Analytical Solutions of Generalised Emden–Fowler Initial and Boundary Value Problems of Higher Order

• Published in: International journal of applied and computational mathematics Vol. 10; no. 2
• Main Author: Awonusika, Richard Olu
• Format: Journal Article
• Language: English
• Published: New Delhi Springer India 01.04.2024; Springer Nature B.V
• Subjects: Applications of Mathematics; Boundary conditions; Boundary value problems; Computational Science and Engineering; Exact solutions; Initial conditions; Mathematical analysis; Mathematical and Computational Physics; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Nuclear Energy; Operations Research/Decision Theory; Original Paper; Power series; Theoretical; Thermal expansion; 35A08; 65L05; Power series method; 34B16; 33C45; Emden–Fowler equation; Generalised Emden–Fowler-type equation; Generalised Cauchy product; 33C05; Series solution; 34A08
• ISSN: 2349-5103; 2199-5796
• DOI: 10.1007/s40819-024-01676-x

In this paper, the analytical solutions of a class of generalised linear and nonlinear Emden–Fowler-type equations of higher order are presented using a power series method. The proposed equations are subject to appropriate initial and boundary conditions. A generalised Cauchy product is applied to reduce the nonlinear terms to power series whose expansion coefficients are expressed recursively as finite sums involving the expansion coefficients of the assumed series solution. In the case of boundary value problems, required initial conditions are assumed and introduced as dummy constants, which are subsequently determined using the corresponding right boundary conditions. Several examples of Emden–Fowler-type initial and boundary value problems of the third, fourth, fifth, and sixth orders are explicitly given to illustrate the proposed method. For comparison purposes, we consider known problems in the literature. Interestingly, in all the examples considered, our series solutions agree excellently with the exact solutions and published results in the literature.