In this section we analyse the solution of the fractional order model and examined its uniqueness, convergence and existence47.Existence of the solutionThis section provides existence of the solution via fixed point theory.
Definition 5.1.114
Let \(\left(k,d\right)\) be a non-empty couchy space and \(\lambda ,0\le \lambda <1\) then consider mapping \(S:X\to X \ni \) for every \(x,\overline{x }\in X\) then then \(d\left(Sx ,S\overline{x } \right)\le \lambda \left(x,\overline{x }\right)\) holds, the \({x}^{*}\in X\) is the unique fixed point of \(S\). If \({S}^{k} \left(k\in N\right)\) the sequence defined by$$\left\{\begin{array}{c}{S}^{k}= S{S}^{k-1} L\in \frac{\mathbb{N}}{\left\{1\right\}},\\ {S}^{1}=S;\end{array}\right.$$
Thus for any \({x}_{0}\in X, {\{ {S}_{{x}_{0}}^{k}\}}_{k=1}^{k=\infty }\) reaches to the fixed \({x}^{\boldsymbol{*}}\).
Definition 5.1.214
Let \(m\in {\mathbb{N}},H\subset {\mathbb{R}}^{m},\left[p,q\right]\subset {\mathbb{R}}and h;\left[p,q\right]\times H\to {\mathbb{R}}\) be a function.
For any \(\left({x}_{1},{x}_{2},\dots {x}_{m}\right)\left({x}_{1}^{*},{x}_{2}^{*},\dots {x}_{m}^{*}\right)\in H\), it satisfies the generalized Lipschitzian condition.$$\left|h\left(S,{x}_{1},{x}_{2},\dots {x}_{m}\right)-h\left({S,x}_{1}^{*},{x}_{2}^{*},\dots {x}_{m}^{*}\right)\right|\le {A}_{1}\left|{x}_{1}-{x}_{1}^{*}\right|+{A}_{2}\left|{x}_{2}-{x}_{2}^{*}\right|+\dots {+A}_{m}\left|{x}_{m}-{x}_{m}^{*}\right|,$$$${A}_{j}\ge 0, j=\text{0,1},\text{2,3}\dots m$$Specifically, h satisfies the Lipschitzian condition.If \(\forall \zeta \epsilon \left(p,q\right] \text{and for any} x, {x}^{*}\epsilon G,\)$$\left|h\left[\zeta ,x\right]-h\left[\zeta ,{x}^{*}\right]\right|\le A\left|x-{x}^{*}\right|,A>0,$$Let us consider the system of equation$${}_{0}{}^{c}{D}_{t}^{\alpha }S\left(t\right)={\phi }_{1}\left(x,t,S\right),$$$${}_{0}{}^{c}{D}_{t}^{\alpha }F\left(t\right)={\phi }_{2}\left(x,t,F\right),$$$${}_{0}{}^{c}{D}_{t}^{\alpha }B\left(t\right)={\phi }_{3}\left(x,t,B\right),$$$${}_{0}{}^{c}{D}_{t}^{\alpha }P\left(t\right)={\phi }_{4}\left(x,t,P\right).$$Now using above equation we have$$S\left(x,t\right)-S\left(x,0\right)={}_{0}{}{I}_{t}^{\alpha }\left\{\lambda B-\beta SF\right\},$$$$F\left(x,t\right)-F\left(x,0\right)={}_{0}{}{I}_{t}^{\alpha }\left\{\left(\lambda +\omega \right)B-\beta SF\right\},$$$$B\left(x,t\right)-B\left(x,0\right)={}_{0}{}{I}_{t}^{\alpha }\left\{\beta SF-\left(\lambda +\omega \right)B\right\},$$$$P\left(x,t\right)-P\left(x,0\right)={}_{0}{}{I}_{t}^{\alpha }\left\{\omega B\right\}.