A new approach based on embedding Green’s functions into fixed-point iterations for highly accurate solution to Troesch’s problem

A novel Picard–Ishikawa–Green's iterative scheme for solving third‐order boundary value problems

“…$$$ {a}_1{y}_1^{\prime \prime }(s)+{b}_1{y}_2^{\prime \prime }(s)+{c}_1{y}_3^{\prime \prime }(s)+\frac{1}{p(s)}={a}_2{y}_1^{\prime \prime }(s)+{b}_2{y}_2^{\prime \prime }(s)+{c}_2{y}_3^{\prime \prime }(s) $$$ and alongside the fact that $$$ G\left(x,s\right) $$$ is a solution of $$$ L\left[G\left(x,s\right)\right]=\delta \left(t-s\right) $$$ as in [17].…”

“…Based on how worthwhile the fixed point iterative scheme has been in approximating the solution of IVPs and BVPs, a new method for approximating the solution of BVPs was introduced. This new method is based on embedding Green's function in a traditional fixed point iterative scheme (see [17][18][19][20][21][22] and other references therein).…”

Iteration with Bisection to Approximate the Solution of a Boundary Value Problem

“…, we need k ≥ 20. Below, we show how to find ω 20 for the nonlinear second-order right focal boundary value problem (7), (8).…”

“…See Zeidler [4] or Granas and Dugundji [5] for a thorough treatment of the classical techniques. For papers using Green's function approaches, see Kafri and Khuri [6], Kafri, Khuri, and Sayfy [7], Khuri and Louhichi [8], or Khuri and Sayfy [9]. See also Duffy [10] for a review of Green's function techniques.…”

Comparison of Basic Iterative Methods Used to Solve of Heat and Fluid Flow Problems

“…Mann and his colleagues also generalised the variational iteration method for proper treatment of the boundary value problem in [10]. Kafri et al, [11] recently presented a new approach for obtaining a numerical solution to Troesch's nonlinear boundary value problem by utilising Green's functions and manipulating fixed-point iterations such as Picard's and Kranoselkii-schemes. Mann's To approximate the solution of a two-point boundary value problem, we propose a fixed-point iteration method like the Mann iteration process in this paper.…”