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This paper presents a high-order computational scheme for numerical solution of a time-fractional diffusion equation (TFDE). This scheme is discretized in time by means of L1-scheme and discretized in space using a compact finite difference method. Stability analysis of the method is discussed. Further, convergence analysis of the present numerical scheme is established and we show that this scheme is of O (Δ t 2 - α + Δ x 4) convergence, where α ∈ (0 , 1) is the order of fractional derivative (FD) appearing in the governing equation and Δ t and Δ x are the step sizes in temporal and spatial direction, respectively. Three numerical examples are considered to illustrate the accuracy and performance of the method. In order to show the advantage of the proposed method we compare our results with those obtained by finite element method and B-spline method. Comparison reveals that the proposed method is fast convergent and highly accurate. Moreover, the effect of α on the numerical solution of TFDE is investigated. The CPU time of the present method is provided. [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]

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 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]

In this paper, we propose a computational technique based on a combination of optimal homotopy analysis method (OHAM) and an iterative finite difference method (FDM) for a class of derivative-dependent doubly singular boundary value problems: (p (x) y ′) ′ = q (x) f (x , y (x) , y ′ (x)) , 0 ≤ x ≤ 1 , y ′ (0) = 0 , α y (1) + β y ′ (1) = B or y (0) = A , α y (1) + β y ′ (1) = B. The principal idea of this approach is to decompose the domain of the problem D = [ 0 , 1 ] into two subdomains as D = D 1 ∪ D 2 = [ 0 , γ ] ∪ [ γ , 1 ] (γ is the vicinity of the singularity). In the first domain D 1 , we use OHAM to overcome the singularity behaviour at x = 0 . In the second domain D 2 , a FDM is designed for solving the resulting regular boundary value problem. Convergence analysis of the method is carried out. Three nonlinear examples are considered to demonstrate the performance and accuracy of the proposed method. It is shown that the computational order of convergence of the FDM is two. [ABSTRACT FROM AUTHOR]

In the present paper, we design a Bessel collocation method for solving one-dimensional nonlinear Bratu's boundary value problem. Two numerical examples are provided to validate the efficiency and accuracy of the proposed method. Numerical results reveal that the present method yields very accurate approximation to the exact solution of Bratu's problem. Moreover, the numerical results are compared with those obtained by two other existing methods, namely, optimal homotopy analysis method and the modified variational iteration method. [ABSTRACT FROM AUTHOR]

In this paper, we describe an optimal B-spline collocation method to solve one-dimensional non-linear Bratu problem. Convergence result of the method is established through Green's function technique and it is proved that the method has sixth order convergence. The applicability and accuracy of the method are demonstrated through two numerical examples. It is shown that the computational result is consistent with theoretical prediction. Moreover, the numerical results obtained by the present method are compared with those obtained by two other B-spline collocation approaches reported in the literature. The computed results reveal that our method is accurate and easy to implement. [ABSTRACT FROM AUTHOR]