The use of Hermite wavelet collocation method for fractional cancer dynamical system

Figures & data

Figure 1. Comparison of compartment of Cancer dynamical model with two numerical method with integer order $\alpha =1$ and m = 256 with time t = 5 days. (a) Comparison plot of Tumour cells $U\left(t\right)$. (b) Comparison plot of Immune effector cells $V\left(t\right)$ and (c) Comparison plot of Immune effector cells $W\left(t\right)$.

Figure 2. Comparison of compartment of Cancer dynamical model with two numerical method with fractional order $\alpha =0.95$ and m = 256 with time t = 40 days. (a) Time sequence diagram of Tumour cells $U\left(t\right)$. (b) Time sequence diagram of Immune effector cells $V\left(t\right)$ and (c) Time sequence diagram of Immune effector cells $W\left(t\right)$.

Figure 3. 3D plot of compartment of Cancer dynamical model with Hermite wavelet method with integer order $\alpha =1$ and m = 256. (a) Three-dimensional plot of Tumour cells $U\left(t\right)$. (b) Three-dimensional plot of Immune effector cells $V\left(t\right)$ and (c) Three-dimensional plot of Immune effector cells $W\left(t\right)$.

Figure 4. Phase diagram of compartment of Cancer dynamical model with two numerical method with integer order $\alpha =1$ and m = 256 with time t = 5 days. (a) Phase plot of Tumour cells $U\left(t\right)$ and cytotoxic T cells $V\left(t\right)$. (b) Phase plot of cytotoxic T cells $V\left(t\right)$ and natural killer T cells $W\left(t\right)$ and (c) Phase plot of Tumour cells $U\left(t\right)$ and natural killer T cells $W\left(t\right)$.

Figure 5. Two-dimensional plot of compartment of Cancer dynamical model with different values of alpha using Hermite wavelet method and m = 256 with time t = 5 days. (a) The graphical behaviour of Tumour cells corresponding to different derivative order α. (b) The graphical behaviour of natural killer T cells corresponding to different derivative order α and (c) The graphical behaviour of cytotoxic T cells corresponding to different derivative order α.

Figure 6. Chaotic phase plot of compartment of Cancer model with Caputo derivative with new numerical scheme with different value of α.

Figure 7. The maximal bifurcation for the Cancer model interpreting Tumour cell $U\left(t\right)$ versus the parameter $k1,d2$ and $d1$. (a) Maximal bifurcation plot of $U\left(t\right)$ at $k_2$. (b) Maximal bifurcation plot of $U\left(t\right)$ at $k_1$ and (c) Maximal bifurcation plot of $U\left(t\right)$ at $d_1$.

Table 1. Comparison between the Tumour cells $U\left(t\right)$ and Immune effectors $V\left(t\right)$, Immune effectors $W\left(t\right)$ using the Hermite wavelet method (HWM) and Toufik-Atangana method $\left(\mathrm{TA}\right)$ for m = 256 and $\alpha =1$.

t	$U_{\mathrm{HWM}}$	$U_{\mathrm{TA}}$	$V_{\mathrm{HWM}}$	$V_{\mathrm{TA}}$	$W_{\mathrm{HWM}}$	$W_{\mathrm{TA}}$
0.1	0.3678	0.3616	0.3501	0.3499	0.3216	0.3305
0.2	0.3809	0.3736	0.3507	0.3502	0.3042	0.3131
0.3	0.3952	0.3867	0.3520	0.3511	0.2881	0.2970
0.4	0.4105	0.4009	0.3539	0.3526	0.2731	0.2820
0.5	0.4271	0.4160	0.3564	0.3546	0.2593	0.2680
0.6	0.4447	0.4323	0.3597	0.3572	0.2464	0.2551
0.7	0.4636	0.4496	0.3637	0.3606	0.2346	0.2431
0.8	0.4835	0.4679	0.3686	0.3646	0.2237	0.2319
0.9	0.5046	0.4873	0.3744	0.3695	0.2136	0.2216

Data availability statement

Data available on request from the authors.