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This article aims to develop an optimal superconvergent numerical method for approximating the solution of the nonlinear time-fractional generalized Fisher's (TFGF) equation. The time-fractional derivative in the model problem is considered in the sense of Caputo and is approximated using the L 2 - 1 σ scheme. Spatial discretization is performed using an optimal superconvergent quintic B-spline (OSQB) technique. To derive the method, a high-order perturbation of the semi-discretized equation of the original problem is generated using spline alternate relations. Convergence and stability of the method are analyzed, demonstrating that the method converges with O (Δ t 2 + Δ x 6) , where Δ x and Δ t are step sizes in space and time, respectively. Three numerical examples are provided to demonstrate the robustness of the proposed method. Our method is compared with an existing method in the literature and the elapsed computational time for the present scheme is provided. [ABSTRACT FROM AUTHOR]

This article aims at developing a computational scheme for solving the time fractional reaction-subdiffusion (TFRSD) equation in two space dimensions. The Caputo fractional derivative is used to describe the time-fractional derivative appearing in the problem and it is approximated by using the L1 scheme. A compact difference scheme of order four is utilized for discretization of the spatial derivatives. Some test problems are solved to investigate the accuracy of the scheme. The computed results confirm that the scheme has convergence of order four in space and an order of min { 2 - α , 1 + α } in the time direction, where α ∈ (0 , 1) is the order of fractional derivative. Moreover, the computed results are compared with those obtained by other methods in order to justify the advantage of proposed algorithm. [ABSTRACT FROM AUTHOR]

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]

In this article, we develop and analyze an efficient numerical technique to solve the nonlinear temporal fractional Burgers' equation (TFBE). The temporal fractional derivative is considered in the terms of Caputo and approximated by using the L 2 - 1 σ scheme. The quintic B-spline (QBS) basis function is employed for discretization of the space derivative to obtain a fully discrete scheme. The proposed method is analyzed for its convergence and stability. Four nonlinear problems are considered to illustrate the advantage and applicability of the present method. The proposed scheme has an order of convergence O (Δ t 2 + Δ x 4) , where Δ t and Δ x are the step sizes in time and space directions, respectively. The comparison with the corresponding results of an existing method based on cubic parametric spline functions demonstrates that the proposed method is more accurate when solving the nonlinear time-fractional Burgers' equation. The CPU time is provided to show the computational efficiency of the method. The obtained stable and highly-accurate numerical results and low computational time collectively underscore the significance of the proposed technique in solving the nonlinear time-fractional Burgers' equation. [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 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 Roul and Warbhe (2016) J. Comp. Appl. Math. 296: 661–676, Roul and Warbhe proposed a computational technique for solving a class of doubly singular boundary value problems (DSBVP). This method approximates the solution of DSBVP in the form of a series but requires a large number of components in the series to achieve a reasonably good accuracy. In this paper, a fast and computationally efficient approach is introduced to approximate the solution to the same DSBVP. Additionally, convergence of the suggested scheme is rigorously proven. Two test problems are considered to demonstrate the efficiency and accuracy of the method. Comparison is performed between the proposed method and the method in Roul and Warbhe (2016) J. Comp. Appl. Math. 296: 661–676. The execution time of the present method is provided. [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]

This paper deals with the design and analysis of a high-order numerical scheme for the two-dimensional time-fractional reaction-subdiffusion equation. The time-fractional derivative of order α, (α ∈ (0,1)) in the model problem is approximated by means of the L2 − 1σ scheme and the space derivatives are discretized by means of a compact alternating direction implicit (ADI) finite difference scheme. Convergence and stability of the method are analyzed. Five numerical examples are provided to demonstrate the applicability and accuracy of the method. It is shown that the method is unconditionally stable and is of O (Δ t 1 + α + h x 4 + h y 4) accuracy, where Δt is the temporal step size and hx and hy are the spatial step sizes. The computed results are in good agreement with the theoretical analysis. Moreover, the comparison with the corresponding results of existing methods demonstrates that our method has advantages in convergence order and error improvement when solving the time-fractional reaction-subdiffusion equation. The CPU time consumed by the proposed 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 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]

