A fourth-order non-uniform mesh optimal B-spline collocation method for solving a strongly nonlinear singular boundary value problem describing electrohydrodynamic flow of a fluid

In this paper, we develop a B-spline collocation method using artificial viscosity for solving a class of singularly perturbed reaction–diffusion equations. We use the artificial viscosity to capture the exponential features of the exact solution on a uniform mesh, and use the B-spline collocation method, which leads to a tridiagonal linear system. The convergence analysis is given and the method is shown to have uniform convergence of second order. The design of an artificial viscosity parameter is confirmed to be a crucial ingredient for simulating the solution of the problem. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show the behavior of the method, with emphasis on treatment of boundary conditions. Results shown by the method are found to be in good agreement with the exact solution.

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The numerical study of advection–diffusion equations by the fourth-order cubic B-spline collocation method

Mathematical Sciences ◽

10.1007/s40096-020-00352-7 ◽

2020 ◽

Vol 14 (4) ◽

pp. 409-423

Author(s):

R. C. Mittal ◽

Rajni Rohila

Keyword(s):

Collocation Method ◽

Fourth Order ◽

Numerical Study ◽

Diffusion Equations ◽

Spline Collocation ◽

B Spline ◽

Spline Collocation Method ◽

Advection Diffusion

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A fourth-order B-spline collocation method for nonlinear Burgers–Fisher equation

Mathematical Sciences ◽

10.1007/s40096-019-00317-5 ◽

2020 ◽

Vol 14 (1) ◽

pp. 75-85 ◽

Cited By ~ 1

Author(s):

Aditi Singh ◽

Sumita Dahiya ◽

S. P. Singh

Keyword(s):

Collocation Method ◽

Fourth Order ◽

Numerical Study ◽

Fisher Equation ◽

Spline Collocation ◽

Von Neumann ◽

B Spline ◽

Spline Collocation Method ◽

The Stability ◽

Nicolson Scheme

AbstractA fourth-order B-spline collocation method has been applied for numerical study of Burgers–Fisher equation, which illustrates many situations occurring in various fields of science and engineering including nonlinear optics, gas dynamics, chemical physics, heat conduction, and so on. The present method is successfully applied to solve the Burgers–Fisher equation taking into consideration various parametric values. The scheme is found to be convergent. Crank–Nicolson scheme has been employed for the discretization. Quasi-linearization technique has been employed to deal with the nonlinearity of equations. The stability of the method has been discussed using Fourier series analysis (von Neumann method), and it has been observed that the method is unconditionally stable. In order to demonstrate the effectiveness of the scheme, numerical experiments have been performed on various examples. The solutions obtained are compared with results available in the literature, which shows that the proposed scheme is satisfactorily accurate and suitable for solving such problems with minimal computational efforts.

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