Original Articles
A boundary shape function iterative method for solving nonlinear singular boundary value problems

https://doi.org/10.1016/j.matcom.2021.03.030Get rights and content

Highlights

  • The proposed scheme is totally different from the conventional shooting method.

  • The numerical algorithm is accurate in the solutions of nonlinear singular BVPs.

  • This developed approach only need a few iterations to solve nonlinear singular BVPs.

Abstract

In this paper, a novel iterative algorithm is developed to solve second-order nonlinear singular boundary value problem, whose solution exactly satisfies the Robin boundary conditions specified on the boundaries of a unit interval. The boundary shape function is designed such that the boundary conditions can be fulfilled automatically, which renders a new algorithm with the solution playing the role of a boundary shape function. When the free function is viewed as a new variable, the original singular boundary value problem can be properly transformed to an initial value problem. For the new variable the initial values are given, whereas two unknown terminal values are determined iteratively by integrating the transformed ordinary differential equation to obtain the new terminal values until they are convergent. As a consequence, very accurate solutions for the nonlinear singular boundary value problems can be obtained through a few iterations. The present method is different from the traditional shooting method, which needs to guess initial values and solve nonlinear algebraic equations to approximate the missing initial values. As practical applications of the present method, we solve the Blasius equation for describing the boundary layer behavior of fluid flow over a flat plate, where the Crocco transformation is employed to transform the third-order differential equation to a second-order singular differential equation. We also solve a nonlinear singular differential equation of a pressurized spherical membrane with a strong singularity.

Introduction

The singular boundary value problem (BVP) occurs in many areas for the applications in science and engineering, including thermal explosions [3], tumor growth models [1], electroosmotic flows [4], [5], [6], modeling of heat sources in human head [13], oxygen diffusion [26], physiology [15], electrohydrodynamic flow of a fluid [36], viscous flows [40], science and engineering [32], as well as various physical models [37].

The current paper designs a novel numerical method to find the singular solution of nonlinear BVP, exactly satisfying the prescribed boundary conditions on the boundaries of a unit interval. When a singular point happens at the boundary, it might be a difficult task to solve this sort problem. Many computational methods have been developed for solving the nonlinear BVPs of the singular type [2], [12], [15], [19], [29], [30], [33], [34].

The boundary shape function method (BSFM) was first introduced by Liu and Chang [24] to find the periodic solution of the nonlinear jerk equations, and by Liu et al. [25] to solve the optimal control problems of nonlinear Duffing oscillators. Later, the methodology of the BSFM was extended to solve the BVP with multipoint boundary conditions [22] and the singularly perturbed BVP with Robin boundary conditions [23]. The present problem with nonlinear BVP and with a singular point at the boundary is more difficult to be solved.

We arrange the paper as follows. First, we give a brief sketch of the shooting method to solve the nonlinear singular BVP in Section 2, which is recast to be an initial value problem (IVP) but with unknown initial conditions. In Section 3, we introduce the boundary shape function (BSF) for satisfying the specified Robin boundary conditions automatically. In Section 4, we take the BSF as the solution, such that we can exactly transform the nonlinear singular BVP to an IVP for the new variable with known initial conditions, and from it the fulfillment of the Robin boundary conditions is automatic for the original variable. The iterative algorithm is developed to find the unknown terminal values of the new variable, which can be iteratively determined with a fast convergence speed. In Section 5, we solve some numerical examples by the proposed iterative algorithm based on the BSFM and compare to that obtained from the shooting method. In Section 6, using the BSFM we solve the Blasius equation and a nonlinear singular differential equation of a pressurized spherical membrane with a strong singularity. Finally, the conclusions are drawn in Section 7.

Section snippets

Shooting method

In the fields of science and engineering, some axisymmetrical diffusion problems in the steady state can be modeled by a class of two-point BVPs [8], [9]: u(x)=f(x,u,u)mxu(x)F(x,u,u),0<x<1,a1u(0)+b1u(0)=c1,ifm=0,oru(0)=0,ifm0,a2u(1)+b2u(1)=c2, where a12+b12>0 and a22+b22>0. If m0, we encounter a nonlinear singular boundary value problem. With certain conditions on f(x,u,u), the existence and uniqueness of the solution of Eqs. (1)–(3) have been discussed in [27], [28], [31].

