$A\Phi (t)+\int\limits {0} {t}{K(x,t-\tau )\Phi }(\tau )d\tau =H$

Lemma 2. Let ${{\tau }_{3}}(t),{{\tau }_{4}}(t)\in CVL(0,T)$. Then
$\underset{x\to a-0}{\mathop{\lim }}\,{{w}_{3}}(x,t)=\frac{1}{3}{{\tau }_{3}}(t)$, $\underset{x\to a+0}{\mathop{\lim }}\,{{w}_{3}}(x,t)=-\frac{2}{3}{{\tau }_{3}}(t)$, $\underset{x\to a+0}{\mathop{\lim }}\,{{w}_{4}}(x,t)=0$.
Lemma 3. Let ${{\tau }_{5}}(x)\in C\left[ a,b \right]$.Then function ${{w}_{5}}(x,t)$ is the fundamental solution of equation (7) and
$\underset{t\to 0}{\mathop{\lim }}\,D_{0t}^{\alpha -1}{{w}_{5}}(x,t)={{\tau }_{5}}(x)$.
Lemma 4. The equation $\partial _{0t}^{\alpha }u(x,t)-\frac{{{\partial }^{3}}}{\partial {{x}^{3}}}u(x,t)=f(x,t)$ with initial condition
$D_{0t}^{\alpha -1}u{{(x,t)}_{t=0}}=0$
has a solution in the form
${{w}_{6}}(x,t)=\int\limits_{0}^{t}{d\eta \int\limits_{a}^{b}{G_{\alpha }^{\frac{2\alpha }{3}}(x-\xi ,t-\eta )f(\xi ,\eta )}}d\xi $.
Existence of solutions.
Let us find solutions in the form
${{u}_{1}}(x,t)=\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{2{{\alpha }_{1}}}{3}}(x-{{L}_{1}},t-\tau ){{\alpha }_{1}}(\tau )}d\tau +\int\limits_{0}^{t}{V_{{{\alpha }_{1}}}^{\frac{2{{\alpha }_{1}}}{3}}(x-{{L}_{1}},t-\tau ){{\beta }_{1}}(\tau )}d\tau +$
$+\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{2{{\alpha }_{1}}}{3}}(x-0,t-\tau ){{\gamma }_{1}}(\tau )}d\tau +{{F}_{1}}(x,t),$
${{u}_{2}}(x,t)=\int\limits_{0}^{t}{G_{{{\alpha }_{2}}}^{\frac{2{{\alpha }_{2}}}{3}}(x-{{L}_{2}},t-\tau ){{\alpha }_{2}}(\tau )}d\tau +\int\limits_{0}^{t}{V_{{{\alpha }_{2}}}^{\frac{2{{\alpha }_{2}}}{3}}(x-0,t-\tau ){{\rho }_{2}}(\tau )}d\tau +$
$+\int\limits_{0}^{t}{G_{{{\alpha }_{2}}}^{\frac{2{{\alpha }_{2}}}{3}}(x-0,t-\tau ){{\gamma }_{2}}(\tau )}d\tau +{{F}_{2}}(x,t),$
${{u}_{3}}(x,t)=\int\limits_{0}^{t}{G_{{{\alpha }_{3}}}^{\frac{2{{\alpha }_{3}}}{3}}(x-{{L}_{3}},t-\tau ){{\alpha }_{3}}(\tau )}d\tau +\int\limits_{0}^{t}{V_{{{\alpha }_{3}}}^{\frac{2{{\alpha }_{3}}}{3}}(x-0,t-\tau ){{\rho }_{3}}(\tau )}d\tau +$
$+\int\limits_{0}^{t}{G_{{{\alpha }_{3}}}^{\frac{2{{\alpha }_{3}}}{3}}(x-0,t-\tau ){{\gamma }_{3}}(\tau )}d\tau +{{F}_{3}}(x,t),$
where functions ${{\alpha }_{i}}(t),{{\gamma }_{i}}(t),{{\beta }_{i}}(t),{{\rho }_{i}}(t)$ $i=\overline{1,3}$ are unknown functions ${{\beta }_{2}}(t)={{\beta }_{3}}(t)={{\rho }_{1}}(t)=0$ and
${{F}_{i}}(x,t)=\int\limits_{{{B}_{i}}}{{{u}_{0i}}(\xi )D_{0t}^{\alpha -1}G_{{{\alpha }_{i}}}^{\frac{2{{\alpha }_{i}}}{3}}}(x-\xi ,t-0)d\xi +\int\limits_{0}^{t}{\int\limits_{{{B}_{i}}}{G_{{{\alpha }_{i}}}^{\frac{2{{\alpha }_{i}}}{3}}(x-\xi ,t-0){{f}_{i}}(\xi ,\tau )d\xi d\tau }}$.
