An efficient numerical technique for the solution of nonlinear singular boundary value problems

Positive solutions of nonlinear singular third-order two-point boundary value problem

This article is concerned with a class of singular nonlinear fractional boundary value problems with p‐Laplacian operator, which contains Riemann–Liouville fractional derivative and Caputo fractional derivative. The boundary conditions are made up of two kinds of Riemann–Stieltjes integral boundary conditions and nonlocal infinite‐point boundary conditions, and different fractional orders are involved in the boundary conditions and the nonlinear term, respectively. Based on the method of reducing the order of fractional derivative, some properties of the corresponding Green’s function, and the fixed point theorem of mixed monotone operator, an interesting result on the iterative sequence of the uniqueness of positive solutions is obtained under the assumption that the nonlinear term may be singular in regard to both the time variable and space variables. And we obtain the dependence of solution upon parameter. Furthermore, two numerical examples are presented to illustrate the application of our main results.

In this work, the electrohydrodynamic flow equation of a fluid in an ion drag configuration in a circular cylindrical conduit is considered. The differential equation of the problem is a highly nonlinear and singular boundary value problem (BVP). An analytical solution of the problem is presented using the differential transform method (DTM). In this method, the differential equation and related boundary conditions transformed into a recurrence set of equations and at the end, the coefficients of Taylor series are calculated based on the solution of this set of equations. DTM results are compared with the numerical solution of the problem and an excellent accuracy is observed. It showed that the proposed method overcame the nonlinearity and singularity of the problem without any need to restrict assumptions or linearization. Most of the scientific problems and engineering phenomena are usually defined based on the nonlinear differential equations of both initial value problems (IVP) and boundary value problems (BVP). It is important to remember that except a limited number of these problems, finding exact analytical solution is difficult. Therefore, researchers use semi-analytical techniques and numerical methods. The semi-analytical methods such as the perturbation method (PM), Adomian decomposition method (ADM), variational iteration method (VIM), homotopy perturbation method (HPM), homotopy analysis method (HAM), and differential transform method (DTM) are useful mathematical ones, which present the analytical solutions for nonlinear differential equations. In this paper, solution of electrohydrodynamic flow equation of a fluid in an ion drag configuration in a circular cylindrical conduit is presented using DTM. Already, the electrohydrodynamic flow of a fluid and its governing equations is considered (1). The electrohydrodynamic flow is important in analysis of the flow meters, accelerators, pumps and magnetohydrodynamic generators. The differential equation of the problem is the nonlinear singular boundary value problem. 2 2

Let T be a time scale. Let T be a time scale. We study the existence of positive solutions for the nonlinear three-point singular boundary value problem with higher-order p-Laplacian dynamic delay differential equations on time scales, subject to some boundary conditions. By using the fixed-point index theory, the existence of positive solution and many positive solutions for nonlinear four-point singular boundary value problem with p-Laplacian operator are obtained.

Let T be a time scale. In this paper, we study the existence of multiple positive solutions for the following nonlinear singular m-point boundary value problem dynamic equations with sign changing coefficients on time scales

Existence and uniqueness of solutions of a class of two-point singular nonlinear boundary value problems

The classical homotopy perturbation method proposed by J. H. He, Comput. Methods Appl. Mech. Eng. 178, 257 (1999) is useful for obtaining the approximate solutions for a wide class of nonlinear problems in terms of series with easily calculable components. However, in some cases, it has been found that this method results in slowly convergent series. To overcome the shortcoming, we present a new reliable algorithm called the domain decomposition homotopy perturbation method (DDHPM) to solve a class of singular two-point boundary value problems with Neumann and Robin-type boundary conditions arising in various physical models. Five numerical examples are presented to demonstrate the accuracy and applicability of our method, including thermal explosion, oxygen-diffusion in a spherical cell and heat conduction through a solid with heat generation. A comparison is made between the proposed technique and other existing seminumerical or numerical techniques. Numerical results reveal that only two or three iterations lead to high accuracy of the solution and this newly improved technique introduces a powerful improvement for solving nonlinear singular boundary value problems (SBVPs).

In this paper, a collocation method based on Laguerre wavelets is proposed for the numerical solutions of linear and nonlinear singular boundary value problems. Laguerre wavelet expansions together with operational matrix of integration are used to convert the problems into systems of algebraic equations which can be efficiently solved by suitable solvers. Illustrative examples are given to demonstrate the validity and applicability of this technique, and the results have been compared with the exact solutions.

In this article, we consider the fourth order non‐self‐adjoint singular boundary value problem with λ as a parameter measures the speed at which new particles are deposited. This differential equation is non‐self‐adjoint, so finding its solutions is not easy by usual methods. Also, this does not have a unique solution; therefore, finite differences and discrete methods may not be applicable. Here, to find approximate solutions, we propose to use the Adomian decomposition method (ADM) and compute the approximate solutions which are fully dependent on the size of the parameter λ. Here, we prove theoretically that the approximate solutions converge to the exact solutions. For small positive values of parameter λ, the above equation has two solutions. No solutions could be found for large positive values of the parameter. The uniqueness of the solution may not be guaranteed for a fixed positive value of the parameter. For negative values of parameter existences of the solution are always guaranteed.

An existence theorem for a class of nonlinear singular boundary value problems

A fourth-order non-uniform mesh optimal B-spline collocation method for solving a strongly nonlinear singular boundary value problem describing electrohydrodynamic flow of a fluid

A collocation method for solving a class of singular nonlinear two-point boundary value problems

We present sufficient conditions for the existence of at least twin or triple positive solutions of a nonlinear four-point singular boundary value problem with a -Laplacian dynamic equation on a time scale. Our results are obtained via some new multiple fixed point theorems.

A note on existence-uniqueness results for a class of doubly singular boundary value problems

A novel method for nonlinear singular fourth order four-point boundary value problems