Extending the Usage of Newton’s Method with Applications to the Solution of Bratu’s Equation

We use Newton’s method to solve previously unsolved problems, expanding the applicability of the method. To achieve this, we used the idea of restricted domains which allows for tighter Lipschitz constants than previously seen, this in turn led to a tighter convergence analysis. The new developments were obtained using special cases of functions which had been used in earlier works. Numerical examples are used to illustrate the superiority of the new results.


Introduction
Let F : Ω ⊂ E 1 → E 2 be a differentiable operator in the sense of Fréchet, where E 1 and E 2 are Banach spaces and Ω is a nonempty and open set. A plethora of problems from many diverse disciplines are formulated using modeling which looks like Hence, the problem of locating a solution x * for Equation (1) is very important. Most people develop iterative algorithms approximating x * under some conditions, since a closed form solution cannot easily be obtained in general. The most widely used iterative method is Newton's method defined for an initial point x 0 ∈ Ω by    x 0 ∈ Ω, x n+1 = x n − F (x n ) −1 F(x n ) for all n = 0, 1, 2 . . . (2) Numerous convergence results appear in the literature based on which lim n→+∞ x n = x * . In this article, we introduce new semilocal convergence results based on our idea of restricted convergence region through which we locate a more precise set containing x n . This way, the majorizing constants and scalar functions are tighter leading to a finer convergence analysis.
To provide the semilocal convergence analysis Kantorovich used the condition [1] F (x) ≤ k, x ∈ Ω.
We shall find a tighter domain than Ω, where Equation (4) is satisfied. This way the new convergence analysis shall be at least as precise.
The layout of the rest of the article involves the semilocal convergence of Newton's method (Equation (2)) given in Section 2. Some numerical examples are also given in Section 2, whereas Section 3 contains the work on Bratu's equation.

Semilocal Convergence
Theorem 1 (Kantorovich's theorem [1]). Let E 1 and E 2 be Banach spaces. Let also F : Ω ⊆ E 1 → E 2 be a twice continuously differentiable operator in the sense of Fréchet where Ω is a non-empty open and convex region. Assume: Then, Newton's sequence defined in Equation (2) converges to a solution x * of the equation Furthermore, the following error bounds hold The Kantorovich theorem can be improved as follows: Theorem 2. Let E 1 and E 2 be Banach spaces. Let F : Ω ⊆ E 1 → E 2 be a twice continuously differentiable operator in the sense of Fréchet. Assume: and (iv) of Theorem 2 holds on Ω 1 with k replacing k, then Theorem 2 can be extended even further with Ω 1 , k = max{k 0 , k}, replacing Ω 0 andk, respectively, since Ω 1 ⊆ Ω 0 , sok ≤k.
Concerning majorizing sequences, define According to the proofs, {r n } and {r n } are majorizing sequences tighter than {s n } and {s n }, respectively, and as such, they converge under the same convergence criteria. Notice also that r * = lim n→+∞ r n ≤ s * and r * = lim n→+∞ r n ≤ s * .
In this case, we obtain from Therefore, we must have thatk γη ≤ 1 2 and k 0 γη < 1 which are true for p ∈ 0.42973177 . . . , 1 2 since k 0 < k, sok = k. Hence, we have extended the convergence interval of the previous cases.
The sufficient convergence criterion for the modified Newton's method x 0 given in Ω, is the same as the Kantorovich condition (iv). In [7], though we proved that this condition can be replaced by k 0 γη ≤ 1 2 which is weaker if k 0 < k. In the case of the example at hand, we have that this condition is satisfied as in the previous case interval. Therefore, by restricting the convergence domain, sufficient convergence criteria can be obtained for Newton's method identical to the ones required for the convergence of the modified Newton's method. The same advantages are obtained if the preceding Lipschitz constants are replaced by the ψ functions that follow.
It is worth noting that the center-Lipschitz condition (not introduced in earlier studies) makes it possible to restrict the domain from Ω to Ω 0 (or Ω 1 ), where the iterates actually lie and where can be used instead of the less tight estimate used in Theorem 1 and in other related earlier studies using only condition (iv) in Theorem 1.
Next, the condition Next, we show how to improve these results by relaxing Equation (5) using even weaker conditions where v : [t 0 , +∞) ∪ {0} → R is a non-decreasing continuous function satisfying v(t 0 ) ≥ 0. Suppose that equation γv(t − t 0 ) = 1 has at least one positive solution. Denote by ρ 1 the smallest such solution.
Moreover, suppose that or Equation (6) and If function v is strictly increasing, then we can choose ρ 1 = v −1 1 γ + t 0 . Notice that Equation (5) implies Equations (6) and (7) or Equations (6) and (8) and Next, we show that ψ 1 or ψ 2 can replace ψ in the results obtained in Reference [4]. Then, in view of Equations (9)-(11), the new results are finer and are provided without additional cost, since ψ requires the computation functions v, ψ 1 and ψ 2 as special cases. Notice that function v is needed to determine ρ 1 (i. e., Ω 0 and Ω 1 ) and that Ω 0 ⊆ Ω and Ω 1 ⊆ Ω 0 .

