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A high-order numerical scheme based on collocation of a quintic B-spline over finite element is proposed for the numerical solution of a class of nonlinear singular boundary value problems (SBVPs) arising in various physical models in engineering and applied sciences. Five illustrative examples are presented to illustrate the applicability and accuracy of the method. In order to justify the advantage of the proposed numerical scheme, the computed results are compared with the results obtained by two other fourth-order numerical methods, namely the finite difference method (Chawla et al. in BIT 28(1):88–97, 1988) and B-spline collocation method (Goh et al. in Comput Math Appl 64:115–120, 2012). [ABSTRACT FROM AUTHOR]

The aim of the present study is to describe and demonstrate a new computational procedure based on nonuniform mesh compact finite difference method to approximate the solution of a nonlinear singular boundary value problem which describes the electrohydrodynamic flow (EHF) of a fluid in a circular cylindrical conduit. The EHF problem under consideration is highly nonlinear and the nonlinearity confronted in the problem is in the form of a rational function and has a singularity at the point x = 0 . To construct a non-uniform grid, we use a grading function. This function generates a finer mesh near the singular point. The efficiency and applicability of the new method are demonstrated by applying it to the EHF equation for small and large values of two relevant parameters, namely the strength of nonlinearity α and the electric Hartmann number H ~ . The velocity field of EHF flow of a fluid in radial direction is computed. It is shown that the proposed new scheme produces a fourth-order numerical approximation to the solution of the considered EHF problem and the velocity field is clearly influenced by the parameters H ~ and α . The computed results are compared with some published numerical results. [ABSTRACT FROM AUTHOR]

This paper deals with the design and analysis of a robust numerical scheme based on an improvised quartic B-spline collocation (IQBSC) method for a class of nonlinear derivative dependent singular boundary value problems (DDSBVP). The convergence analysis of the method is studied by means of Green's function approach. It should be pointed out that the numerical order of convergence of standard quartic B-spline collocation (SQBC) scheme for second-order boundary value problems (BVPs) is four, however, our proposed IQBSC method is shown to be sixth order convergence. To illustrate the applicability and accuracy of the method, we consider eight test problems. The obtained results are compared to those from some existing numerical schemes in order to show the advantage of present method. It is shown that the rate of convergence of present numerical scheme is higher than that of some of existing numerical methods. The CPU time of the present numerical method is provided. [ABSTRACT FROM AUTHOR]

This work aims to find the numerical solution to a class of nonlinear singular boundary value problems (SBVPs). The considered problem has a singularity at x = 0. We introduce a computational technique comprising an optimal homotopy analysis (OHA) approach and exponential compact finite difference method (ECFDM) to solve this SBVPs. In this technique, the domain of the problem I = [ 0 , 1 ] is divided into two subintervals as I = I 1 ∪ I 2 = [ 0 , ξ ] ∪ [ ξ , 1 ] (the point x = ξ is chosen sufficiently close to the singularity). In interval I 1 , we employ the OHA scheme to overcome the singularity. In interval I 2 , an ECFDM is designed to solve the resultant boundary value problem (BVP). Convergence analysis of the ECFDM is discussed. Furthermore, numerical experiments are performed to confirm the theoretical claims. The proposed ECFDM is shown to be fourth-order convergence. [ABSTRACT FROM AUTHOR]

In this paper, a high-order numerical scheme based on quartic B-spline functions is proposed to solve derivative-dependent nonlinear singular boundary value problems. Convergence of the method is analysed. Five test problems are considered to illustrate the accuracy and efficiency of the method. The results obtained by present method are compared with those obtained by the uniform mesh cubic B-spline collocation (UCS) method [P. Roul and V.M.K. Prasad Goura, B-spline collocation methods and their convergence for a class of nonlinear derivative-dependent singular boundary value problems, Appl. Math. Comput. 341 (2019), pp. 428–450] and non-uniform mesh cubic B-spline collocation (NCS) method [P. Roul and V.M.K. Prasad Goura, B-spline collocation methods and their convergence for a class of nonlinear derivative-dependent singular boundary value problems, Appl. Math. Comput. 341 (2019), pp. 428–450]. The CPU time of proposed method is compared with that of the NCS method. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel numerical approach to solve nonlinear singular boundary value problems (SBVP) that arise in various physical models. In this method, we decompose the original domain of the problem into two subdomains. We first employ a series-based method in the first domain to remove the singularity at x = 0 . A B-spline collocation approach is then considered for solving the resulting boundary value problems (BVP) on the second domain. The method is illustrated by four numerical examples, including thermal explosions, heat conduction model of the human head, reaction-diffusion process inside a porous catalyst and stress distribution on a rotationally shallow membrane cap. The computed results are compared with other numerical techniques. Comparison reveals that our approach provides more accurate solution than the methods given in Ravikanth (Appl Math Comput 189:2017–2022, 2007), Cohen and Jones (J Inst Math Appl 13:379–384, 1974), Roul and Warbhe (J Math Chem 54:1255–1285, 2016) and Pandey and Singh (J Comput Appl Math 166:553–564, 2004). [ABSTRACT FROM AUTHOR]

In this work, a fast numerical scheme is proposed for numerical solution of a general class of nonlinear singular boundary value problems (SBVPs), which describe various physical phenomena in applied science and engineering. It is worth noting that the standard cubic B-spline collocation method provides a second order convergent approximation to the solution of a second-order SBVP, while the proposed method provides a sixth-order convergent approximation. To demonstrate the applicability and accuracy of the method, we consider six nonlinear examples, including five real-life problems. It is shown that the experimental rate of convergence of present scheme is six and our method produces significantly more accurate results than the existing ones. Additionally, the CPU time for the method is provided. [ABSTRACT FROM AUTHOR]