We

Boundary shape function

The following results are required to construct the boundary shape function [23].

Theorem 1

s1(x),s2(x)C1[0,1] are shape functions and satisfy a1s1(0)+b1s1(0)=1,a2s1(1)+b2s1(1)=0,a1s2(0)+b1s2(0)=0,a2s2(1)+b2s2(1)=1.

Proof

Refer [23] for the proof of the existence of s1(x) and s2(x). 

Theorem 2

Suppose that y(x)C1[0,1] is a free function. With Eqs. (13) and (14) for s1(x) and s2(x) and G(x)[a1y(0)+b1y(0)c1]s1(x)+[a2y(1)+b2y(1)c2]s2(x),there exists a boundary shape function B(x)=y(x)G(x),which automatically

The boundary shape function method

In Theorem 2, we can replace B(x) by u(x) as a boundary shape function to automatically satisfy the Robin boundary conditions (2), (3). G(x) is defined by Eq. (15) and thus the variable transformation from u(x) to a new variable y(x) is given by u(x)=y(x)G(x),which is inserted into Eq. (1) to obtain a new ordinary differential equation (ODE) for y(x): y(x)=H(x,y(x),y(x);y(1),y(1))G(x)+F(x,y(x)G(x),y(x)G(x)).We place y(1),y(1) in H to stress that they are unknown parameters in the

Numerical tests

We compare the shooting method in Section 2 and the BSFM in Section 4 with the following examples.

Two practical nonlinear singular problems

In the application of the BSFM to solve practical nonlinear singular problems, we may encounter the situation which is more complex than that in Eq. (1). In such problems the singularities with xu(x) and x2u2(x) and u(x)=0 at the boundary point are more difficult to be treated than the singularity u(x)x appeared in Eq. (1) at the boundary point x=0. We solve two such problems in the below by using the BSFM.

Conclusions

For the second-order nonlinear singular boundary value problems, the boundary shape functions, exactly satisfying the given Robin boundary conditions, are derived. Because the singularity is appeared on the boundary, the numerical method is designed to exactly and automatically preserve the Robin boundary conditions. In the shooting method, we have applied the fictitious time integration method to solve the residual equation for satisfying the right-boundary condition. The present paper derived

References (40)

Cited by (11)

  • An explicit representation of the three-point Hermite interpolant for the numerical solution of singular boundary value problems

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    Wavelet Galerkin method (WGM) was proposed by Nosrati Sahlan and Hashemizadeh [51] to approximate the solution of a class of practical SBVPs. Furthermore, the simplified reproducing kernel method (SRKM) [32], Haar wavelet collocation method (HWCM) [53], B-spline functions method (BSM) [4], boundary shape function method (BSFM) [28], He's variational iteration method (HVIM) [20], neural networks [49,50], and many other numerical techniques have been recently utilized to solve different types of nonlinear SBVPs [18,25,26,40,48,56,58,59,61,62]. In addition to the analytical importance of the Hermite interpolation, the widespread utilization of this interpolation method in solving a variety of differential equations demonstrates its practical importance [16,17,27,29,46,55].

  • An adaptive pair of one-step hybrid block Nyström methods for singular initial-value problems of Lane–Emden–Fowler type

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    Many researchers have widely studied the model in (1) and provided different strategies to solve it. To indicate some of the existing methods to give numerical solutions to the SSIVPs given in (1) and related problems, we can mention the implicit Euler method proposed by Koch et al. [7], the homotopy-perturbation technique by [5], the pseudospectral method presented by Mehrpouya [10], the explicit one-step strategies reported by Kutniv et al. [8], the collocation method by Bhrawy and Alofi [4], the Haar Wavelet resolution technique in [24], the heuristic approach in [22], an iterative method in [9], the quasilinearization method by Singh et al. [23], the hybrid block Nyström methods in [20], the analytical methods presented in [26], the Adomian decomposition method in [6], the spline techniques in [16,17] or a compact finite difference method in [18] and the references enclosed in those manuscripts. Another possibility to deal with (1) is to transform the differential equation in an equivalent system of first order differential equations and use any of the methods for this kind of problems, as the one in [11], although this is usually more expensive computationally.

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