$$Then by the definition of Riemann–Liouville fractional integral, we get$$S\left(x,t\right)-S\left(x,0\right)=\frac{1}{\Gamma \left(\alpha \right)}\underset{0}{\overset{t}{\int }}{\left(t-v\right)}^{\alpha -1}{\phi }_{1}\left(x,v,S\right)dv,$$$$F\left(x,t\right)-F\left(x,0\right)=\frac{1}{\Gamma \left(\alpha \right)}\underset{0}{\overset{t}{\int }}{\left(t-v\right)}^{\alpha -1}{\phi }_{2}\left(x,v,F\right)dv,$$$$B\left(x,t\right)-B\left(x,0\right)=\frac{1}{\Gamma \left(\alpha \right)}\underset{0}{\overset{t}{\int }}{\left(t-v\right)}^{\alpha -1}{\phi }_{3}\left(x,v,B\right)dv,$$$$P\left(x,t\right)-P\left(x,0\right)=\frac{1}{\Gamma \left(\alpha \right)}\underset{0}{\overset{t}{\int }}{\left(t-v\right)}^{\alpha -1}{\phi }_{4}\left(x,v,P\right)dv.$$In this section we have provided the existence of solution. By the aid of fixed point theory, Cauchy space and with the help of Lipschitzian condition, Riemann–Liouville fractional integral we have shown the existence of solution for the considered model. Hence we can conclude that their exist unique solution for the proposed time fractional mathematical model.Convergence of the solutionConsider a mapping \(H:G\to G\) is non-linear with Banach space G. Let us take$$\Vert H\left(u\right)-H(v)\Vert \le {\mu }_{i}\Vert u-v\Vert , \forall u,v\in G$$Hence their exist fixed point converge to a singular point is H and$$\Vert {v}_{m}-{v}_{p}\Vert \le \frac{{\mu }_{i}^{p}}{1-{\mu }_{i}}\Vert {v}_{1}-{v}_{0}\Vert , i=\text{1,2},\text{3,4}\dots $$
Proof
Let \(\left(c\left[j\right],\Vert \bullet \Vert \right)\) be a Banach space with norm specified as \(\Vert g\left(t\right)\Vert ={max}_{t\in j}\left|g(t)\right|\) function on J.
Now we verify \(\left\{{S}_{p}\right\},\left\{{F}_{p}\right\},\left\{{B}_{p}\right\},\left\{{P}_{p}\right\}\) is a Cauchy sequence in \(\left( {c\left[ j \right],\parallel \cdot \parallel } \right)\)For S consider,$$\Vert {S}_{m}-{S}_{p}\Vert =\underset{\mathit{t\epsilon J}}{\text{max}}S\left|{S}_{m}-{S}_{p}\right|,$$$$=\underset{\mathit{t\epsilon J}}{\text{max}}\int \left({K}_{p}+h\right)\left({S}_{m-1}-{S}_{p-1}\right)-h{L}^{-1}\left(\frac{1}{{S}^{\alpha }}\left(\left(\lambda {B}_{m-1}-\lambda {B}_{p-1}\right)- \left(\beta {S}_{m-1}{F}_{m-1}- \beta {S}_{p-1}{F}_{p-1}\right)\right)\right)$$$$ \mathop { \le {\text{max}}}\limits_{t\varepsilon J} \left| {\left( {K_{p} + h} \right)\left( {S_{m – 1} – S_{p – 1} } \right)} \right| – h\left| {\mathop \smallint \limits_{0}^{t} \left( {\left( {\lambda B_{m – 1} – \lambda B_{p – 1} } \right) – \left( {\beta S_{m – 1} F_{m – 1} – \beta S_{p – 1} F_{p – 1} } \right)} \right)\frac{{\left( {t – v} \right)^{\alpha } }}{{\Gamma \left( {1 + \alpha } \right)}}dv} \right|\,\left( {\text{By convolution theorem}} \right) $$$$\le \left|\left({K}_{p}+h\right)\left({S}_{m-1}-{S}_{p-1}\right)\right|-h\underset{0}{\overset{t}{\int }}\lambda {\delta }_{1}+\beta \left({\delta }_{2}+{\delta }_{3}\right)\frac{{\left(t-v\right)}^{\alpha }}{\Gamma \left(1+\alpha \right)}\left|{s}_{m-1}-{S}_{p-1}\right|dv$$The above inequality is reduced to$$\Vert {S}_{m}-{S}_{p}\Vert \le {\mu }_{1}\Vert {S}_{m-1}-{S}_{p-1}\Vert ,$$where \({{\delta }_{1}=B}_{m-1}-{B}_{p-1 }, {{\delta }_{2}=S}_{m-1}-{S}_{p-1}, {{\delta }_{3}=F}_{m-1}-{F}_{p-1}\)Then takes \(m=p+1\) it yields$$\Vert {S}_{p+1}-{S}_{p}\Vert \le {\mu }_{1}\Vert {S}_{p}-{S}_{p-1}\Vert \le {\mu }_{1}^{2}\Vert {S}_{p-1}-{S}_{p-2}\Vert \dots {\mu }_{1}^{p}\Vert {S}_{1}-{S}_{0}\Vert .