A Successive Linearization Method Approach to Solve Lane-Emden Type of Equations

We propose a new application of the successive linearization method for solving singular initial and boundary value problems of Lane-Emden type. To demonstrate the reliability of the proposed method, a comparison is made with results from existing methods in the literature and with exact analytical solutions. It was found that the method is easy to implement, yields accurate results, and performs better than some numerical methods

On the Numerical Integration of Singular Initial and Boundary Value Problems for Generalised Lane-Emden and Thomas-Fermi Equations

We propose a geometric approach for the numerical integration of singular initial value problems for (systems of) quasi-linear differential equations. It transforms the original problem into the problem of computing the unstable manifold at a stationary point of an associated vector field and thus into one which can be solved in an efficient and robust manner. Using the shooting method, our approach also works well for boundary value problems. As examples, we treat some (generalised) Lane-Emden equations and the Thomas-Fermi equation.Comment: 29 pages, 9 figure

Vieta-Lucas Wavelet based schemes for the numerical solution of the singular models

In this paper, numerical methods based on Vieta-Lucas wavelets are proposed for solving a class of singular differential equations. The operational matrix of the derivative for Vieta-Lucas wavelets is derived. It is employed to reduce the differential equations into the system of algebraic equations by applying the ideas of the collocation scheme, Tau scheme, and Galerkin scheme respectively. Furthermore, the convergence analysis and error estimates for Vieta-Lucas wavelets are performed. In the numerical section, the comparative analysis is presented among the different versions of the proposed Vieta-Lucas wavelet methods, and the accuracy of the approaches is evaluated by computing the errors and comparing them to the existing findings.Comment: 23 pages, 4 figures, 2 Table

A new approach for solving nonlinear singular boundary value problems

In this paper, an e_cient method based on Quasi-Newton's method and the simpli_ed reproducing kernel method is proposed for solving nonlinear singular boundary value problems. For the Quasi-Newton's method the convergence order is studied. The uniform convergence of the numerical solution as well as its derivatives are also proved. Numerical examples are given to demonstrate the e_ciency and stability of the proposed method. The numerical results are compared with exact solutions and the outcomes of other existing numerical methods

Asymptotics of conduction velocity restitution in models of electrical excitation in the heart

We extend a non-Tikhonov asymptotic embedding, proposed earlier, for calculation of conduction velocity restitution curves in ionic models of cardiac excitability. Conduction velocity restitution is the simplest non-trivial spatially extended problem in excitable media, and in the case of cardiac tissue it is an important tool for prediction of cardiac arrhythmias and fibrillation. An idealized conduction velocity restitution curve requires solving a non-linear eigenvalue problem with periodic boundary conditions, which in the cardiac case is very stiff and calls for the use of asymptotic methods. We compare asymptotics of restitution curves in four examples, two generic excitable media models, and two ionic cardiac models. The generic models include the classical FitzHugh–Nagumo model and its variation by Barkley. They are treated with standard singular perturbation techniques. The ionic models include a simplified “caricature” of Noble (J. Physiol. Lond. 160:317–352, 1962) model and Beeler and Reuter (J. Physiol. Lond. 268:177–210, 1977) model, which lead to non-Tikhonov problems where known asymptotic results do not apply. The Caricature Noble model is considered with particular care to demonstrate the well-posedness of the corresponding boundary-value problem. The developed method for calculation of conduction velocity restitution is then applied to the Beeler–Reuter model. We discuss new mathematical features appearing in cardiac ionic models and possible applications of the developed method

Bernstein operational matrix of differentiation and collocation approach for a class of three-point singular BVPs: error estimate and convergence analysis

Singular boundary value problems (BVPs) have widespread applications in the field of engineering, chemical science, astrophysics and mathematical biology. Finding an approximate solution to a problem with both singularity and non-linearity is highly challenging. The goal of the current study is to establish a numerical approach for dealing with problems involving three-point boundary conditions. The Bernstein polynomials and collocation nodes of a domain are used for developing the proposed numerical approach. The straightforward mathematical formulation and easy to code, makes the proposed numerical method accessible and adaptable for the researchers working in the field of engineering and sciences. The priori error estimate and convergence analysis are carried out to affirm the viability of the proposed method. Various examples are considered and worked out in order to illustrate its applicability and effectiveness. The results demonstrate excellent accuracy and efficiency compared to the other existing methods