Discrete Galerkin and Iterated Discrete Galerkin Methods for Derivative-Dependent Fredholm–Hammerstein Integral Equations with Green’s Kernel

In this article, we look at a class of two-point nonlinear boundary value problems and transform this into derivative-dependent Fredholm–Hammerstein integral equations, i.e., the integral equation, where the kernel is of Green’s type, and the nonlinear function inside the integral is dependent on the derivative. We obtain the error analysis by replacing all the integrals in the Galerkin method with numerical quadrature. We propose the discrete Galerkin and iterated discrete Galerkin methods by piecewise polynomials to obtain the convergence analysis of these derivative-dependent Fredholm–Hammerstein integral equations. By choosing the numerical quadrature rule appropriately, the convergence rates in Galerkin and iterated Galerkin methods are preserved. We show that the iterated discrete Galerkin method improves over the discrete Galerkin method in terms of the order of convergence. To demonstrate the theoretical results, several numerical examples are provided.

Nonlinear boundary value problems for ordinary differential equations can be found in the chemical reaction, nuclear physics, atomic structure, atomic calculation, and gas dynamics. Boundary value problems are difficult to address analytically in the majority of circumstances. Therefore, different numerical methods must be used to tackle these problems. The numerical methods for solving boundary value problems, such as the decomposition technique, the Adomian decomposition method, and the modified decomposition method, are widely documented in the literature (see [1,7,8,12,13,21,23]), where the authors considered the boundary value problem (1.1) using a nonlinear function φ that is not affected by derivatives. Although these numerical approaches have numerous advantages, they demand a tremendous amount of computer work since they need the computation of indeterminate coefficients in a series of nonlinear algebraic or more complex transcendental equations, which adds to the computational load (see [9,17,18]). Furthermore, in some circumstances, the indeterminate coefficients may not be uniquely determined. This could be the most significant drawback of the Adomian decomposition method for solving nonlinear boundary value problems. As a result, instead of addressing boundary value problems, an equivalent integral equation can be solved, which leads to derivative-dependent nonlinear Hammerstein integral equation: where the function g(·) and the Green's kernel κ(·, ·) are known, the unknown function ϑ has yet to be determined. In general, integral equations (1.3) reformulate nonlinear boundary value problems. The nonlinear Fredholm-Hammerstein integral equations, in which the nonlinear function is independent of the derivative, have enormous literature (see [10,14,16,19]). In Refs. [3][4][5], projection and iterated projection methods are used for nonlinear Fredholm integral equation with certain classes of kernels. Atkinson et al. [6] discussed the discrete Galerkin and iterated discrete Galerkin methods for linear Fredholm integral equations. In Ref. [4], Atkinson
Here we let the approximation subspaces where P r be the space of polynomials of degree ≤ r, (r ≥ 1). The Galerkin method for Eq. (2.9) is as follows: Find ν h ∈ X h such that In general, the integrations in the system (2.12) that arise due to the inner products and Ke i cannot be evaluated precisely. The replacement of these integrals by numerical quadrature leads to the discrete Galerkin method, which we describe below. We will first introduce some convenient notations Let q be a positive integer such that where ρ is the number of quadrature point on the interval [0, 1]. It is convenient to introduce the sets of indices Also, let k i , k = 1, 2, . . . , q, denote the elementary Lagrange polynomials associated with the nodes in τ i , so that the Lagrange interpolation polynomial of a function x at these nodes can be written as . . , q, then (2.16) can be written as Newton form of the interpolating polynomial: The actual evaluation is done by nested multiplication. In fact, we use (2.17) to construct a discrete analogue K h and L h of K and L, respectively.
It is convenient to introduce the following functions: With the above notation, we can define the discrete operator As a simpler computational definition for t ∈ Δ i , we can write , then we have the following: We define the discrete inner product Put l = (i − 1)p + j, i = 1, . . . , m and j = 1, 2, . . . , p, and t l=(i−1)p+j = x j i and w (i−1)p+j = w j i , and set R = mp. We define For convenient, we can write Q h : X → X h by where Qϑ i is the discrete orthogonal projection of ϑ i ∈ C(Δ i ) on the polynomial of degree less than r on Δ i , and Q h satisfies Now, we quote some crucial properties of Q h from Sloan [22], which we need in the convergence analysis.

Lemma 1.
Let Q h : X → X h be the discrete orthogonal projection operator given by (2.29). Then, there hold (i) For any ϑ ∈ X, where c is a general constant not dependent on h.
The Galerkin method for Eq. (2.9) is as follows: find ν h ∈ X h such that Let T h be the operator defined by Then, Eq. (2.36) can be written as Corresponding approximate solution ϑ h of ϑ is defined by The approximate solution for (2.9) in the iterated discrete Galerkin method is defined asν Applying Q h on both sides of Eq. (2.40), we have Using Q hνh = ν h in Eq. (2.40), we havẽ

Convergence Analysis
The existence and convergence of approximate and iterated approximate solutions in the discrete Galerkin method are discussed in this section. To accomplish so, we start with Vainikko theorem [24], which gives us the condition under which the solvability of one equation leads to the solvability of another.
We have chosen the Gauss two-point quadrature rule which gives degree of precision d = 4, hence the expected orders of convergence for r = 1 are a = 2 and b = 3.
From Tables 1 and 2, we can see that the iterated approximate solution has a higher convergence rate than the discrete Galerkin method's approximate solution. Now, we compare our numerical results with Galerkin and iterated Galerkin methods as discussed by Moumita et. al. [20]. We are using here Gauss two-point quadrature rule, which preserve the same order of convergence as in Galerkin and iterated Galerkin methods (Tables 3, 4) but we got better error in discrete Galerkin and iterated discrete Galerkin methods.