Singular Two-point Boundary Value Problems Research Articles

Order-preserving fuzzy transform for singular boundary value problems of polytropic gas flow and sewage diffusion

We describe a fuzzy transform method to analyze two-point singular and non-singular boundary value problems with high-order solution accuracies. This framework implements a fuzzy transform that approximates solutions with fourth-order accuracy at the interior mesh points. The fuzzy components and the triangular base function are locally arranged with a three-point linear combination of solution values. This yields a tri-diagonal Jacobian matrix that can be easily computed in a space-time efficient manner. Since a linear system relates solution values and fuzzy components, it is easy to obtain solution approximations via fuzzy components by a tri-diagonal matrix inversion. In addition to the numerical solution, it is easy to determine an approximate analytic solution using a cubic spline interpolating polynomial from the data available with fuzzy components. The error estimates for approximate analytical solutions and numerical solutions are obtained by integrated absolute error and maximum absolute errors. The new mechanism is analyzed for convergence using matrix theory. Several linear and nonlinear equations of practical importance related to sewage diffusion and polytropic gas flow models are simulated to corroborate the new scheme's utility and fourth-order convergence. Mathematics subject classification03B80, 65L10, 65L12

Advanced numerical scheme and its convergence analysis for a class of two-point singular boundary value problems

In the past decades, many applications related to applied physics, physiology and astrophysics have been modelled using a class of two-point singular boundary value problems (SBVPs). In this article, a novel approach based on the shooting projection method and the Legendre wavelet operational matrix formulation for approximating a class of two-point SBVPs with Dirichlet and Neumann–Robin boundary conditions is proposed. For the new approach, an initial guess is postulated in contrast to the boundary conditions in the first step. The second step deals with the usage of the Legendre wavelet operational matrix method to solve the initial value problem (IVP). Further, the resulting solution of the IVP is utilized at the second endpoint of the domain of a differential equation in a shooting projection method to improve the initial condition. These two steps are repeated until the desired accuracy of the solution is achieved. To support the mathematical formulation, a detailed convergence analysis of the new approach is conducted. The new approach is tested against some existing methods such as various types of the variational iteration method, considering several numerical examples to which it provides high-quality solutions.

Hybrid Cubic B-spline Method for Solving A Class of Singular Boundary Value Problems

In this paper, a class of singular two-point boundary value problems are solved using hybrid cubic B-spline method. In this algorithm, a free parameter γ plays an important role to give accurate converge results for the solution. This parameter are chosen via optimization. Numerical examples are displayed to prove that our suggested method is flexible and effective, and the numerical results are compared with other numerical methods from the literature

A sinc-collocation approximation solution for strongly nonlinear class of weakly singular two-point boundary value problems

In this study, an efficient collocation method based on Sinc function coupled with double exponential transformation is developed. This approach is used for solving a class of strongly nonlinear regular or weekly singular two-point BVPs with homogeneous or non homogeneous boundary conditions. The properties of the Sinc-collocation scheme were used to reduce the computations of the problem to the nonlinear system of equations. To use the Newton method in solving the nonlinear system, its vectormatrix form was obtained. The convergence analysis of the method is discussed. The analysis show that the method is convergent exponential. In order to investigate the capability and accuracy of the method, it is applied to solve several existing problems chosen from the open literature. The numerical results compared with other existing methods. The obtained results indicate high capacity and rapid convergence of the proposed method.

Existence of positive solutions for a singular third-order two-point boundary value problem on the half-line

In this paper, we consider the following singular third-order two-point boundary value problem on the half-line of the form {x‴+ϕ(t)f(t,x,x′,x″)=0,0<t<+∞,x(0)=0,x′(0)=a1,x′(+∞)=b1,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extstyle\\begin{cases} x'''+\\phi (t)f(t,x,x',x'')=0, \\quad 0< t< +\\infty , \\\\ x(0)=0, \\qquad x'(0)=a_{1},\\qquad x'(+\\infty )=b_{1}, \\end{cases} $$\\end{document} where phi in C[0,+infty ), fin C([0,+infty )times (0,+infty )times mathbb{R}^{{2}},mathbb{R}) may be singular at x=0, and a_{1}, b_{1} are positive constants. Using the Leray–Schauder nonlinear alternative and the diagonalization method together with the truncation function technique, we obtain the existence and qualitative properties of positive solutions for the problem. As applications, an example is given to illustrate our result.

