An Efficient Numerical Technique for Solving Time-Fractional Generalized Fisher's Equation

Abstract

This paper extends the existing Fisher's equation by adding the source term and generalizing the degree β of the non-linear part. A numerical solution of a modified Fisher's equation for different values of β using the cubic B-spline collocation scheme is also investigated. The fractional derivative in a time dimension is discretized in Caputo's form based on the L1 formula, while cubic B-spline basis functions are used to interpolate the spatial derivative. The non-linear part in the model is linearized by the modified formula. The efficiency of the proposed scheme is examined by simulating four test examples with different initial and boundary conditions. The effect of different parameters is discussed and presented in tables and graphics form. Moreover, by using the Von Neumann stability formula, the proposed scheme is shown to be unconditionally stable. The results of error norms reflect that the present scheme is suitable for non-linear time fractional differential equations.