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

Math Methods in App Sciences

2020

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A high-accuracy numerical approach for a nonhomogeneous time-fractional diffusion equation with Neumann and Dirichlet boundary conditions is described in this paper. The time-fractional derivative is described in the sense of Riemann-Liouville and discretized by the backward Euler scheme. A fourth-order optimal cubic B-spline collocation (OCBSC) method is used to discretize the space variable. The stability analysis with respect to time discretization is carried out, and it is shown that the method is unconditionally stable. Convergence analysis of the method is performed. Two numerical examples are considered to demonstrate the performance of the method and validate the theoretical results. It is shown that the proposed method is of order O(Δx 4 + Δt 2 −) convergence, where ∈ (0,1). Moreover, the impact of fractional-order derivative on the solution profile is investigated. Numerical results obtained by the present method are compared with those obtained by the method based on standard cubic B-spline collocation method. The CPU time for present numerical method and the method based on cubic B-spline collocation method are provided.

“…Proof. Taking into account the arguments used in the proof of Theorem 2 of Roul and Goura, 16 one can proof this theorem.…”

Section: Cubic Spline Interpolationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A high‐order B‐spline collocation scheme for solving a nonhomogeneous time‐fractional diffusion equation

Math Methods in App Sciences

2020

Self Cite

“…The proposed method is fourth order convergence in space and (2 − )-th order convergence in time. Various numerical schemes based on B-spline collocation methods have been applied to solve a wide variety of problems, see [24][25][26][27][28][29][30][31][32][33][34]. Besides B-spline collocation techniques, there are other methods that can be used to solve classic or fractional differential equations, such as finite difference methods [6,7,14], finite element method [35], boundary element methods [36,37], meshless methods [38,39] etc.…”

Section: Introductionmentioning

confidence: 99%

A high order numerical scheme for solving a class of non‐homogeneous time‐fractional reaction diffusion equation

Numerical Methods Partial

2020

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This paper is concerned with the development of a high order numerical technique for solving time‐fractional reaction–diffusion equation. The fractional derivative in the governing equation is described in the Caputo sense and a collocation method based on quintic B‐spline basis function is used to discretize the space variable. The stability and convergence analysis of the method are investigated, and it is shown that the proposed method converges to the exact solution of the problem with order of convergence O(Δt2 − α + Δx4), where α is the order of fractional derivative (0 < α < 1). Three test problems are considered to illustrate the efficiency and accuracy of present numerical scheme and to verify the theoretical result. It is shown that the order of fractional derivative has a profound effect on the solution of time‐fractional reaction–diffusion equation.

“…In recent years, nonlinear singular boundary value problems (SBVPs) have been extensively studied in the literature 1‐5 . A wide variety of physical phenomena, such as the thermal behaviour of a spherical cloud of gas, oxygen diffusion in a cell, isothermal gas sphere, electrohydrodynamic flow of a fluid in a cylinder, and thermal explosion are modelled by SBVPs, see previous studies 6‐10 and the references therein.…”

Section: Introductionmentioning

confidence: 99%

An efficient numerical method based on exponential B‐spline basis functions for solving a class of nonlinear singular boundary value problems with Neumann and Robin boundary conditions

Kumari

Math Methods in App Sciences

2020

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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.