This study deals with the numerical solution of a general class of nonlinear singular boundary value problems (SBVPs). Firstly, we modify the original model problem at the singular point and then we construct a numerical technique based on quartic trigonometric B-spline functions to solve the resulting problem. Numerical experiments are performed to demonstrate the applicability and efficiency of the method. More specifically, we consider three real-life problems: (1) thermal explosion in a cylindrical vessel; (2) isothermal gas sphere; (3) oxygen diffusion in a spherical cell. The computed results are compared with the results obtained by the compact finite difference method (CFDM) (Roul et al. in Appl Math Comput 350:283–304, 2019) and the B-spline collocation method (Thula and Roul in Mediterr J Math 15(4):176, 2018) in order to justify the advantage of present method. The proposed method is a promising one to handle the general class of SBVPs. [ABSTRACT FROM AUTHOR]

This paper is concerned with the development of a collocation method based on the Bessel polynomials for numerical solution of a general class of nonlinear singular boundary value problems (SBVPs). Due to the existence of singularity at the point x = 0 , we first modify the problem at the singular point. The proposed method is then developed for solving the resulting regular boundary value problem. To demonstrate the effectiveness and accuracy of the method, we apply it on several numerical examples. The numerical results obtained confirm that the present method has an advantage in terms of numerical accuracy over the uniform mesh cubic B-spline collocation (UCS) method (Roul and Goura in Appl Math Comput 341:428–450, 2019), non-standard finite difference (NSFD) method (Verma and Kayenat in J Math Chem 56:1667–1706, 2018), three-point finite difference methods (FDMs) (Pandey and Singh in Int J Comput Math 80:1323–1331, 2003; Pandey and Singh in J Comput Appl Math 205:469–478, 2007) and the cubic B-spline collocation (CBSC) method (Caglar et al. in Chaos Solitons Fractals 39:1232–1237, 2009) [ABSTRACT FROM AUTHOR]

• A new sixth order computational method is developed for EHF problem. • The method is highly accurate, computationally efficient and a fast convergent. • The velocity field is computed for small and large values of two relevant parameters H and β. • It is shown that the parameters H and β both have a profound impact on the velocity profile. This paper deals with the construction of a computational approach based on B-spline functions for solving a nonlinear boundary value problem describing the electro-hydrodynamic flow (EHF) of a fluid in a circular cylindrical conduit. The radial dependence of the velocity field emerging in the EHF is computed. We study the effects of two relevant parameters, namely the Hartmann electric number H and the strength of the nonlinearity β , on the velocity field. Computational results show that the method is of sixth-order accuracy. It is shown that the Hartmann electric number (HEN) and the strength of the nonlinearity both have a profound impact on the velocity profile of EHF and that these effects can be understood from analytical considerations. In particular, quantitative results include: The velocity, taking its maximum at the center of the conduit, does not exceed the value 1 / (1 + β) (this confirms a previous result). At large HEN, a boundary layer develops near the outer radial boundary of the conduit (r = 1). Its thickness is proportional to 1 / (H 1 + β) , being determined by both the HEN and the nonlinearity. Moreover, when a boundary layer is present, the flow velocity has a plug-like profile approaching the plateau value 1 / (1 + β) (from below) for r values smaller than those of the boundary layer. If both the nonlinearity and the HEN are too small for a boundary layer to develop, then the flow profile is essentially parabolic and describable via a modified Bessel function. The CPU time for our method is provided. [ABSTRACT FROM AUTHOR]

In this paper, we develop a numerical scheme to approximate the solution of a general class of nonlinear singular boundary value problems (SBVPs) subject to Neumann and Robin boundary conditions. The original differential equation has a singularity at the point x=0, which is removed via L'Hospital's law with an assumption about the derivative of the solution at the point x=0. An exponential B‐spline collocation approach is then constructed to solve the resulting boundary value problem. Convergence analysis of the method is discussed. Numerical examples are provided to illustrate the applicability and efficiency of the method. Our results are compared with those obtained by other three numerical methods such as uniform mesh cubic B‐spline collocation (UCS) method, nonstandard finite difference method, and finite difference method based on Chawla's identity. [ABSTRACT FROM AUTHOR]

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 (h 4) convergent approximation. [ABSTRACT FROM AUTHOR]

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=D1UD2=[0,δ]U[δ,1] (δ is vicinity of the singularity). In the first domain D1, 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 D2, 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. [ABSTRACT FROM AUTHOR]

• A new sixth order computational method is developed for DDSBVP. • Convergence of the method is analyzed. • The method is highly accurate, computationally efficient and a fast convergent. In the present work, a high-order numerical scheme based on B-spline functions is developed for solving a class of nonlinear derivative dependent singular boundary value problems (DDSBVP). To derive the method, we first generate a high order perturbation of the original problem by using spline alternate relations. Then, we determine the approximate solution by forcing it to satisfy the resulting perturbed problem at the grid points of the spline. Convergence analysis of the method is established through matrix approach. Four nonlinear examples are considered to demonstrate the accuracy and robustness of the method. The proposed method provides O (h 6) superconvergent approximation to the solution of the problem under consideration, where h is the step size. This method produces significantly more accurate results than the two newly developed numerical schemes using the same B-spline functions as used in the present method, namely UCS method and NCS method. Moreover, the computational time of present method is compared with that of NCS method. [ABSTRACT FROM AUTHOR]