochastic equations and others. Several numerical methods have been developed for the solution of the integral equations. Recently, the applications of reproducing kernel method (RKM) have become of great interest for scholars. We use a reproducing kernel Hilbert space approach that allows us to formulate the estimation problem as an unconstrained numeric maximization problem easy to solve. (Nosrati Sahlan, Marasi, & Ghahramani, 2015), Bernstein polynomials (Alipour & Rostamy, 2011) and Legendre wavelets (Yousefi, 2006). For further see Kilbas and Saigo (1999), Kumar, Singh, and Dixit (2011), Pandey, Singh, and Singh (2009), Wang, Zhu, and Fečkan (2014).
The aim of this paper is to introduce the reproducing kernel method to solve nonlinear Abel's integral equation. The standard form of equation (Wazwaz, 1997) is given by where the function f(x) is a given real-valued function, and F(u(x)) is a nonlinear function of u(x). Recall that the unknown function u(x) occurs only inside the integral sign for the Abel's integral equation.
This paper is organized as six sections including the introduction. In the next section, we introduce construction of the method in the reproducing kernel space for solving Equation (1.1). The analytical solution is presented in Section 3. The implementations of the method is provided in Section 4. Numerical findings demonstratig the accuracy of the new numerical scheme are reported in Section 5. The last section is a brief conclusion.

Construction of the method
In this section, we construct the space W m 2 [0, 1] and then formulate the reproducing kernel function R x (y) in the space W m 2 [0, 1]. The dimensional space is finite. First, we present some necessary definitions from reproducing kernel theory.
is a real-valued function or complex function, x ∈ X, X is a abstract set} be a Hilbert space, with inner product is an absolutely continuous real-valued function on [0, 1] and u (m) The inner product and norm in W m 2 [0, 1] are given respectively by

Definition 2.3
If ∀x ∈ X, there exists a unique function R x (y) in ℍ, for each fixed y ∈ X then R x (y) ∈ ℍ and any u(x) ∈ ℍ, which satisfies ⟨u(y), R x (y)⟩ W m 2 = u(x). Then, Hilbert space ℍ is called the reproducing kernel space and R x (y) is called the reproducing kernel of ℍ.

A transformation of the Equation (1.1)
Using modified Taylor series, the nonlinear Abel's integral equations with weakly singular kernel transform into nonlinear differential equations that can be solved easily.
With the Taylor series expansion of F(u(t)) expanded about the given point x belonging to the interval [0, 1], we have the Taylor series approximation of F(u(t)) in the following form We use the truncated Taylor series and substitute it instead of the nonlinear term of Equation (1.1),

The analytical solution
In this section, we present a nonlinear differential operator and a normal orthogonal system of the First of all, we define an invertible bounded linear operator as such that Next, we construct an orthogonal function system of W m 2 [0, 1].
and continuity of u(x) , then we have u(x) ≡ 0.
by using the Gram-Schmidt algorithm, and then the approximate solution will be obtained by calculating a truncated series based on these functions, such that where ik are orthogonal coefficients. However, Gram-Schmidt algorithm has some drawbacks such as high volume of computations and numerical instability, to fix these flaws see Moradi et al. (2014).

) has a unique solution, then the solution satisfies the form
) depend on u and its derivatives, then its solution can be obtained by the following iterative method.
By truncating the series of the left-hand side of (3.5), we obtain the approximate solution of (3.6)

Convergence of method
It is proved that u(x) is the solution of Equation (1.1).

Hence, as x N ⟶ y it shows that
It follows that Consequently, the method mentioned is convergent. ✷

Applications and numerical results
The reproducing kernel method for solving nonlinear Abel's integral equations with weakly singular Example 5.1 Consider the following nonlinear Abel integral equation (Wazwaz, 2011): The approximate solution by the proposed method for n = 2 is computed. The Taylor series approximation of ln(u(t)) is used in the following form The absolute errors obtained in spaces W 6 2 [0, 1], W 8 2 [0, 1] are given in Table 1. This is an indication of accuracy on the reproducing Kernel space. However, by increasing m, the approximate solution improves.
The comparisons between the exact solution and the numerical solutions for m = 8 are shown in Figure 1. We can see clearly that the numerical solutions and exact solution coincide completely. Figure 2 reveals the absolute errors in spaces W 6 2 [0, 1], W 8 2 [0, 1], respectively.  Example 5.2 In the second example, we solve the nonlinear Abel integral equation (Wazwaz, 2011): the exact solution is u(x) = cos(x + 1).
The approximate solution by the proposed method for n = 1 is computed. The Taylor series approximation of cos −1 (u(t)) is used in the following form The absolute errors obtained in spaces W 5 2 [0, 1], W 7 2 [0, 1] are given in Table 2. This is an indication of accuracy on the reproducing Kernel space. However, by increasing m, the approximate solution improves.
The comparisons between the exact solution and the numerical solutions for m = 7 are shown in Figure 3. We can see clearly that the numerical solutions and exact solution coincide completely. Figure 4 reveals the absolute errors in spaces W 5 2 [0, 1], W 7 2 [0, 1], respectively.
Example 5.3 Let us consider the nonlinear Abel integral equation (Wazwaz, 2011): The approximate solution by the proposed method for n = 2 is computed. The Taylor series approximation of u 2 (t) is used in the following form The absolute errors obtained in spaces W 5 2 [0, 1], W 7 2 [0, 1] are given in Table 3. This is an indication of accuracy on the reproducing Kernel space. However, by increasing m, the approximate solution improves.   The comparisons between the exact solution and the numerical solutions for m = 7 are shown in Figure 5. We can see clearly that the numerical solutions and exact solution coincide completely. Figure 6 reveals the absolute errors in spaces W 5 2 [0, 1], W 7 2 [0, 1], respectively.  Example 5.4 Now, we consider the singular nonlinear Abel integral equation (Wazwaz, 2011): the exact solution is u(x) = 1 + ln(x + 1).
The approximate solution by the proposed method for n = 1 is computed. The Taylor series approximation of e u(t) is used in the following form The absolute errors obtained in spaces W 5 2 [0, 1], W 8 2 [0, 1] are given in Table 4. This is an indication of accuracy on the reproducing Kernel space. However, by increasing m, the approximate solution improves.   The comparisons between the exact solution and the numerical solutions for m = 8 are shown in Figure 7. We can see clearly that the numerical solutions and exact solution coincide completely. Figure 8 reveals the absolute errors in spaces W 5 2 [0, 1], W 8 2 [0, 1], respectively.

Conclusions
To numerically solve nonlinear Abel's integral equations by means of the reproducing kernel method, the reproducing kernel functions as a basis and Taylor series to remove singularity were used. The absolute errors in two spaces were computed. By increasing m, the accuracy of the approximate solution improves. So, to get the more accurate result, it is sufficient to increase m. As seen from the examples, the method can be accurate and stable. 1.5 10 9 2. 10 9 2.5 10 9 3. 10 9 3.5 10 9