It follows from Lemma 4 and the results given in [19] that these functions are the solutions of equations (7) and they satisfy initial conditions (8).
Taking into account condition (9),we have
$\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{2{{\alpha }_{1}}}{3}}(-{{L}_{1}},t-\tau ){{\alpha }_{1}}(\tau )d\tau +}\int\limits_{0}^{t}{V_{{{\alpha }_{1}}}^{\frac{2{{\alpha }_{1}}}{3}}(-{{L}_{1}},t-\tau ){{\beta }_{1}}(\tau )d\tau }+\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{2{{\alpha }_{1}}}{3}}(0,t-\tau ){{\gamma }_{1}}(\tau )d\tau +}{{F}_{1}}(0,t)=$
$={{a}_{1}}\int\limits_{0}^{t}{G_{{{\alpha }_{2}}}^{\frac{2{{\alpha }_{2}}}{3}}(-{{L}_{2}},t-\tau ){{\alpha }_{2}}(\tau )d\tau +}{{a}_{1}}\int\limits_{0}^{t}{G_{{{\alpha }_{2}}}^{\frac{2{{\alpha }_{2}}}{3}}(0,t-\tau ){{\gamma }_{2}}(\tau )d\tau +}{{a}_{1}}\int\limits_{0}^{t}{V_{{{\alpha }_{2}}}^{\frac{2{{\alpha }_{2}}}{3}}(0,t-\tau ){{\rho }_{2}}(\tau )d\tau +}{{a}_{1}}{{F}_{2}}(0,t)$,
$\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{2{{\alpha }_{1}}}{3}}(-{{L}_{1}},t-\tau ){{\alpha }_{1}}(\tau )d\tau +}\int\limits_{0}^{t}{V_{{{\alpha }_{1}}}^{\frac{2{{\alpha }_{1}}}{3}}(-{{L}_{1}},t-\tau ){{\beta }_{1}}(\tau )d\tau }+\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{2{{\alpha }_{1}}}{3}}(0,t-\tau ){{\gamma }_{1}}(\tau )d\tau +}{{F}_{1}}(0,t)=$
$={{a}_{2}}\int\limits_{0}^{t}{G_{{{\alpha }_{3}}}^{\frac{2{{\alpha }_{3}}}{3}}(-{{L}_{3}},t-\tau ){{\alpha }_{3}}(\tau )d\tau +}{{a}_{2}}\int\limits_{0}^{t}{G_{{{\alpha }_{3}}}^{\frac{2{{\alpha }_{3}}}{3}}(0,t-\tau ){{\gamma }_{3}}(\tau )d\tau +}{{a}_{2}}\int\limits_{0}^{t}{V_{{{\alpha }_{3}}}^{\frac{2{{\alpha }_{3}}}{3}}(0,t-\tau ){{\rho }_{3}}(\tau )d\tau +}{{a}_{2}}{{F}_{3}}(0,t)$.
