## 2 Materials and Methods Direct Formulation of the Problem on Interval (\(0,t\))  
We considered the case of the heat equation on a segment with inhomogeneous boundary data. \[\frac{\partial u_{1}(x,t)}{\partial t} = \frac{\partial^{2}u_{1}(x,t)}{\partial x^{2}},\;0<x<1,\;t\in(0,T], \tag{1}\] \[u_{1}(x,0)=0,\;0\leq x\leq 1,\] (2) \[u_{1}(0,t)=0,\;0\leq t\leq T,\] (3) \[u_{1}(1,t)=u(t),\;0\leq t\leq T, \tag{4}\]  
Assume the \(q(t)\) function is defined as the following  
\[q_{1}(t)\in C^{3}[0,T],q_{1}(0)={q^{\prime}}_{1}(0)={q^{\prime\prime}}_{1}(0) ={q^{\prime}}_{1}(T)={q^{\prime\prime}}_{1}(T)=0, \tag{5}\]  
by using [PERSON]'s principle method ([24], p. 109)  
\[u_{1}(x,t)=\int\limits_{0}^{t}w_{t}(x,t-\tau)q_{1}(\tau)d\tau, \tag{6}\]  
integration by parts for the right part for (6) once, we obtain  
\[u_{1}(x,t)=\int\limits_{0}^{t}w(x,t-\tau){q^{\prime}}_{1}(\tau)d\tau+q_{1}(0) w(x,t). \tag{7}\]  
Now, we can decide to obtain the solution for \(w(x,t)\) as the following  
\[w(x,t)=x+2{\sum\limits_{n=1}^{\infty}{\frac{(-1)^{n}}{n\pi}}}e^{-(n\pi)^{2}t} \sin(n\pi x), \tag{8}\]  
by substituting (8) in (7), to obtain a solution to a non-stationary problem, from (5) \(q_{1}(0)=0\),  
\[u_{1}(x,t)=x+{\sum\limits_{n=1}^{\infty}{(-1)^{n}C_{n}(t)\sin(n\pi x)}}, \tag{9}\]  
\[C_{n}(t)=\frac{2}{n\pi}{\int\limits_{0}^{t}{e^{-(n\pi)^{2}(t-\tau)}q^{\prime} }_{1}(\tau)d\tau}, \tag{10}\]  
where \(x\in(0,1)\), and \(t\in(0,T]\). **Lemma 1**. : _Let \(q_{1}(t)\) satisfy condition (5). Then, there exists a solution \(u_{1}(x,t)\) for problem (1)-(5) such that \(u_{1}(x,t)\) satisfies the Equation (1) on the set \((0,1)\times(0,T]\), initial condition (2), boundary conditions (3), (4) and \(u_{1}(x,t)\in C([0,1]\times[0,T])\cap C^{2,1}((0,1)\times(0,T])\)._  
Proof. : By integrating the right side of the Formula (10) in parts twice, we obtain \[u_{1}(x,0)=0,\;0\leq x\leq 1,\]0  
since \(|C_{n}(t)\sin(n\pi x)|\leq|C_{n}(t)|\) for any \(n\) and from the Cauchy-Bunyakovsky inequality  
\[u_{1}(x,0)=0,\;0\leq x\leq 1,\]1  
by means of (5), (4) and (12) for any \(t\in(0,T]\) and for any \(n\) we obtain  
\[u_{1}(x,0)=0,\;0\leq x\leq 1,\]2  
Using Equations (11)-(13) and convergence of the series \(\sum\limits_{n-1}^{\infty}\frac{1}{n^{3}}\), \(\sum\limits_{n-1}^{\infty}\frac{1}{n^{5}}\), \(\sum\limits_{n-1}^{\infty}\frac{1}{n^{6}}\) with the Weierstrass criterion follows the unchanging convergence of the above series on \((0,1)\times(0,T]\). Since the functions \(e^{-(n\pi)^{2}(\tau)}q^{\prime\prime\prime}{}_{1}(\tau)\in L_{2}(0,T]\), obtaining  
\[u_{1}(x,0)=0,\;0\leq x\leq 1,\]3  
Thus, with \(q_{1}(t),q^{\prime}{}_{1}(t)\) and \(q^{\prime\prime}{}_{1}(t)\in C(0,T]\) in addition to Equations (11) and (14), we take \(C_{n}(t)\in C(0,T]\). From this condition and the convergence of (19) in domain \((0,1)\times(0,T]\), we have \(u_{1}(x,t)\in C((0,1)\times(0,T])\). Differentiating a \(C_{n}(t)\sin(n\pi x)\) with \(x\) and by using (13), we obtain  
\[u_{1}(x,0)=0,\;0\leq x\leq 1,\]4  
From the above relation, we obtain the convergence of the \(\sum\limits_{n-0}^{\infty}\left(C_{n}(t)\sin n\pi x\right)^{\prime}{}_{x}\) in \([0,1]\times[0,T]\), from (8) we have \(\frac{\partial u_{1}(x,t)}{\partial x}=\sum\limits_{n-0}^{\infty}\left(C_{n} (t)\sin n\pi x\right)^{\prime}{}_{x}\) in \((0,1)\times(0,T]\) and \(\frac{\partial u_{1}(x,t)}{\partial x}\in((0,1)\times(0,T])\). Now, let us examine the function \(\frac{\partial^{2}u_{1}(x,t)}{\partial x^{2}}\).