The authors of Kaur and Natesan 2023 [A novel numerical scheme for time-fractional Black-Scholes PDE governing European options in mathematical finance, (Numerical Algorithms, 94, (2023) 1519–1549)] proposed a numerical scheme, which is based on a combination of L1 scheme for time discretization and spine method for spatial discretization, for solving a Caputo temporal-fractional Black-Scholes (TBS) equation governing European options. This method is (2-α)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(2-\alpha )$$\end{document}-th order accurate in time and second-order accurate in space. The present work deals with the construction of a high-order numerical scheme for solving the same time-fractional problem. In this method, the L2-1σ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L2-1_{\sigma }$$\end{document} scheme is employed to approximate the Caputo temporal-fractional derivative, while a high-order compact finite difference (CFD) method is proposed for the approximation of space derivatives. The existence and uniqueness of the solution to the TBS model are investigated through the application of the maximum-minimum principle and the Sumudu decomposition technique. The stability and convergence of the method are investigated. The proposed method is shown to be second-order accuracy in time and fourth-order accuracy in space. Numerical experiments are performed to illustrate the accuracy of the suggested numerical scheme and validate the theoretical results. Moreover, the present numerical scheme is used for pricing the European double barrier knock-out (EDBK) option. The execution time of the present numerical algorithm is provided. In order to justify the advantage of present method, the computed results are compared with the results obtained by the methods in Kaur and Natesan 2023 [A novel numerical scheme for time-fractional Black-Scholes PDE governing European options in mathematical finance, (Numerical Algorithms, 94, (2023) 1519-1549)] and Zhang et al. 2016 [Numerical solution of the time-fractional Black-Scholes model governing European options, Comput. Math. Appl. 71 (2016) 1772-1783]. [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]

This paper presents adaptive, graded and uniform mesh schemes to approximate the solution of a fractional order advection-diffusion model, which generally shows a weak singularity at the initial time level. The temporal fractional derivative in the underlying problem is described in a Caputo form and is discretized by means of L1 scheme on a nonuniform mesh. The space derivative is discretized on a uniform mesh employing a fourth-order compact finite difference scheme. The adaptive grid is generated via equidistribution of a positive monitor function. Stability and convergence results for the proposed method on graded mesh are established. Numerical examples are provided to study the accuracy and efficiency of the proposed techniques and to support the theoretical results. A discussion about the advantages of the graded and adaptive meshes over the uniform one is also presented. The CPU times for the proposed numerical schemes are provided. [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]

In this paper, we present a fast-converging iterative scheme to approximate the solution of a system of Lane-Emden equations arising in catalytic diffusion reactions. In this method, the original system of differential equations subject to a Neumann boundary condition at X=0 and a Dirichlet boundary condition at X=1, is transformed into an equivalent system of Fredholm integral equations. The resulting system of integral equations is then efficiently treated by the optimized homotopy analysis method. A numerical example is provided to verify the effectiveness and accuracy of the method. Results have been compared with those obtained by other existing iterative schemes to show the advantage of the proposed method. It is shown that the residual error in the present method solution is seven orders of magnitude smaller than in Adomian decomposition method and five orders of magnitude smaller than in the modified Adomian decomposition method. The proposed method is very simple and accurate and it converges quickly to the solution of a given problem. [ABSTRACT FROM AUTHOR]

Very Recently, Das et al. (J Math Chem 54:527-551, 2016) proposed a method based on variational iteration method for solving Bratu-type problems. In this work, we design a fast iterative scheme based on the optimal homotopy analysis method to obtain series solution for such problem. The proposed method contains a convergence control parameter which adjusts the interval of convergence of the series solution. This parameter is computed by minimizing the sum of the squared residual of the governing equation and with the help of which we obtain an optimal series solution to the problem. Further, our method avoids the usual time-consuming procedure to find the root of a nonlinear algebraic or transcendental equation for the undetermined coefficients. Besides the design of the method, convergence analysis and error estimate of the proposed method are supplemented. Two Bratu type problems are solved to demonstrate the efficiency and accuracy of the proposed method. Numerical results reveal the superiority of the proposed iterative scheme over a newly developed iterative method based on the variational iteration method (Das et al. 2016). [ABSTRACT FROM AUTHOR]

In this paper, we consider the following class of singular two-point boundary value problem posed on the interval x 휖 (0, 1] A recursive scheme is developed, and its convergence properties are studied. Further, the error estimation of the method is discussed. The proposed scheme is based on the integral equation formalism and optimal homotopy analysis method in which a recursive scheme is established without any undetermined coefficients. The original differential equation is transformed into an equivalent integral equation to remove the singularity. The integral equation is then made free of undetermined coefficients by imposing the boundary conditions on it. Finally, the integral equation without any undetermined coefficients is efficiently treated by using optimal homotopy analysis method for finding the numerical solution. The optimal control-convergence parameter involved in the components of the series solution is obtained by minimizing the squared residual error equation. The present method is applied to obtain numerical solution of singular boundary value problems arising in various physical models, and numerical results show the advantages of our method over the existing methods. [ABSTRACT FROM AUTHOR]