Furthermore
$\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{2{{\alpha }_{1}}}{3}}(-{{L}_{1}},t-\tau ){{\alpha }_{1}}(\tau )d\tau +}\int\limits_{0}^{t}{V_{{{\alpha }_{1}}}^{\frac{2{{\alpha }_{1}}}{3}}(-{{L}_{1}},t-\tau ){{\beta }_{1}}(\tau )d\tau }+\int\limits_{0}^{t}{\frac{\phi (-\frac{{{\alpha }_{1}}}{3},\frac{2{{\alpha }_{1}}}{3};0)}{3{{(t-\tau )}^{1-\frac{2{{\alpha }_{1}}}{3}}}}{{\gamma }_{1}}(\tau )d\tau +}{{F}_{1}}(0,t)=$
$={{a}_{1}}\int\limits_{0}^{t}{G_{{{\alpha }_{2}}}^{\frac{2{{\alpha }_{2}}}{3}}(-{{L}_{2}},t-\tau ){{\alpha }_{2}}(\tau )d\tau +}{{a}_{1}}\int\limits_{0}^{t}{\frac{\phi (-\frac{{{\alpha }_{2}}}{3},\frac{2{{\alpha }_{2}}}{3};0)}{3{{(t-\tau )}^{1-\frac{2{{\alpha }_{2}}}{3}}}}{{\gamma }_{2}}(\tau )d\tau +}$
$+\operatorname{Im}\left[ {{a}_{1}}\int\limits_{0}^{t}{\frac{{{e}^{\frac{2\pi i}{3}}}\phi (-\frac{{{\alpha }_{2}}}{3},\frac{2{{\alpha }_{2}}}{3};0)}{3{{(t-\tau )}^{1-\frac{2{{\alpha }_{2}}}{3}}}}{{\rho }_{2}}(\tau )d\tau } \right]+{{a}_{1}}{{F}_{2}}(0,t)$.
So, we have
$\frac{1}{3}{{a}_{1}}{{\gamma }_{2}}(t)+\frac{\sqrt{3}}{6}{{a}_{1}}{{\rho }_{2}}(t)=D_{0t}^{\frac{2{{\alpha }_{2}}}{3}}(\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{2{{\alpha }_{1}}}{3}}(-{{L}_{1}},t-\tau ){{\alpha }_{1}}(\tau )d\tau }+\int\limits_{0}^{t}{V_{{{\alpha }_{1}}}^{\frac{2{{\alpha }_{1}}}{3}}(-{{L}_{1}},t-\tau ){{\beta }_{1}}(\tau )d\tau }+$
$+\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{2{{\alpha }_{1}}}{3}}(0,t-\tau ){{\gamma }_{1}}(\tau )d\tau -}{{a}_{1}}\int\limits_{0}^{t}{G_{{{\alpha }_{2}}}^{\frac{2{{\alpha }_{2}}}{3}}(-{{L}_{2}},t-\tau ){{\alpha }_{2}}(\tau )d\tau )+}D_{0t}^{\frac{2{{\alpha }_{2}}}{3}}({{F}_{1}}(0,t)-{{F}_{2}}(0,t))$
$\frac{1}{3}{{a}_{2}}{{\gamma }_{3}}(t)+\frac{\sqrt{3}}{6}{{a}_{2}}{{\rho }_{3}}(t)=D_{0t}^{\frac{2{{\alpha }_{3}}}{3}}(\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{2{{\alpha }_{1}}}{3}}(-{{L}_{1}},t-\tau ){{\alpha }_{1}}(\tau )d\tau }+\int\limits_{0}^{t}{V_{{{\alpha }_{1}}}^{\frac{2{{\alpha }_{1}}}{3}}(-{{L}_{1}},t-\tau ){{\beta }_{1}}(\tau )d\tau }+$
$+\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{2{{\alpha }_{1}}}{3}}(0,t-\tau ){{\gamma }_{1}}(\tau )d\tau -}{{a}_{2}}\int\limits_{0}^{t}{G_{{{\alpha }_{2}}}^{\frac{2{{\alpha }_{3}}}{3}}(-{{L}_{3}},t-\tau ){{\alpha }_{3}}(\tau )d\tau )+}D_{0t}^{\frac{2{{\alpha }_{3}}}{3}}({{F}_{1}}(0,t)-{{F}_{3}}(0,t))$
From above relation we obtain
