A note on variation iteration method with an application on Lane–Emden equations

PurposeIn this article, the authors consider the following nonlinear singular boundary value problem (SBVP) known as Lane–Emden equations, −u″(t)-(α/t) u′(t) = g(t, u), 0 < t < 1 where α ≥ 1 subject to two-point and three-point boundary conditions. The authors propose to develop a novel method to solve the class of Lane–Emden equations.Design/methodology/approachThe authors improve the modified variation iteration method (VIM) proposed in [JAAC, 9(4) 1242–1260 (2019)], which greatly accelerates the convergence and reduces the computational task.FindingsThe findings revealed that either exact or highly accurate approximate solutions of Lane–Emden equations can be computed with the proposed method.Originality/valueNovel modification is made in the VIM that provides either exact or highly accurate approximate solutions of Lane-Emden equations, which does not exist in the literature.

Applications of a Generalized Singular Boundary Value Problem for the Exact Solutions of Some Temperature/Concentration Equations

In the field of fluid mechanics, the temperature distribution and the nanoparticles concentration are usually described by singular boundary value problems (SBVPs). Such SBVPs are also used to describe various models with applications in engineering and other areas. Generally, obtaining the analytic solutions of such kind of problems is a challenge due to the singularity involved in the governing equations. In this paper, a class of SBVPs is analyzed. The solution of this class is analyzed and investigated through developing several theorems and lemmas. In addition, the theoretical results are invested to construct several solutions for various models/problems in fluid mechanics in the literature. Moreover, the published results are recovered as special cases of our analysis.

Three-Order Multipoint Boundary Value Problems for p-Laplacian Operator on Time Scales

In this paper, the existence of positive solutions for the nonlinear four-point singular BVP for there-order with p-Laplacian operator on time scales will be studied. By using the fixed-point theory, the existence of positive solutions for nonlinear singular boundary value problem with p-Laplacian operator on time scales is obtained.

Numerical solution of the problem of recovering parameters for the limit model of an uninsulated vacuum plane diode

The model of a vacuum diode under the influence of a strong external magnetic field is considered. The uninsulated variant, when a part of electrons emitted from the cathode reaches the anode, is investigated. The model is described by a singular boundary value problem for a system of ordinary differential equations. The sensitivity of the problem solution to the change of input parameters is investigated. A coordinate descent method to restore parameters of the boundary value problem is implemented numerically.

A new application of Schrödinger-type identity to singular boundary value problem for the Schrödinger equation

In this paper, we present a modified Schrödinger-type identity related to the Schrödinger-type boundary value problem with mixed boundary conditions and spatial heterogeneities. This identity can be regarded as an $L^{1}$ L 1 -version of Fisher–Riesz’s theorem and has a broad range of applications. Using it and fixed point theory in $L^{1}$ L 1 -metric spaces, we prove that there exists a unique solution for the singular boundary value problem with mixed boundary conditions and spatial heterogeneities. We finally provide two examples, which show the effectiveness of the Schrödinger-type identity method.