# Efficient Solution Criteria for a Coupled Fractional Laplacian System on Some Infinite Domains
## 4 Application to the Turbulent Flow System
\begin{array}{l}\mathcal{W}_{1}\big{(}t,\mathcal{V}(t),\mathcal{D}_{0^{ \text{\tiny{th}}}}^{\text{\tiny{th}}}\,\mathcal{V}(t)\big{)}=\frac{1}{(1+t^{ \text{\tiny{th}}})}\bigg{[}\frac{1}{9\epsilon^{\text{\tiny{th}}}}+\frac{1}{15+t} \sin(\mathcal{V}(t))+\frac{1}{9+t}\frac{D_{0^{\text{\tiny{th}}}}^{\text{\tiny {th}}}\,\mathcal{V}(t)}{(1+D_{0^{\text{\tiny{th}}}}^{\text{\tiny{th}}}\, \mathcal{V}(t))}\bigg{]},\\ \mathcal{W}_{2}\big{(}t,\mathcal{U}(t),\mathcal{D}_{0^{\text{\tiny{th}}}}^{ \text{\tiny{th}}}\,\mathcal{U}(t)\big{)}=\frac{1}{(1+t^{\text{\tiny{th}}})} \bigg{[}\frac{1}{5t^{\text{\tiny{th}}}}+\frac{1}{20+t^{2}}\cos(\mathcal{U}(t)) +\frac{1}{16+t^{2}}\frac{D_{0^{\text{\tiny{th}}}}^{\text{\tiny{th}}}\, \mathcal{U}(t)}{(1+D_{0^{\text{\tiny{th}}}}^{\text{\tiny{th}}}\,\mathcal{U}(t) )}\bigg{]}.\end{array} \right. \tag{37}\]  
Then, one can see that  
\[ \begin{array}{l}|\mathcal{W}_{1}\big{(}t,\mathcal{V}_{1},\mathcal{V}_{2} \big{)}|\leq\frac{1}{(1+t^{32})}\bigg{[}\frac{1}{9\epsilon^{\text{\tiny{th}}} }+\frac{1}{15+t}+\frac{1}{9+t}\bigg{]},\\ |\mathcal{W}_{2}\big{(}t,\mathcal{U}_{1},\mathcal{U}_{2}\big{)}|\leq\frac{1}{(1 +t^{54})}\bigg{[}\frac{1}{5e^{2t}}+\frac{1}{20+t^{2}}+\frac{1}{16+t^{2}}\bigg{]},\end{array} \]  
yielding \(f_{1}(t)=\frac{1}{(1+t^{32})}\Big{[}\frac{1}{9\epsilon^{\text{\tiny{th}}}}+ \frac{1}{15+t}+\frac{1}{9+t}\Big{]},\quad f_{2}(t)=\frac{1}{(1+t^{34})}\Big{[} \frac{1}{5e^{2t}}+\frac{1}{20+t^{2}}+\frac{1}{16+t^{2}}\Big{]}.\)  
We also obtain  
\[ \begin{array}{l}|\mathcal{W}_{1}\big{(}t,\mathcal{V}_{1},\mathcal{V}_{2} \big{)}-\mathcal{W}_{1}\big{(}t,\mathcal{\bar{V}}_{1},\mathcal{\bar{V}}_{2} \big{)}|\leq\frac{1}{(1+t^{32})}\frac{1}{9+t}\big{(}|\mathcal{V}_{1}-\mathcal{ \bar{V}}_{1}|+|\mathcal{V}_{2}-\mathcal{\bar{V}}_{2}|\big{)},\\ |\mathcal{W}_{2}\big{(}t,\mathcal{U}_{1},\mathcal{U}_{2}\big{)}-\mathcal{W}_{2} \big{(}t,\mathcal{\bar{U}}_{1},\mathcal{\bar{U}}_{2}\big{)}|\leq\frac{1}{(1+t^ {54})}\frac{1}{16+t^{2}}\big{(}|\mathcal{U}_{1}-\mathcal{\bar{U}}_{1}|+| \mathcal{U}_{2}-\mathcal{\bar{U}}_{2}|\big{)}.\end{array} \]  
\[ \begin{array}{l}\mathcal{N}_{1}(t)=\frac{1}{(1+t^{52})}\frac{1}{9+t},\quad \mathcal{N}_{2}(t)=\frac{1}{(1+t^{54})}\frac{1}{16+t^{2}},\quad\mathcal{W}_{1} ^{*}=19,\text{ and }\mathcal{W}_{2}^{*}=15.\text{ Therefore, we can infer that}\end{array} \]  
\[ \begin{array}{l}|\Lambda_{1}|=1,\ \mathcal{A}_{1}\approx 0.0377694<1,\ \mathcal{A}_{2} \approx 0.276993<\infty,\ \mathcal{A}_{3}\approx 1.72682\times 10^{106}<\infty,\\ |\Lambda_{2}|=0.709106,\ \mathcal{B}_{1}\approx 0.000171751<1,\mathcal{B}_{2} \approx 0.0553989<\infty,\ \mathcal{B}_{3}\approx 1.7349\times 10^{59}<\infty,\\ \text{and}\quad\mathcal{G}=\max\{\mathcal{A}_{1},\mathcal{B}_{1}\}<1.\end{array} \]  
Hence, all conditions of Theorem 3 hold. Then, the coupled \(\varphi\)-\(\mathbb{R}\mathbb{L}\) fractional Laplacian implicit differential system (34) possesses a unique solution \((\mathcal{\bar{U}},\mathcal{\bar{V}})\) on unbounded domain \([0,\infty)\times[0,\infty)\). Additionally, it has an explicit iterated unbounded solution, represented as follows:  
\[