$\frac{1}{3}{{a}_{1}}{{\gamma }_{2}}(t)+\frac{\sqrt{3}}{6}{{a}_{1}}{{\rho }_{2}}(t)=\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{2({{\alpha }_{1}}-{{\alpha }_{2}})}{3}}(-{{L}_{1}},t-\tau ){{\alpha }_{1}}(\tau )d\tau }+\int\limits_{0}^{t}{V_{{{\alpha }_{1}}}^{\frac{2({{\alpha }_{1}}-{{\alpha }_{2}})}{3}}(-{{L}_{1}},t-\tau ){{\beta }_{1}}(\tau )d\tau }+$
$+\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{2({{\alpha }_{1}}-{{\alpha }_{2}})}{3}}(0,t-\tau ){{\gamma }_{1}}(\tau )d\tau -}{{a}_{1}}\int\limits_{0}^{t}{G_{{{\alpha }_{2}}}^{0}(-{{L}_{2}},t-\tau ){{\alpha }_{2}}(\tau )d\tau +}D_{0t}^{\frac{2{{\alpha }_{2}}}{3}}({{F}_{1}}(0,t)-{{F}_{2}}(0,t))$
$\frac{1}{3}{{a}_{2}}{{\gamma }_{3}}(t)+\frac{\sqrt{3}}{6}{{a}_{2}}{{\rho }_{3}}(t)=\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{2({{\alpha }_{1}}-{{\alpha }_{3}})}{3}}(-{{L}_{1}},t-\tau ){{\alpha }_{1}}(\tau )d\tau }+\int\limits_{0}^{t}{V_{{{\alpha }_{1}}}^{\frac{2({{\alpha }_{1}}-{{\alpha }_{3}})}{3}}(-{{L}_{1}},t-\tau ){{\beta }_{1}}(\tau )d\tau }+$
$+\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{2({{\alpha }_{1}}-{{\alpha }_{3}})}{3}}(0,t-\tau ){{\gamma }_{1}}(\tau )d\tau -}{{a}_{2}}\int\limits_{0}^{t}{G_{{{\alpha }_{2}}}^{0}(-{{L}_{3}},t-\tau ){{\alpha }_{3}}(\tau )d\tau +}D_{0t}^{\frac{2{{\alpha }_{3}}}{3}}({{F}_{1}}(0,t)-{{F}_{3}}(0,t))$
In a similar manner, we obtain from condition (10) that
$\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{{{\alpha }_{1}}}{3}}(-{{L}_{1}},t-\tau ){{\alpha }_{1}}(\tau )}d\tau +\int\limits_{0}^{t}{V_{{{\alpha }_{1}}}^{\frac{{{\alpha }_{1}}}{3}}(-{{L}_{1}},t-\tau ){{\beta }_{1}}(\tau )}d\tau +\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{{{\alpha }_{1}}}{3}}(0,t-\tau ){{\gamma }_{1}}(\tau )}d\tau +{{F}_{1x}}(0,t)=$
$={{b}_{1}}\int\limits_{0}^{t}{G_{{{\alpha }_{2}}}^{\frac{{{\alpha }_{2}}}{3}}(-{{L}_{2}},t-\tau ){{\alpha }_{2}}(\tau )}d\tau +{{b}_{1}}\int\limits_{0}^{t}{G_{{{\alpha }_{2}}}^{\frac{{{\alpha }_{2}}}{3}}(0,t-\tau ){{\gamma }_{2}}(\tau )}d\tau +{{b}_{1}}\int\limits_{0}^{t}{V_{{{\alpha }_{2}}}^{\frac{{{\alpha }_{2}}}{3}}(0,t-\tau ){{\rho }_{2}}(\tau )}d\tau +{{b}_{1}}{{F}_{2x}}(0,t),$
$\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{{{\alpha }_{1}}}{3}}(-{{L}_{1}},t-\tau ){{\alpha }_{1}}(\tau )}d\tau +\int\limits_{0}^{t}{V_{{{\alpha }_{1}}}^{\frac{{{\alpha }_{1}}}{3}}(-{{L}_{1}},t-\tau ){{\beta }_{1}}(\tau )}d\tau +\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{{{\alpha }_{1}}}{3}}(0,t-\tau ){{\gamma }_{1}}(\tau )}d\tau +{{F}_{1x}}(0,t)=$