A highly accurate and computationally efficient technique for solving the electrohydrodynamic flow in a circular cylindrical conduit

The present work proposes a new, highly accurate and efficient computational technique for numerical solution of a strongly nonlinear singular two-point boundary value problem which governs electrohydrodynamic flow (EHF) of a fluid in an ion drag configuration in a circular cylindrical conduit. The nonlinearity confronted in the underlying problem is in the form of rational function and the governing equation has a singularity at the point z=0. The proposed method tackles the singularity behavior at the origin. Analysis of the proposed method is carried out. Numerical experiments were performed to demonstrate the applicability and robustness of the new method. The velocity field of EHF of a fluid in radial direction is computed. In order to justify the advantage of the method, the computed results are compared with the results obtained from some recently developed computational techniques in the literature. The effects of two relevant parameters, namely the strength of nonlinearity (σ) and the Hartmann electric number (H) on the velocity profile are investigated.

Neuron analysis through the swarming procedures for the singular two-point boundary value problems arising in the theory of thermal explosion

Neuron analysis through the swarming procedures for the singular two-point boundary value problems arising in the theory of thermal explosion

A New Accurate Approximate Solution of Singular Two-Point Boundary Value Problems

In this paper, A new accurate solutions are obtained for classes of singular two-point boundary value problems (BVPs) using a new solution procedure based on the construction of the auxiliary functions of the standard optimal homotopy asymptotic method (OHAM). This procedure give us accurate approximate solutions using only one order of approximation compared with the solutions obtained previously by the standard OHAM approximation of three order. The obtained numerical results which are displayed in tables and plotted graphically in figures leads to concluded that this procedure is efficient and of third order reliable for finding the solutions of singular two BVPs.

A Novel Neuroevolutionary Paradigm for Solving Strongly Nonlinear Singular Boundary Value Problems in Physiology

In the present research work, we designed a hybrid stochastic numerical solver to investigate nonlinear singular two-point boundary value problems with Neumann and Robin boundary conditions arising in various physical models. In this method, we hybridized Harris Hawks Optimizer with Interior Point Algorithm named HHO-IPA. We construct artificial neural networks (ANNs) model for the problem, and this model is tuned with the proposed scheme. This scheme overcomes the singular behavior of problems. The accuracy and applicability of the method are illustrated by finding absolute errors in the solution. The outcomes are compared with the results present in the literature to demonstrate the effectiveness and robustness of the scheme by considering four different nonlinear singular boundary value problems. Further, the convergence of the scheme is proved by performing computational complexity analysis. Moreover, the graphical overview of statistical analysis is added to our investigation to elaborate further on the scheme’s stability, accuracy, and consistency.

The series expansion and Chebyshev collocation method for nonlinear singular two-point boundary value problems

The solution of singular two-point boundary value problem is usually not sufficiently smooth at one or two endpoints of the interval, which leads to a great difficulty when the problem is solved numerically. In this paper, an algorithm is designed to recognize the singular behavior of the solution and then solve the equation efficiently. First, the singular problem is transformed to a Fredholm integral equation of the second kind via Green’s function. Second, the truncated fractional series of the solution about the singularity is formulated by using Picard iteration and implementing series expansion for the nonlinear function. Third, a suitable variable transformation is performed by using the known singular information of the solution such that the solution of the transformed equation is sufficiently smooth. Fourth, the Chebyshev collocation method is used to solve the deduced equation to obtain approximate solution with high precision. Fifth, the convergence analysis of the collocation method is conducted in weighted Sobolev spaces for linear singular equations. Sixth, numerical examples confirm the effectiveness of the algorithm. Finally, the Thomas–Fermi equation and the Emden–Fowler equation as some applications are accurately solved by the method.

