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

Amechi Okeke G., Victor Udo A., Rasulov Z.

Mathematical Methods in the Applied Sciences, vol.47, no.9, pp.7255-7269, 2024 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 47 Issue: 9
  • Publication Date: 2024
  • Doi Number: 10.1002/mma.9971
  • Journal Name: Mathematical Methods in the Applied Sciences
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Applied Science & Technology Source, Communication Abstracts, Compendex, INSPEC, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Page Numbers: pp.7255-7269
  • Keywords: boundary value problem, JG-stability, Picard–Ishikawa iterative scheme, strong convergence
  • Yıldız Technical University Affiliated: Yes


The purpose of this paper is to introduce a novel fixed point iterative scheme based on Green's function, called the Picard–Ishikawa–Green's iterative scheme and use it in approximating the solution of boundary value problems (BVPs). It is proved that Picard–Ishikawa–Green's scheme converges strongly for an integral operator which represents the solution of BVP and the scheme is stable. Moreover, we prove that the integral operator is a contraction. Furthermore, it is shown that the novel scheme converges faster than all of Ishikawa–Green's, Khan–Green's, and Mann–Green's schemes. Finally, numerical examples are given to substantiate the validity of our results for third-order BVPs. Our results extend and generalize several other results in literature.