$={{b}_{2}}\int\limits_{0}^{t}{G_{{{\alpha }_{3}}}^{\frac{{{\alpha }_{3}}}{3}}(-{{L}_{3}},t-\tau ){{\alpha }_{3}}(\tau )}d\tau +{{b}_{2}}\int\limits_{0}^{t}{G_{{{\alpha }_{3}}}^{\frac{{{\alpha }_{3}}}{3}}(0,t-\tau ){{\gamma }_{3}}(\tau )}d\tau +{{b}_{2}}\int\limits_{0}^{t}{V_{{{\alpha }_{3}}}^{\frac{{{\alpha }_{3}}}{3}}(0,t-\tau ){{\rho }_{3}}(\tau )}d\tau +{{b}_{2}}{{F}_{3x}}(0,t)$
From above relation we obtain
$\frac{{{b}_{1}}}{3}{{\gamma }_{2}}(t)-\frac{{{b}_{1}}\sqrt{3}}{6}{{\rho }_{2}}(t)=\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{{{\alpha }_{1}}-{{\alpha }_{2}}}{3}}(-{{L}_{1}},t-\tau ){{\alpha }_{1}}(\tau )}d\tau +\int\limits_{0}^{t}{V_{{{\alpha }_{1}}}^{\frac{{{\alpha }_{1}}-{{\alpha }_{2}}}{3}}(-{{L}_{1}},t-\tau ){{\beta }_{1}}(\tau )}d\tau +$
$+\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{{{\alpha }_{1}}-{{\alpha }_{2}}}{3}}(0,t-\tau ){{\gamma }_{1}}(\tau )}d\tau -{{b}_{1}}\int\limits_{0}^{t}{G_{{{\alpha }_{2}}}^{0}(-{{L}_{2}},t-\tau ){{\alpha }_{2}}(\tau )}d\tau +D_{0t}^{\frac{{{\alpha }_{2}}}{3}}{{F}_{1x}}(0,t)-D_{0t}^{\frac{{{\alpha }_{2}}}{3}}{{b}_{1}}{{F}_{2x}}(0,t)$
$\frac{{{b}_{2}}}{3}{{\gamma }_{3}}(t)-\frac{{{b}_{2}}\sqrt{3}}{6}{{\rho }_{3}}(t)=\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{{{\alpha }_{1}}-{{\alpha }_{3}}}{3}}(-{{L}_{1}},t-\tau ){{\alpha }_{1}}(\tau )}d\tau +\int\limits_{0}^{t}{V_{{{\alpha }_{1}}}^{\frac{{{\alpha }_{1}}-{{\alpha }_{3}}}{3}}(-{{L}_{1}},t-\tau ){{\beta }_{1}}(\tau )}d\tau +$
$+\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{{{\alpha }_{1}}-{{\alpha }_{3}}}{3}}(0,t-\tau ){{\gamma }_{1}}(\tau )}d\tau -{{b}_{2}}\int\limits_{0}^{t}{G_{{{\alpha }_{3}}}^{0}(-{{L}_{3}},t-\tau ){{\alpha }_{3}}(\tau )}d\tau +D_{0t}^{\frac{{{\alpha }_{3}}}{3}}{{F}_{1x}}(0,t)-D_{0t}^{\frac{{{\alpha }_{3}}}{3}}{{b}_{1}}{{F}_{3x}}(0,t)$
Taking into account condition (11) and using Lemmas given above, we have
$\frac{1}{3}{{\gamma }_{1}}(t)+\frac{1}{3{{a}_{1}}}{{\gamma }_{2}}(t)+\frac{1}{3{{a}_{2}}}{{\gamma }_{3}}(t)=\int\limits_{0}^{t}{\frac{1}{{{a}_{1}}}\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\partial }^{2}}}{{{x}^{2}}}G_{{{\alpha }_{2}}}^{\frac{2{{\alpha }_{2}}}{3}}(x-{{L}_{2}},t-\tau )}{{\alpha }_{2}}(\tau )d\tau +$
$+\int\limits_{0}^{t}{\frac{1}{{{a}_{2}}}\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\partial }^{2}}}{{{x}^{2}}}G_{{{\alpha }_{3}}}^{\frac{2{{\alpha }_{3}}}{3}}(x-{{L}_{3}},t-\tau )}{{\alpha }_{3}}(\tau )d\tau -\int\limits_{0}^{t}{\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\partial }^{2}}}{{{x}^{2}}}G_{{{\alpha }_{1}}}^{\frac{2{{\alpha }_{1}}}{3}}(x-{{L}_{1}},t-\tau )}{{\alpha }_{1}}(\tau )d\tau -$
