Abstract
This paper extends the existing Fisher equation by adding the source term and generalizing the degree $\beta$ of non-linear part. Numerical solutions of modified Fisher's equation for different values of $\beta$ using cubic B-spline collocation scheme are investigated. The fractional derivative in time dimension is discretized in Caputo's form based on $L1$ formula, while cubic B-spline basis functions are used to interpolate the spatial derivative. The non-linear part in the equation is linearized by the modified formula. The efficiency of the proposed scheme is examined by considering four test examples with different initial and boundary conditions. The effect of different parameters is discussed and presented in the form of tables and graphics. Moreover, by Von Neumann stability formula, proposed scheme is shown to be unconditionally stable. The results of error norms reflect that present scheme is suitable for non-linear time fractional differential equations.
Highlights
Fractional calculus-based models have been used in different fields of engineering and science
We have presented a cubic B-spline (CBS) algorithm for numerical simulation of the time-fractional generalized Fisher’s equation
0.0811 0.1209 0.1606 by the proposed scheme are presented in the form of tables and graphics
Summary
Fractional calculus-based models have been used in different fields of engineering and science. Rawashdeh [25] introduced the fractional natural decomposition method (FNDM) to find the analytical and approximate solutions of the non-linear time-fractional. We have presented a cubic B-spline (CBS) algorithm for numerical simulation of the time-fractional generalized Fisher’s equation. The present approach is novel for the numerical results of fractional order PDEs and, to the best of our knowledge, any spline solution of the time-fractional generalized Fisher’s equation has never yet been studied. This scheme is effective for homogeneous and non-homogeneous boundary conditions.
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