Existence of a positive solution for a singular fractional boundary value problem with fractional boundary conditions using convolution and lower order proble

Existence of a positive solution is shown for two singular two-point fractional boundary value problems with fractional boundary conditions using fixed point theory, lower order problems, and convolution of Green's functions. A nontrivial example is included.

A Legendre reproducing kernel method with higher convergence order for a class of singular two-point boundary value problems

We study a higher-order Legendre reproducing kernel method (LRKM) for singular two-point boundary value problems (BVPs). Our method relies upon Legendre polynomials. We carry out error estimatation by using interpolation theory. Various numerical tests including linear and nonlinear problems are performed to reveal the stability and efficiency of the new scheme. Therewith, as an application of the proposed approach, a famous time-dependent partial differential equation, namely Allen–Cahn test problem, is transfomed into the associated system of linear BVPs by localizing in temporal variable and is challenged numerically in details by the LRKM.

Parametrization of Solutions to the Emden–Fowler Equation and the Thomas–Fermi Model of Compressed Atoms

For the nonlinear Emden–Fowler equation, a singular Cauchy problem and singular two-point boundary value problem on the half-line $$r \in [0, + \infty )$$ and on an interval $$r \in [0,R]$$ with a Dirichlet boundary condition at the origin and with a Robin boundary condition at the right endpoint of the interval are considered. For special parameter values, the given boundary value problem corresponds to the Thomas–Fermi model of the charge density distribution inside a spherically symmetric cooled heavy atom occupying a confined or infinite space, where $$R$$ denotes the boundary of a compressed atom and grows to infinity for a free atom. For the boundary value problem on the half-line, a new parametric representation of the solution is obtained that covers the entire range of argument values, i.e., the half-line $$r \in [0, + \infty )$$ , with the parameter $$t$$ running over the unit interval. For analytic functions involved in this representation, an algorithm for explicit computation of their Taylor coefficients at $$t = 0$$ is described. As applied to the Thomas–Fermi problem for a free atom, corresponding Taylor series expansions are given and they are shown to converge exponentially on the unit interval $$t \in [0,1]$$ at a rate higher than that for an earlier constructed similar representation. An efficient analytical-numerical method is presented that computes the solution of the Thomas–Fermi problem on the half-line with any prescribed accuracy not only in an neighborhood of $$r = + \infty $$ , but also at any point of the half-line $$r \in [0, + \infty )$$ . For the Cauchy problem set up at the origin, a new formula for the critical value of the derivative that corresponds to the solution of the problem on the half-line is derived. It is shown in a numerical experiment that this formula is more efficient than the Majorana formula. For the solution of the Cauchy problem with a positive derivative at the origin, a parametrization is obtained that ensures that the boundary conditions of the singular boundary value problem on the interval $$r \in [0,R]$$ are satisfied with a suitable $$R > 0$$ . An efficient analytical-numerical method for solving this Cauchy problem is constructed and numerically implemented.

Solutions of classes of differential equations using modified differential transform method

This paper proposed a numerical approximation scheme for solving singular two-point boundary value problems. This approach is based on a truncated series obtained by differential transform (DT) method, the Laplace transformation (LT) approach and process of Pade approximants. The modified differential transform method ´ (MDTM) aims to overcome the difficulty of solving these types of problems and gives a good approximation for the true solution in a large region. Some illustrative examples are presented to demonstrate the effectiveness and applicability of the new method and the obtained results are compared with the exact solution.

