Extending the applicability of Newton’s method for a class of boundary value problems using the shooting method

doi:10.1016/j.amc.2020.125378

Applied Mathematics and Computation

Volume 384, 1 November 2020, 125378

https://doi.org/10.1016/j.amc.2020.125378 Get rights and content

Abstract

We use Newton’s method to approximate locally unique solutions for a class of boundary value problems by applying the shooting method. The utilized operator is Fréchet-differentiable between Banach spaces. These conditions are more general than those that appear in previous works. In particular, we show that the old semilocal and local convergence criteria for Newton’s method involving Banach space value operators can be replaced by weaker ones. Hence, extending the applicability of the method. Several numerical examples are developed to test the new convergence criteria and also compare them to the old ones.

Introduction

In many scientific and engineering problems the main objective is to solve an equation. Equivalently, the principal aim may be to find a good local approximation x* for the equation: $f (x) = 0,$ using the determined Fréchet–differentiable operator f defined in a nonempty, open and convex domain E of a Banach space T₁ with correspondence in a Banach space T₂.

We have focused, in this work, on problems such as the following: $u^{″} = φ (v, u, u^{'}), a \leq v \leq b,$ with $u (a) = α, u (b) = β .$

Although there are equations appearing in these type of problems that can be solved by applying an analytical way, the solution is often approximated numerically through an iterative method due to its high level of complexity. Normally, Newton’s method is used due to its easily, efficient and simple implementation.

A common technique used to solve problems involving differential equations with boundary conditions is the so called shooting method. This method acquires this name because of the desire, to arrive to the boundary final point, starting from the point on the frontier boundary, or in case of it not being exact, to approximate it as much as possible. Other methods can also be used to obtain solutions. The main complication with this method is that in order to ensure convergence, the steps involved must be deeply studied. There is also the possibility that it may not even be convergent.

The type of convergence usually analyzed is the semilocal type, which is focused on the information around a starting point in order to turn on conditions of convergence to the iterative process towards the sought solution. There is a plethora of studies that analyze Newton’s method as in Amat et al. [1], Argyros and Hilout [3], Argyros and Magreñán [4], Burden et al. [6], Kantorovich and Akilov [7], Kelley [9], 18 [10], Parhi and Gupta [11], Rheinboldt [12], Sánchez [13], Traub [14].

The problem appearing in this method and in others is that the starting point is needed, so that it approximates the solution, being such starting point unknown at first. Once conditions are given on this point, there are specific conditions that can ensure convergence.

The layout of this work is: in Section 2 the shooting method of resolution for problems with border values is detailed. A study of the general convergence of Newton’s method that appears in these problems is performed as well depending on whether the function involved is increasing or not. A semilocal and local convergence analysis is presented for Newton’s method involving Banach space valued operators once the method is defined, in Section 3. The convergence criteria are improved. Finally, in Section 4, examples of the new developed theory are included.

Section snippets

Shooting method

Consider the boundary value problem ${\begin{matrix} \frac{d^{2} \bar{u}}{d v^{2}} + g (\bar{u}) = 0, \\ \bar{u} (a) = A, \bar{u} (b) = B, \end{matrix}$ where $\bar{u} (v)$ is a real valued function and $g (\bar{u})$ is an arbitrary function. The shooting method is a usual way to approximate a solution of a boundary value problem like Eq. (2.1). Here, we fix an initial aproximation of the initial slope, α₁, and we compute the solution $\bar{u} (v, α_{1})$ to obtain $\bar{u} (b, α_{1})$ (see Fig. 1).

We have several techniques if the value of $| \bar{u} (b, α_{1}) - B |$ is larger than desired. One method, for example requires

Convergence analysis

Consider H: E ⊂ T₁ → T₂ to be continuously differentiable in the Fréchet sense.

We present the semi–local followed by the local convergence analysis of Newton’s method $v_{n + 1} = v_{n} - H^{'} {(v_{n})}^{- 1} H (v_{n}) .$

Definition 3.1

We say that operator H′ satisfies the center–Lipschitz condition on E, if there exists L₀ > 0 such that for each v ∈ E, and some fixed v₀ ∈ E $∥ H^{'} (v) - H^{'} (v_{0}) ∥ \leq L_{0} ∥ v - v_{0} ∥ .$ Suppose that μ > 0 is such that $∥ H^{'} {(v_{0})}^{- 1} ∥ \leq μ$ . Define the set $E_{0} = D \cap U (v_{0}, \frac{1}{μ L_{0}})$ .

Definition 3.2

We say that operator H′ satisfies the restricted–Lipschitz condition

Numerical examples

In this section we show several examples where the previous conditions are analyzed.

Example 4.1

As an academic and motivational example, let us set $T_{1} = T_{2} = R,$ $E = \bar{U} (1, 1 - p),$ $p \in [0, \frac{1}{2})$ . Consider function ϕ as $ϕ (v) = v^{3} - p .$ Using (4.1), we get $μ = \frac{1}{3},$ $η = \frac{1}{3} (1 - p),$ $L_{0} = 3 (3 - p),$ $L = 6 (1 + \frac{1}{μ L_{0}})$ and $L_{1} = 6 (2 - p)$ . Notice that for each $p \in [0, \frac{1}{2})$ $L_{0} < L < L_{1} .$ The Kantorovich convergence criterion (3.12) is not satisfied, since $h_{1} > \frac{1}{2} for each p \in [0, \frac{1}{2}) .$ However, our condition (3.9) gives $h \leq \frac{1}{2} for each p \in I : = [0.4262, \frac{1}{2}) .$

Example 4.2

Let $E = \bar{U} (0, 1)$ and $v_{*} = {(0, 0, 0)}^{T}$ .

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Extending the applicability of Newton’s method for a class of boundary value problems using the shooting method

Abstract

Introduction

Section snippets

Shooting method

Convergence analysis

Numerical examples

Appl. Math. Comput.

Appl. Math. Comput.

Convergence and Applications of Newton–Type Iterations

Computational Methods in Nonlinear analysis

A Contemporary Study of Iterative Method

Computer Methods for Ordinary Differential Equations and Differential Algebraic Eccuations

Numerical Analysis