$-\int\limits_{0}^{t}{\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\partial }^{2}}}{{{x}^{2}}}V_{{{\alpha }_{1}}}^{\frac{2{{\alpha }_{1}}}{3}}(x-{{L}_{1}},t-\tau )}{{\beta }_{1}}(\tau )d\tau +\frac{1}{{{a}_{1}}}{{F}_{2xx}}+\frac{1}{{{a}_{3}}}{{F}_{3xx}}-{{F}_{1xx}}$
Using conditions (12), we have
$\frac{1}{3}{{\alpha }_{1}}(t)+\frac{\sqrt{3}}{6}{{\beta }_{1}}(t)=-\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{0}({{L}_{1}},t-\tau ){{\gamma }_{1}}(\tau )d\tau +D_{0t}^{\frac{2{{\alpha }_{1}}}{3}}({{\varphi }_{1}}(t)-{{F}_{1}}({{L}_{1}},t))}$

$\frac{1}{3}{{\alpha }_{2}}(t)=-\int\limits_{0}^{t}{G_{{{\alpha }_{2}}}^{0}({{L}_{2}},t-\tau ){{\gamma }_{2}}(\tau )d\tau -\int\limits_{0}^{t}{V_{{{\alpha }_{2}}}^{0}({{L}_{2}},t-\tau )}{{\rho }_{2}}(\tau )d\tau +D_{0t}^{\frac{2{{\alpha }_{2}}}{3}}({{\varphi }_{2}}(t)-{{F}_{2}}({{L}_{2}},t))}$

$\frac{1}{3}{{\alpha }_{3}}(t)=-\int\limits_{0}^{t}{G_{{{\alpha }_{3}}}^{0}({{L}_{3}},t-\tau ){{\gamma }_{3}}(\tau )d\tau -\int\limits_{0}^{t}{V_{{{\alpha }_{3}}}^{0}({{L}_{3}},t-\tau )}{{\rho }_{3}}(\tau )d\tau +D_{0t}^{\frac{2{{\alpha }_{3}}}{3}}({{\varphi }_{3}}(t)-{{F}_{3}}({{L}_{3}},t))}$
$\frac{1}{3}{{\alpha }_{1}}(t)-\frac{\sqrt{3}}{6}{{\beta }_{1}}(t)=-\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{0}({{L}_{1}},t-\tau ){{\gamma }_{1}}(\tau )d\tau +D_{0t}^{\frac{{{\alpha }_{1}}}{3}}({{\psi }_{1}}(t)-{{F}_{1x}}({{L}_{1}},t))}$
We obtain the following system of integral equations (20)-(24) with respect to unknowns $\Phi (t)$
\[\Alpha \Phi (t)+\int\limits_{0}^{t}{\Kappa (t-\tau )}\Phi (\tau )d\tau =H\]
.
Where $\Phi (t)$ is the unknown functions, $A$ is the $9\times 9$ matrix, $K$ is the matrix of potentials. Using above system, the matrices can be written in the form
$\frac{{{a}_{1}}}{3}$ $-\frac{\sqrt{3}}{6}{{b}_{2}}$
$\Phi (t)={{({{\alpha }_{1}}(t),{{\alpha }_{2}}(t),{{\alpha }_{3}}(t),{{\beta }_{1}}(t),{{\gamma }_{1}}(t),{{\gamma }_{2}}(t),{{\gamma }_{3}}(t),{{\rho }_{2}}(t),{{\rho }_{3}}(t))}^{T}}$
$\det A\ne 0$
$\sum\limits_{i=1}^{3}{\int\limits_{{{B}_{i}}}{{{u}_{i}}{{u}_{ixxx}}dx=\sum\limits_{i=1}^{3}{{{u}_{i}}{{u}_{ixx}}{{|}_{{{B}_{i}}}}-\frac{1}{2}\sum\limits_{i=1}^{3}{u_{i}^{2}{{|}_{{{B}_{i}}}}=}}}}$${{u}_{1}}(0,t){{u}_{1xx}}(0,t)-{{u}_{1}}(-{{L}_{1}},t){{u}_{1xx}}(-{{L}_{1}},t)+{{u}_{2}}({{L}_{2}},t){{u}_{2xx}}({{L}_{2}},t)-{{u}_{2}}(0,t){{u}_{2xx}}(0,t)+$
