## 4 Numerical Calculations
### Case Study I: Planar Particle (\(n=0\))  
The parameters associated with the reaction-diffusion model (1) are set as [11]  
\[\alpha=1,\quad h=0.1,\quad s_{h}=1.\]  
Clearly, in this case, the boundary conditions take the form  
\[\frac{d}{dx}M(0)=0,\quad\frac{d}{dx}M(1)+M(1)=1.\]  
In order to compare our results with two existing wavelet-based spectral approaches, we take only three basis functions at the first stage. Using \(H=2\) in the spectral QLM-ChPs technique, we obtain the following approximation  
\[M_{2}^{(5)}(x)=0.00496563\,x^{2}+0.985103,\]  
which can be compared to those approximations obtained by the Chebyshev wavelet method (CWM) and Legendre wavelet method (LWM) developed in [11]. These two approximate solutions are  
\[U_{\text{CWM}}(x)=0.0032\,x^{2}+0.9904,\quad U_{LWM}(x)=0.00496\,x^{2}+0.98734.\]  
Similarly, the solution obtained by the Adomian decomposition method (ADM) is given by [10]  
\[A_{ADM}(x)=1-\left(\frac{h^{2}}{\alpha+1}\right)\left[1+\frac{2}{s_{h}}\right] +\left(\frac{h^{2}}{\alpha+1}\right)x^{2}\approx 0.005\,x^{2}+0.985.\]  
It can be clearly seen that our approximations are closer to the output of LWM and ADM than the solution obtained by the CWM. By running our algorithm, we can obtain theresults with more accuracy using larger values of \(H\). For instance, by taking \(H=4\) and \(H=8\), we obtain  
\[M_{4}^{(5)}(x) =0.00000209205\,x^{4}+8.1598\times 10^{-9}\,x^{3}+0.00496247\,x^{2}+ 0.985102,\] \[M_{8}^{(5)}(x) =1.68201\times 10^{-12}\,x^{8}+4.9749\times 10^{-14}\,x^{7}-1.743 71\times 10^{-9}\,x^{6}\] \[+4.07856\times 10^{-14}\,x^{5}+2.0988\times 10^{-6}\,x^{4}+4.34436 \times 10^{-15}\,x^{3}+0.00496248\,x^{2}\] \[-1.33061\times 10^{-110}\,x+0.985102.\]  
The visualizations of the approximate solutions using different values of \(H=2,4,8\) are plotted in Figure 1. The absolute error of the associated REFs with these values of \(H\) are also exhibited in \[\frac{d}{dx}M(0)=0,\quad\frac{d}{dx}M(1)+M(1)=1.\]0 By looking at this figure demonstrating the closeness of these solutions, we see that the magnitude of REFs are decreased whenever we increase the \(H\). This shows that the proposed method is convergent, and even its tendency to converge is exponential. Using the same parameters as above (fixed \(H=2\) and \(s_{h}=1\)) but with different values of \(h\), we examine the effects of the model parameter \(h\) on the computations. The results by using \(h=0.1,0.2,\ldots,0.5\) are plotted in \[\frac{d}{dx}M(0)=0,\quad\frac{d}{dx}M(1)+M(1)=1.\]1 The associated REFs are also visualized in the same figure. It is evident that by increasing the parameter \(h\), the corresponding REFs are decreasing in magnitude. This implies that for large values of \(h\), we are required to use a large value of \(H\) in the computations to obtain the desired accuracy. Let us consider the case of \(h=0.5\) and compare our obtained approximate solution with the outcomes of some existing ones. We obtain the next approximation  
\[\frac{d}{dx}M(0)=0,\quad\frac{d}{dx}M(1)+M(1)=1.\]2  
which is very close to the output of LWM, reported as [11]  
\[\frac{d}{dx}M(0)=0,\quad\frac{d}{dx}M(1)+M(1)=1.\]3  
However, one can see a gap between our approximation and that obtained by the ADM [10] given in the form \(A_{ADM}(x)=0.125\,x^{2}+0.625\) and the solution of CWM [11] obtained as \(U_{CWM}(x)=0.1024\,x^{2}+0.6928\). Again, more alignment will be seen between the approximations obtained by the QLM-ChPs and LWM. \[\frac{d}{dx}M(0)=0,\quad\frac{d}{dx}M(1)+M(1)=1.\]4 Approximate solutions (**left**) and the related absolute values of REFs (**right**) via the QLM–ChPs approach with \(H=2,4,8,16\), and \(\alpha=1,h=0.1,s_{h}=1\). The next simulation results are devoted to the influence of the parameter \(\alpha\) on the obtained numerical solutions. Towards this end, we first fix \(h=0.5\) and \(s_{h}=1\). By taking \(H=2\) as before, the approximate solutions for diverse values of \(\alpha=0.70,0.75,0.80,0.90\), and \(\alpha=1\) are displayed in Figure 3 (left picture).