A New Iteration Method Based on Green’s Functions for the Solution of PDEs

• Published in: International journal of applied and computational mathematics Vol. 3; no. 4; pp. 3091 - 3103
• Main Authors: Khuri, S. A., Sayfy, A., Zaveri, Aisha
• Format: Journal Article
• Language: English
• Published: New Delhi Springer India 01.12.2017; Springer Nature B.V
• Subjects: Applications of Mathematics; Applied mathematics; Boundary layers; Computational mathematics; Computational Science and Engineering; Differential equations; Fixed points (mathematics); Iterative methods; Mathematical and Computational Physics; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Nuclear Energy; Operations Research/Decision Theory; Operators (mathematics); Original Paper; Partial differential equations; Theoretical; Fixed point iteration schemes; Green’s function; Partial differential equations; Numerical solution
• ISSN: 2349-5103; 2199-5796
• DOI: 10.1007/s40819-016-0289-x

A novel approach that combines iterative methods and Green’s functions is presented for the numerical treatment of a class of partial differential equations (PDEs) subject to specified boundary conditions. The essence of the proposed strategy is to define an integral operator, expressed in terms of Green’s function, and then apply well-known fixed point iterations schemes, including Picard’s and Krasnoselskii–Mann’s. The technique is implemented on a number of examples, including linear and nonlinear PDEs. These numerical experiments elucidate reliability and efficiency of the approach. The results are very promising as they yield highly accurate approximations when compared to closed-form solutions.