Singular Boundary Value Problems Research Articles

A new mixed MADM-Collocation approach for solving a class of Lane–Emden singular boundary value problems

In this paper, a new approach is proposed for solving a class of singular boundary value problems of Lane–Emden type. It is well known that the Adomian decomposition method (ADM) fails to provide a convergent series solution to strongly nonlinear boundary value problem in the wider region and the B-spline collocation method yields unsatisfactory approximation in the presence of singularity. To avoid these shortcomings of both methods, we propose a novel numerical method based on a combination of modified Adomian decomposition method and quintic B-spline collocation method to obtain more accurate solution of the problem under consideration. The principal idea of this approach is to decompose the domain of the problem $$D = [0, 1]$$ into two subdomains as $$D = D_{1} U D_{2}= [0, \delta ] U [\delta , 1]$$ ( $$\delta $$ is vicinity of the singularity). In the first domain $$D_1$$ , the underlying singular boundary value problem is efficiently tackled by a modified Adomian decomposition method. The intent is to apply the ADM in the smaller domain for finding a satisfactory solution. Finally, in the second domain $$D_2$$ , a collocation approach based on quintic B-spline basis function is designed for solving the resulting regular boundary value problem. The error estimation of the quintic B-spline interpolation is supplemented. In addition, six illustrative examples are presented to demonstrate the applicability and accuracy of the new method. It is shown that the resulting solutions appear to be higher accurate when compared to some existing numerical methods.

Iterative Analysis of the Unique Positive Solution for a Class of Singular Nonlinear Boundary Value Problems Involving Two Types of Fractional Derivatives with p‐Laplacian Operator

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.

Special functions, integral transforms with applications

In this study, the author used integral transforms and a method of the exponential nature to deal with the families of fractional differential equations, Stieltjes type singular integral equations and boundary value problems. Certain integrals involving special functions are evaluated. Constructive examples are also provided throughout the paper. The main purpose of this article is to present mathematical results that are useful to researchers in a variety of fields.

A spline-based computational technique applicable for solution of boundary value problem arising in human physiology

Non-polynomial quintic spline functions based algorithms are used for computing an approximation to the nonlinear two point second order singular boundary value problems arising in human physiology. After removing the singularity by L' hospital rule, the resulting boundary value problem is then efficiently treated by employing non-polynomial quintic spline for finding the numerical solution. Two examples have been included and comparison of the numerical results made with cubic extended B-spline method and finite difference method.

A spline-based computational technique applicable for solution of boundary value problem arising in human physiology

Non-polynomial quintic spline functions based algorithms are used for computing an approximation to the nonlinear two point second order singular boundary value problems arising in human physiology. After removing the singularity by L' hospital rule, the resulting boundary value problem is then efficiently treated by employing non-polynomial quintic spline for finding the numerical solution. Two examples have been included and comparison of the numerical results made with cubic extended B-spline method and finite difference method.

Twoweak solutions for a singular (p,q)-Laplacian problem

Here, a singular boundary value problem involving the (p,q)-Laplacian operator in a smooth bounded domain in RN is considered. Using the variational method and critical point theory, the existence of two weak solutions is proved.

Septic B-spline method for solving nonlinear singular boundary value problems arising in physiological models

In this paper, we present a numerical method based on septic B-spline function for nonlinear singular second-order two-point boundary value problems, which depend on different physiological processes as thermal explosions problem and the steady state oxygen diffusion in a spherical cell with Michaelis–Menten uptake kinetics and distribution of heat sources in the human head. Septic B-spline method has a truncation error of O(h^8) and converges to the exact solution with O(h^6). The numerical problems show that our method is very effective. The resulting sets of differential equations are modified at the singular point and are treated by using septic B-spline for finding the numerical solution. The maximum absolute errors and the absolute residual errors are acceptable.

