Multi-peak solutions of non-linear elliptic singularly perturbed reaction-diffusion equations using finite element simulation

• Published in: Journal of the Taiwan Institute of Chemical Engineers Vol. 50; pp. 56 - 68
• Main Authors: Singh, Akhilesh Kumar, Kumar, Manoj
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.05.2015
• Subjects: Finite element method; Newton's method; Non-linear elliptic boundary value problem; Reaction-diffusion equation; Singular perturbation; Finite element method; Newton's method; Reaction-diffusion equation; Non-linear elliptic boundary value problem; Singular perturbation
• ISSN: 1876-1070; 1876-1089
• DOI: 10.1016/j.jtice.2014.12.001

The present paper deals with a review of nonlinear reaction-diffusion models appearing in different branches of science and engineering. The multi-peak solutions of non-linear elliptic singularly perturbed reaction-diffusion equations do not appear to have been previously studied in detail, computationally. The finite element method for the solution of non-linear elliptic singularly perturbed reaction-diffusion equations (subject to appropriate Dirichlet's boundary condition) is discussed, and a variant of Newton's method having fifth order of convergence is used to linearize the nonlinear system of equations. Examples of nonlinear elliptic singularly perturbed reaction-diffusion equation having non-linearity in homogeneous/non-homogeneous form are considered to show the existence of multi-peak solutions.