Abstract
In this piece of work, using three spatial grid points, we discuss a new two-level implicit cubic spline method of O(k2 + kh2 + h4) for the solution of quasi-linear parabolic equation , 0 0 subject to appropriate initial and Dirichlet boundary conditions, where h > 0, k > 0 are grid sizes in space and time-directions, respectively. The cubic spline approximation produces at each time level a spline function which may be used to obtain the solution at any point in the range of the space variable. The proposed cubic spline method is applicable to parabolic equations having singularity. The stability analysis for diffusion- convection equation shows the unconditionally stable character of the cubic spline method. The numerical tests are performed and comparative results are provided to illustrate the usefulness of the proposed method.
Highlights
The use of cubic splines for the numerical solution of linear two point boundary value problems has been discussed by Bickley [1], Fyfe [2], Albasiny and Hoskins [3] and Rubin and Khosla [4]
The cubic spline approximation produces at each time level a spline function which may be used to obtain the solution at any point in the range of the space variable
Jain and Lohar [18] have solved non-linear parabolic equations by using second order cubic spline method
Summary
The use of cubic splines for the numerical solution of linear two point boundary value problems has been discussed by Bickley [1], Fyfe [2], Albasiny and Hoskins [3] and Rubin and Khosla [4]. In 1983, Jain and Aziz [7] have derived fourth order cubic spline method for the solution of general two point non-linear boundary value problems. Monoj Kumar et al [9,10,11], and Rashidinia et al [12,13] have discussed higher order cubic spline finite difference method for singular two point boundary value problems. High order finite difference methods for the solution of non-linear parabolic equations have been discussed by Jain et al [20] and Mohanty et al [21,22]. In the present paper, using three spatial grid points (see Figure 1), a new cubic spline technique similar to that of Jain and Aziz [7] is developed for the solution of one dimensional general quasi-linear parabolic equation.
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