Positive solutions for singular impulsive Dirichlet boundary value problems

In this paper, a class of singular impulsive Dirichlet boundary value problems is considered. By using variational method and critical point theory, different parameter ranges are obtained to guarantee existence and multiplicity of positive classical solutions of the problem when nonlinearity exhibits different growths.

A new iterative algorithm for a strongly nonlinear singular boundary value problem

Recently, Mastroberardino (2011) proposed a homotopy analysis method (HAM) for solving a nonlinear singular boundary value problem (SBVP) governing electrohydrodynamic flow (EHF) of a fluid in an ion drag configuration in a circular cylindrical conduit. The nonlinearity confronted in this problem is in the form of rational function. The HAM (Mastroberardino, 2011) produces a series solution to the problem, but requires a large number of iterations for obtaining a reasonable analytical approximation. In this paper, we propose a new iterative technique, namely the discrete optimized homotopy analysis method (DOHAM), based on a domain decomposition approach and an optimized HAM for solving the same problem. Convergence of the new method is carried out. The uniqueness of solution of the problem is established. Computational results reveal that our method provides better results as compared to some existing iterative methods. The velocity field of electrohydrodynamic flow of a fluid is computed. The influence of the strength of non-linearity and the Hartmann electric number on the velocity profile is investigated.

Sumudu Transform with Modified Homotopy Perturbation Method to Solve Two Point Singular Boundary Value Problems

In this paper, a class of two point singular boundary value problems (SBVPs) has been solved by a new powerful technique. This technique relies on the Sumudu transform (ST) and modified homotopy perturbation method (MHPM). The proposed scheme is based on the freedom of the homotopy perturbation method (HPM) by introducing a suitable initial approximation; in addition, the residual error will be canceled in several points of the interest interval (RECP). Furthermore, only a first order approximation of the modified Sumudu transform homotopy perturbation method (MSTHPM) is required. the proposed method has been applied to solve several examples and the results have been compared with the exact solutions, it is found that the proposed solutions are of high accuracy and, therefore, MHPSTM is extremely efficient, simple and can be applied to other nonlinear problems.

Doubly singular boundary value problems with derivative dependent source function: A fast‐converging iterative approach

In this paper, a fast‐converging recursive scheme is presented to approximate the solution of a class of derivative dependent doubly singular boundary value problems (DDSBVP). First, the original problem is reformulated as an equivalent integral equation. The resulting integral equation is then efficiently tackled by an improved homotopy analysis method (IHAM). This method contains a parameter, which greatly accelerates the convergence of the series solution. The convergence of the method is carried out. To illustrate the efficiency and accuracy of the proposed recursive approach, we consider three nonlinear examples, including one physical model problem, which describes stress distribution on a rotationally shallow membrane cap. Results show that our method excels over the existing methods.

Positive solutions for a singular fractional nonlocal boundary value problem

We investigate a singular fractional differential equation with an infinite-point fractional boundary condition, where the nonlinearity f(t,x) may be singular at x = 0, and g(t) may also have singularities at t= 0 or t=1. We establish the existence of positive solutions using the fixed point index theory in cones.

Comparison of Green's functions for a family of boundary value problems for fractional difference equations

ABSTRACTIn this paper, we obtain sign conditions and comparison theorems for Green's functions of a family of boundary value problems for a Riemann-Liouville type delta fractional difference equation. Moreover, we show that as the length of the domain diverges to infinity, each Green's function converges to a uniquely defined Green's function of a singular boundary value problem.

Numerical Solution of Nonlinear Second Order Singular BVPs Based on Green’s Functions and Fixed-Point Iterative Schemes

This article discusses a numerical iterative scheme for the solution of a class of nonlinear singular boundary value problems. It introduces a recent approach, based on Green’s functions and Picard’s and Mann’s fixed-point iterations procedures, to tackle such problems. The convergence analysis of the proposed method is presented to verify its efficiency. A number of examples are given to demonstrate the applicability of the method. The numerical experiments show that this approach is better than many other existing techniques and that it is reliable, accurate and less time consuming.