$+{{u}_{3}}({{L}_{3}},t){{u}_{3xx}}({{L}_{3}},t)-{{u}_{3}}(0,t){{u}_{3xx}}(0,t)-\frac{1}{2}({{u}_{1x}}(0,t)-{{u}_{2x}}(0,t)-{{u}_{2x}}(0,t))$
=
= =
$\int\limits_{0}^{t}{\partial _{0\tau }^{\alpha }f(\tau )}d\tau =\frac{1}{\Gamma (1-\alpha )}\int\limits_{0}^{t}{\int\limits_{0}^{\tau }{\frac{\frac{\partial }{\partial s}f(s)}{{{(\tau -s)}^{\alpha }}}dsd\tau =}}\frac{1}{\Gamma (1-\alpha )}\int\limits_{0}^{t}{ds}\int\limits_{s}^{t}{\frac{\frac{\partial }{\partial s}f(s)}{{{(\tau -s)}^{\alpha }}}d\tau =}$
$=\frac{1}{\Gamma (1-\alpha )}\int\limits_{0}^{t}{\frac{\partial }{\partial s}f(s)\cdot \frac{{{(\tau -s)}^{1-\alpha }}}{1-\alpha }}|_{\tau =s}^{\tau =t}ds=\frac{1}{(1-\alpha )\Gamma (1-\alpha )}\int\limits_{0}^{t}{\frac{\frac{\partial }{\partial s}f(s}{{{(t-\tau )}^{\alpha -1}}}ds=}$
$=\frac{1}{\Gamma (2-\alpha )}f(s)\cdot {{(t-s)}^{1-\alpha }}|_{s=0}^{s=t}+\frac{1}{\Gamma (1-\alpha )}\int\limits_{0}^{t}{\frac{f(s)}{{{(t-s)}^{\alpha }}}ds=}$
$=D_{0t}^{\alpha -1}f(t)-\frac{{{t}^{1-\alpha }}}{\Gamma (2-\alpha )}f(0)$.
$\int\limits_{a}^{b}{\int\limits_{0}^{t}{u(x,\tau )f(x,\tau )d\tau d}}x=\int\limits_{0}^{t}{\int\limits_{a}^{b}{\frac{u(x,\tau )}{{{(t-\tau )}^{\frac{\alpha }{2}}}}}}{{(t-\tau )}^{\frac{\alpha }{2}}}f(x,\tau )dxd\tau \le $
$\le \varepsilon \int\limits_{0}^{t}{\frac{d\tau }{{{(t-\tau )}^{\alpha }}}}\int\limits_{a}^{b}{{{u}^{2}}(x,\tau )dx+\frac{1}{4\varepsilon }}\int\limits_{0}^{t}{{{(t-\tau )}^{\alpha }}d\tau }\int\limits_{a}^{b}{{{f}^{2}}(x,\tau )dx}=$
$=\varepsilon \Gamma (1-\alpha )D_{0t}^{\alpha -1}\int\limits_{a}^{b}{{{u}^{2}}(x,t)dx}+\frac{1}{4\varepsilon }D_{0t}^{\alpha -1}\int\limits_{a}^{b}{{{f}^{2}}(x,t)dx}$

$F(x,y,{{D}^{\alpha }}u,......)=0$

$\frac{\sqrt{3}}{{{3}^{8}}}{{a}_{2}}{{a}_{3}}b_{2}^{2}b_{3}^{2}$
$\frac{1}{3}{{a}_{3}}{{\gamma }_{3}}(t)=\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{2}{3}({{\alpha }_{1}}-{{\alpha }_{3}})}({{L}_{1}},t-\tau ){{\alpha }_{1}}(\tau )d\tau +}\int\limits_{0}^{t}{V_{{{\alpha }_{1}}}^{\frac{2}{3}({{\alpha }_{1}}-{{\alpha }_{3}})}(0,t-\tau ){{\rho }_{1}}(\tau )d\tau +}$.
$+\int\limits_{0}^{t}{G_{{{\alpha }_{1}}}^{\frac{2}{3}({{\alpha }_{1}}-{{\alpha }_{3}})}(0,t-\tau ){{\gamma }_{1}}(\tau )d\tau -{{a}_{3}}}\int\limits_{0}^{t}{G_{{{\alpha }_{3}}}^{0}({{L}_{3}},t-\tau ){{\alpha }_{3}}(\tau )d\tau -}$.

$D_{0t}^{\frac{{{\alpha }_{1}}}{3}}({{F}_{1x}}(0,t)-{{b}_{2}}{{F}_{2x}}(0,t)-{{b}_{3}}{{F}_{3x}}(0,t))$