A new efficient recursive numerical scheme is presented for solving a class of singular two-point boundary value problems that arise in various physical models. The approach is based on the homotopy perturbation method in which we establish a recursive scheme without any undetermined coefficients to approximate the singular boundary value problems. The convergence analysis of the present method is discussed. Several numerical examples are provided to show the efficiency of our method for obtaining approximate solutions and to analyze its accuracy. The numerical results reveal that the present method yields a very rapid convergence of the solution without requiring much computational effort. The approximate solution obtained by the present method shows its superiority over existing methods. The Mathematica codes for numerical computation of singular boundary value problems are provided in the paper. [ABSTRACT FROM AUTHOR]

This paper presents a new iterative technique for solving nonlinear singular two-point boundary value problems with Neumann and Robin boundary conditions. The method is based on the homotopy perturbation method and the integral equation formalism in which a recursive scheme is established for the components of the approximate series solution. This method does not involve solution of a sequence of nonlinear algebraic or transcendental equations for the unknown coefficients as in some other iterative techniques developed for singular boundary value problems. The convergence result for the proposed method is established in the paper. The method is illustrated by four numerical examples, two of which have physical significance: The first problem is an application of the reaction-diffusion process in a porous spherical catalyst and the second problem arises in the study of steady-state oxygen-diffusion in a spherical cell with Michaelis-Menten uptake kinetics. [ABSTRACT FROM AUTHOR]

The paper deals with a numerical technique for solving nonlinear singular boundary value problems arising in various physical models. First, we convert the original problem to an equivalent integral equation to surmount the singularity and employ afterward the boundary condition to compute the undetermined coefficient. Finally, the integral equation without undetermined coefficient is treated using homotopy perturbation method. The present method is implemented on three physical model examples: i) thermal explosions; ii) steady-state oxygen diffusion in a spherical shell; iii) the equilibrium of the isothermal gas sphere. The results obtained by the present method are compared with that obtained using finite-difference method, B-spline method and a numerical technique based on the direct integration method, and comparison reveals that the proposed method with few solution components produces similar results and the method is computationally efficient than others. [ABSTRACT FROM AUTHOR]

We numerically investigate the effective material properties of aggregates consisting of soft convex polygonal particles, using the discrete element method. First, we construct two types of 'sand piles' by two different procedures. Then we measure the averaged stress and strain, the latter via imposing a 10% reduction of gravity, as well as the fabric tensor. Furthermore, we compare the vertical normal strain tensor between sand piles qualitatively and show how the construction history of the piles affects their strain distribution as well as the stress distribution. In the next step, elastic constants are determined, assuming Hooke's law to be locally valid throughout the sand piles. We determine the relationship between invariants of the stress and strain tensor, observing that the behaviour is nonlinear. While linear elastic behaviour near the centre of the pile is compatible with our data, nonlinearity signals the transition to plastic behaviour near its surface. A similar behaviour was assumed by Cantelaube et al. (Static multiplicity of stress states in granular heaps. Proc R Soc Lond A 456:2569-2588, ). We find that the macroscopic stress and fabric tensors are not collinear in the sand pile and that the elastic behaviour is anisotropic in an essential way. [ABSTRACT FROM AUTHOR]

This work is a contribution to the understanding of the mechanical properties of non-cohesive granular materials in the presence of friction and a continuation of our previous work (Roul et al. ) on numerical investigation of the macroscopic mechanical properties of sand piles. Besides previous numerical results obtained for sand piles that were poured from a localized source ('point source'), we here consider sand piles that were built by adopting a 'line source' or 'raining procedure'. Simulations were carried out in two-dimensional systems with soft convex polygonal particles, using the discrete element method (DEM). First, we focus on computing the macroscopic continuum quantities of the resulting symmetric sand piles. We then show how the construction history of the sand piles affects their mechanical properties including strain, fabric, volume fraction, and stress distributions; we also show how the latter are affected by the shape of the particles. Finally, stress tensors are studied for asymmetric sand piles, where the particles are dropped from either a point source or a line source. We find that the behaviour of stress distribution at the bottom of an asymmetric sand pile is qualitatively the same as that obtained from an analytical solution by Didwania and co-workers (Proc R Soc Lond A 456:2569-2588, ). [ABSTRACT FROM AUTHOR]