B-spline collocation methods and their convergence for a class of nonlinear derivative dependent singular boundary value problems

This paper is concerned with the construction and convergence analysis of two B-spline collocation methods for a class of nonlinear derivative dependent singular boundary value problems (DDSBVP). The first method is based on uniform mesh, while the second method is based on non-uniform mesh. For the second method, we use a grading function to construct the non-uniform grid. We prove that the method based on uniform mesh is of second-order accuracy and the method based on non-uniform mesh is of fourth-order accuracy. Three nonlinear examples with derivative dependent source functions are considered to verify the performance and theoretical rate of convergence of present methods. Moreover, we consider some special cases of the problem under consideration in order to compare our methods with other existing methods. It is shown that our second method based on cubic B-spline basis functions has the same order of convergence as quartic B-spline collocation method [1]. Moreover, our methods yield more accurate results and are computationally attractive than the methods developed in [1–8]. The proposed methods are applied on three real-life problems, the first problem describes the distribution of radial stress on a rotationally shallow membrane cap, the second problem arises in the study of thermal explosion in cylindrical vessel and the third problem arises in astronomy.

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.

New Cubic B-spline Approximation for Solving Non-linear Singular Boundary Value Problems Arising in Physiology

In this article, a new cubic B-spline approximation method is presented for solving second order singular boundary value problems. The proposed numerical technique is based on cubic B-spline collocation method equipped with a new approximation for second order derivative. Several test problems arising in the field of Physiological sciences are considered. The approximate results are compared with the numerical techniques existing in literature. It is found that our new approximation performs superior to current methods due to its simplicity, straight forward interpolation and less computational cost.

A new approximate method and its convergence for a strongly nonlinear problem governing electrohydrodynamic flow of a fluid in a circular cylindrical conduit

In this paper, we propose a new approximate method, namely the discrete Adomian decomposition method (DADM), to approximate the solution of a strongly nonlinear singular boundary value problems describing the electrohydrodynamic flow of a fluid in an iron drag configuration in a circular cylindrical conduit. Convergence of the new method is analyzed. The velocity field of electrohydrodynamic flow of a fluid is determined. The influence of the Hartmann electric number and the strength of nonlinearity on the velocity profile is investigated. It is observed that the velocity field increases with the increase in the Hartmann electric number and decreases with the increase in the strength of nonlinearity. The results obtained by the proposed method are compared with results given in the literature.

The Application of Quartic Trigonometric B-spline for Solving Second Order Singular Boundary Value Problems

In this paper, a quartic trigonometric B-spline collocation approach is described and presented for the numerical solution of the second order singular boundary value problems. Several numerical examples are discussed to exhibit the feasibility and capability of the technique. The unknown coefficients \(C_{i}\), \(i=-4,-3,\ldots,n-1\) are obtained through optimization. The maximum errors \((L_{\infty})\) and norm errors \((L_{2})\) are also computed for different space size steps to assess the performance of the proposed technique. The rate of convergence is discussed numerically to be of fourth-order. The numerical solutions are contrasted with both analytical and other existing numerical solutions that exist in the literature. The comparison shows that the quartic trigonometric B-spline method is superior as it yields more accurate solutions.

A new highly accurate domain decomposition optimal homotopy analysis method and its convergence for singular boundary value problems

We describe a new iterative technique based on domain decomposition optimal homotopy analysis method for solving singular boundary value problems arising in various physical models in applied science and technology. This method produces approximate solution in the form of a series with easily computable components. The method is analyzed for convergence. Some computational examples are presented to illustrate the applicability and efficiency of the proposed method. The advantage in the domain decomposition optimal homotopy analysis method as compared with the classical optimal homotopy analysis method is that fewer number of solution components are required to obtain similar results. Furthermore, this method shows its advantage over the spline method and the finite difference methods.