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

In this paper, a non-uniform mesh optimal B-spline collocation method is presented for the numerical solution of a singular two-point boundary value problem describing electrohydrodynamic flow (EF) of a fluid in a circular cylindrical conduit. The EF problem is highly nonlinear and has a singularity at the point r=0. Further, this problem has a boundary layer near right end of the solution domain. We consider a grading function to construct a non-uniform mesh over the problem domain. The non-uniform mesh is constructed in such a way that the mesh is finer near the right end boundary. Convergence of the proposed method is analyzed. The EF problem is solved for small and large values of the two relevant parameters: (i) strength of non-linearity β, and (ii) Hartmann electric number H. The effects of β and H on the velocity field are investigated. The computed results have been compared with those obtained by two other B-spline collocation methods over uniform mesh to show the advantage of the present method. It is shown that the present method provides O(h4) convergent approximation.

Fractal Cubic Spline Methods for Singular Boundary-Value Problems

In this article, we propose a new numerical method based on fractal cubic spline for the numerical approximate solutions of linear and nonlinear singular two-point boundary-value problems. Error analysis of the proposed method is examined and it is shown that proposed method has second-order convergence. Numerical examples are provided to validate the theoretical results.

A new high order numerical approach for a class of nonlinear derivative dependent singular boundary value problems

This paper is concerned with the design and analysis of a high order numerical method based on B-spline collocation approach to approximate the solution of a class of nonlinear derivative dependent singular two-point boundary value problems subjected to Neumann and Robin boundary conditions. Convergence result for the proposed method is established through Green's function approach. It is proved that the new method provides h6−order convergent approximation to the solution of the underlying problem. To validate this theoretical result, numerical experiments are carried out. It is shown that the rate of convergence of present method is two orders and four orders of magnitude higher when compared respectively to the uniform mesh B-spline collocation method and non-uniform mesh B-spline collocation method [41]. Moreover, the new method has shown its advantage (in terms of accuracy) over other existing methods [10,23,31,36,49] applied to the special cases of the problem.

Highly Accurate Method for Solving Singular Boundary-Value Problems Via Padé Approximation and Two-Step Quartic B-Spline Collocation

We propose a highly accurate method to solve a class of singular two-point boundary-value problems having singular coefficients. These problems often arise when a partial differential equation is reduced to an ordinary differential equation by physical symmetry. The proposed method is based on Pade approximation and two-step collocation. The resolution domain is divided, Pade approximant is used on the subdomain in the vicinity of the singular point and provides a new boundary condition, and then, two-step quartic B-spline is used on the remaining subdomain where the problem is transformed to a regular boundary-value problem. The subdomains are chosen optimally using a suitable optimization procedure. Some examples are presented along with a comparison of the numerical results with other methods.

ADMP: A Maple Package for Symbolic Computation and Error Estimating to Singular Two-Point Boundary Value Problems with Initial Conditions

A two step Adomian decomposition method (TSADM) coupled with pade approximations is applied for solving various two-point boundary value problems with initial conditions. This technique is based on a new definition of the Adomian polynomials and the TSADM. A Maple package ADMP is used to implement the proposed algorithm. The algorithm does not require linearization, perturbation, predicting the initial terms or a set of basis function or other restrictive assumptions and approximates the analytical solution. The validity of the applied package is given by two examples.

A new high-order numerical method for solving singular two-point boundary value problems

Recently, Goh et al. (2012) [ 28 ] proposed a numerical technique based on quartic B-spline collocation for solving a class of singular boundary value problems (SBVP) with Neumann and Dirichlet boundary conditions (BC). This method is only fourth-order accurate. In this paper, we propose an optimal numerical technique for solving a more general class of nonlinear SBVP subject to Neumann and Robin BC. The method is based on high order perturbation of the problem under consideration. The convergence of the proposed method is analyzed. To demonstrate the applicability and efficiency of the method, we consider four numerical examples, three of which arise in various physical models in applied science and engineering. A comparison with other available numerical solutions has been carried out to justify the advantage of the proposed technique. Numerical result reveals that the proposed method is sixth order convergent, which in turn is two orders of magnitude larger than in Goh et al. (2012) [ 28 ].