Bratu's Equation
Bratu's equation is defined by the following nonlinear integral equation where −∞ < α < β < ∞, µ ∈ R + and the kernel T is the Green's function Observe that Equation (12) can also be seen as the following boundary value problem [8]: Let µ > 0 and α = 0. It follows from [8] that Equation (12) has two solutions such that  Bratu's equation appears in connection to many problems: combustion, heat transfer, chemical reactions, and nanotechnology [9].
Using Newton's method, we approximate the solutions of Bratu's equation.
But condition (3) does not hold if operator (13) is defined by Equation (13), since Therefore, it is clear that F (x) is not bounded in a general domain Ω. However, it is hard to find a region containing a solution of F(x) = 0 and such that F (x) is bounded there.
Our aim is to solve F(x) = 0 using Newton's method x 0 given in Ω, Then, we solve Using m nodes in the Gauss-Legendre quadrature formula where the nodes t i and the weights β i are known. We can write We shall relate sequence {x n } with its majorizing sequence To achieve this using Equation (16), we compute F , F and where y ∈ R m , F (x)y z = −µA(e x 1 y 1 z 1 , e x 2 y 2 z 2 , . . . , e x m y m z m ) T , y = (y 1 , y 2 , . . . , y m ), and z = (z 1 , z 2 , . . . , z m ). Let B(x, ρ) = {y ∈ R m ; y − x ≤ ρ} and let B(x, ρ) be its closure. Clearly, Theorems 1 and 2 hold if operator F is defined by Equation (16) and Newton's method in the form of Equation (14) is used.
We shall verify the hypotheses of these theorems, so we can solve our problem. To achieve this, and where u(x 0 ) = (e x 1 , e x 2 , . . . , e x m ) T and x 0 = ( x 1 , x 2 , . . . , x m ) T . Moreover, we have F (x)y z ≤ µ A (e x 1 y 1 z 1 , e x 2 y 2 z 2 , . . . , e x m y m z m ) T and F (x) ∞ ≤ µ A ∞ e x ∞ , where we used the infinity norm. Notice that F (x) ∞ is not bounded, since e x ∞ is increasing as a function of x ∞ . Hence, Theorem 1 or Theorem 2 cannot be used.
as an auxiliary function to construct majorazing sequence {s n }. We also use the sequence Note then that lim lim n→+∞ r n = r * = 0.1420714278 . . . and lim n→+∞ r n = r * = 0.1415728924 . . .. We also obtain the a priori error estimates shown in Table 2, which shows that the error bounds are improved under our new approach.  In this section, we consider the alternative to Equation (4) condition since ψ 1 is non-decreasing. Then, we look for a function f 1 The solution of Equation (21) is given by Define also Otherwise, i.e., if f (t) ≤ f 0 (t), then the following results hold with f 0 replacing f 1 . Notice that f 1 is the function obtained by Kantorovich if t 0 = s 0 and ψ 1 ( (22) is reduced to with γ and η defined in Equations (17) and (18), respectively. Next, we need the auxiliary results for function f 1 .
Lemma 1. Let f 1 be the function defined in Equation (23) and Then: (a) α 1 is the unique minimum of f 1 in [t 0 , +∞).
Next, we define the scalar sequence
We need an auxiliary result relating sequence {x n } to {t n }.  (26) is satisfied, then x n ∈ B(x 0 , t * − t 0 ), for n ≥ 1, where t * is the smallest positive root of f 1 (t) = 0. Then, sequence (25) is majorizing for the sequence {x n }:

Proof.
Observe that We prove the following four items for all n ≥ 1: Firstly, from Γ 1 exists and Secondly, from Taylor's series and Equation (14), it follows that Thirdly, Fourthly, Then, if (i)-(iv) hold for all n = 0, 1, 2, . . . , k, we show in an analogous way that these items hold for n = k + 1 too.
Proof. Sequence {x n } converges, since {t n } is its majorizing sequence. Then, if x * = lim n→+∞ x n , x * − x n ≤ t * − t n , for all n ≥ 0. Moreover, the sequence { F (x n ) } is bounded. By the continuity of F, we have F(x * ) = 0, since F(x n ) = F (x n )(x n+1 − x n ) ≤ F (x n ) x n+1 − x n and lim n→+∞ x n+1 − x n = 0.
To show the uniqueness of x * , let y * be another solution of Equation (16) it follows that x * = y * , provided that the operator Q = 1 0 F (x * + t(y * − x * )) dt is invertible. To prove that Q is invertible, we prove equivalently that there exists the operator P −1 , where P = Γ 0 1 0 F (x * + θ(y * − x * )) dθ. Indeed, as so P −1 exists.

Remark 3.
We have by Equation (22) (a) If v = ψ = ψ 1 , the results in this study coincide with the ones in [4]. Moreover, if inequality in Equations (9)-(11) is strict, then, the new results have the following advantages: weaker sufficient convergence conditions, tighter error estimates on x n+1 − x n , x n − x * and at least as precise information on the location of the solution x * . (b) These results can be improved even further, if we simply use the condition and majorizing function f 2 (as in f 1 with ψ 1 = ψ 2 , t 0 = t 1 ) (also see the numerical section).

Remark 5.
(a) It is worth noting that there are alternative approaches to the root-finding other than Newton's method [10,11], where the latter one has cubic order of convergence, whereas Newton's is only quadratic. (b) If the solution is sufficiently smooth, then one can use generalized Gauss quadrature rules for splines.
This way, instead of projecting f into a space of higher-degree polynomials as is done in our article, one can project it to a spline space (see [12][13][14]). These quadratures in general do not affect the convergence order, but they do make the computation more efficient, since fewer quadrature points are required to reach a certain error tolerance.
In Section 2, we have seen that F (x) ∞ ≤ A ∞ e x ∞ , so that F (x) ∞ is not bounded. Then, any solution x * of the particular system given by Equation (16) should satisfy x * ∞ ≤ A ∞ e x * ∞ . We can take the region B(0, ρ), with ρ ∈ (r 1 , r 2 ) and r 1 = 0.14247951 . . . and r 2 = 3.27838858 . . ., where F (x) ∞ is bounded and contains the solution x * (see Figure 2). The convergence of Newton's method to x * follows Kantorovich's Theorem 1.