$$Using triangular inequality$$\Vert {S}_{p}-{S}_{p-1}\Vert \le \Vert {S}_{p+1}-{S}_{p}\Vert +\Vert {S}_{p+2}-{S}_{p+1}\Vert +\dots \Vert {S}_{p}-{S}_{p-1}\Vert ,$$$$\le \left[{ \mu }_{1}^{p}+{\mu }_{1}^{p-1}+{\mu }_{1}^{p-2}+\dots +{\mu }_{1}^{m-1}\right]\Vert {S}_{1}-{S}_{0}\Vert ,$$$$\le { \mu }_{1}^{p}\left[\frac{1-{\mu }_{1}^{m-p-1}}{1-{\mu }_{1}}\right]\Vert {S}_{1}-{S}_{0}\Vert .$$As \(0<\mu <1, so 1-{\mu }_{1}^{m-p-1}<1 ,\) then we have$$\Vert {S}_{p+1}-{S}_{p}\Vert \le \frac{{ \mu }_{1}^{p}}{1-{\mu }_{1}}\Vert {S}_{1}-{S}_{0}\Vert $$But \(\Vert {S}_{1}-{S}_{0}\Vert <\infty \) consequently as \(m\to \infty \) then \(\Vert {S}_{p+1}-{S}_{p}\Vert \to 0\) proves \(\left\{{S}_{p}\right\}\) is a Cauchy sequence.This proves theorem.Simultaneously we have,$$\Vert {F}_{p+1}-{F}_{p}\Vert \le \frac{{ \mu }_{2}^{p}}{1-{\mu }_{1}}\Vert {F}_{1}-{F}_{0}\Vert ,$$$$\Vert {B}_{p+1}-{B}_{p}\Vert \le \frac{{ \mu }_{3}^{p}}{1-{\mu }_{3}}\Vert {B}_{1}-{B}_{0}\Vert ,$$$$\Vert {P}_{p+1}-{P}_{p}\Vert \le \frac{{ \mu }_{4}^{p}}{1-{\mu }_{4}}\Vert {P}_{1}-{P}_{0}\Vert .$$Uniqueness of the solutionThe solution of considered fractional differential equation via q-HATM is unique,Whenever \(0<{\mu }_{i }<1, i=\text{1,2},\text{3,4}\)
Proof
The solution of equation is illustrated as follows
In general,$$v\left(x,t\right)=\sum_{p=0}^{\infty }{v}_{p}\left(x,t\right).$$
For i = 1, suppose \(S, {S}^{*}\) be two different values \(\ni S-{S}^{*}\le \underset{t\in J}{\text{max}}S-{S}^{*},\)$$\le \left|\left({K}_{p}+h\right)\left(S-{S}^{*}\right)-h{L}^{-1}\left(\frac{1}{{S}^{\alpha }}\left(\lambda B-\beta SF\right)\right)\right|,$$
\(\le \left|\left({K}_{p}+h\right)\left(S-{S}^{*}\right)\right|-h\left|\underset{0}{\overset{t}{\int }}\left(\lambda B-\beta SF\right)\frac{{\left(t-v\right)}^{\alpha }}{\Gamma \left(1+\alpha \right)} dv\right|\)
(By convolution theorem)$$\le \left|\left({K}_{p}+h\right)\left(S-{S}^{*}\right)\right|-h\underset{0}{\overset{t}{\int }}\left(\lambda {S}_{1}+\left({S}_{2}+{S}_{3}\right)\beta \right)\frac{{\left(t-v\right)}^{\alpha }}{\Gamma \left(1+\alpha \right)}\left|S-{S}^{*}\right|dv$$The above inequality related to$$\left|S-{S}^{*}\right|\le {\mu }_{S}\left|S-{S}^{*}\right|$$$${\mu }_{S}=\left({K}_{p}+h\right)\left(S-{S}^{*}\right)-h\underset{0}{\overset{t}{\int }}\left(\lambda {\delta }_{1}+\left({\delta }_{2}+{\delta }_{3}\right)\beta \right)\frac{{\left(t-v\right)}^{\alpha }}{\Gamma \left(1+\alpha \right)} dv,$$where \({\delta }_{1}= {B}_{m-1}-{B}_{p-1}, {\delta }_{2}= {S}_{m-1}-{S}_{p-1}, {\delta }_{3}= {F}_{m-1}-{F}_{p-1},\)We get$$\left(1-{\mu }_{5}\right)\left|S-{S}^{*}\right|\le 0,$$$$\left|S-{S}^{*}\right|=0, 0<\mu <1,$$Similarly,