## 3 Reduction of the IP1, IP2, and IP3 to Nonclassical Problems  
In this section, we formulate the three inverse coefficient-source problems. Let us differentiate \(E(t)\) with respect to \(t\), and using Equation (1) and the first boundary conditions in (3), we find  
\[f(t)=\big{(}\
u(u_{x}(X,t)-u_{x}(0,t))+\beta(t)h(t)-E^{\prime}(t)\big{)}/X,\ \ \ h(t)=B_{r}(t)-B_{l}(t), \tag{9}\]  
where \(E^{\prime}=\frac{d}{dt}E,\ \ u_{x}=\frac{\partial u}{\partial x}\). Next, we differentiate \(F(t)\) with respect to \(t\), and using Equation (2) and the second boundary conditions in (3), we obtain  
\[\beta(t)=(F^{\prime}(t)-\
u_{m}(B_{x}(X,t)-B_{x}(0,t)))/g(t),\ \ \ g(t)=u_{r}(t)-u_{l}(t), \tag{10}\]  
where \(F^{\prime}=\frac{d}{dt}F\) and \(B_{x}=\frac{\partial B}{\partial x}\). Let us introduce new solution functions to obtain zero boundary conditions:  
\[U(x,t) = u(x,t)-\Big{(}(1-\frac{x}{X})u_{l}(t)+\frac{x}{X}u_{r}(t)\Big{)}, \tag{11}\] \[b(x,t) = B(x,t)-\Big{(}(1-\frac{x}{X})B_{l}(t)+\frac{x}{X}B_{r}(t)\Big{)}. \tag{12}\]  
Applying these transformations to the problem (1)-(4), we arrive at the following nonclassical forms of (1)-(4). **IP1. ** Let us suppose that \(\beta(t)\) and \(f(t)\) are unknowns and the overdeterminations (7), (8) are given. Then the reduced (nonclassical) problem reads as follows:  
\[\frac{\partial U}{\partial t} - \
u\frac{\partial^{2}U}{\partial x^{2}}-\frac{F^{\prime}(t)-\
u_ {m}(b_{x}(X,t)-b_{x}(0,t))}{g(t)}\frac{\partial b}{\partial x}\] \[= \frac{E^{\prime}(t)}{X}-\frac{\
u}{X}(U_{x}(X,t)-U_{x}(0,t))- \Big{(}(1-\frac{x}{X})u_{l}^{{}^{\prime}}(t)+\frac{x}{X}u_{r}^{{}^{\prime}}( t)\Big{)}\,,\] \[\frac{\partial b}{\partial t} - \
u_{m}\frac{\partial^{2}b}{\partial x^{2}}-\frac{F^{\prime}(t)- \
u_{m}(b_{x}(X,t)-b_{x}(0,t))}{g(t)}\frac{\partial U}{\partial x}\] \[= \frac{F^{\prime}(t)}{X}-\frac{\
u_{m}}{X}(b_{x}(X,t)-b_{x}(0,t))- \Big{(}(1-\frac{x}{X})B_{l}^{{}^{\prime}}(t)+\frac{x}{X}B_{r}^{{}^{\prime}}( t)\Big{)}\]  
with new initial conditions  
\[U(x,0) = u_{0}(x)-\Big{(}(1-\frac{x}{X})u_{l}(0)+\frac{x}{X}u_{r}(0)\Big{)} \equiv U_{0}(x), \tag{15}\] \[b(x,0) = B_{0}(x)-\Big{(}(1-\frac{x}{X})B_{l}(0)+\frac{x}{X}B_{r}(0)\Big{)} \equiv b_{0}(x) \tag{16}\]  
and zero Dirichlet boundary conditions. **IP2. ** Now, let us suppose that \(\beta(t)\) is a known function, but \(f(t)\) is unknown, and the overdetermination (7) is given. Then the nonclassical problem reads as the following parabolic system:  
\[\beta(t)=(F^{\prime}(t)-\
u_{m}(B_{x}(X,t)-B_{x}(0,t)))/g(t),\ \ \ g(t)=u_{r}(t)-u_{l}(t), \tag{10}\]0 \[\beta(t)=(F^{\prime}(t)-\
u_{m}(B_{x}(X,t)-B_{x}(0,t)))/g(t),\ \ \ g(t)=u_{r}(t)-u_{l}(t), \tag{10}\]1 \[\beta(t)=(F^{\prime}(t)-\
u_{m}(B_{x}(X,t)-B_{x}(0,t)))/g(t),\ \ \ g(t)=u_{r}(t)-u_{l}(t), \tag{10}\]2 \[\beta(t)=(F^{\prime}(t)-\
u_{m}(B_{x}(X,t)-B_{x}(0,t)))/g(t),\ \ \ g(t)=u_{r}(t)-u_{l}(t), \tag{10}\]3  
with initial conditions (15), (16) and zero Dirichlet boundary conditions. **IP3. ** Finally, let us suppose that \(\beta(t)\) is an unknown function, but \(f(t)\) is known and the overdetermination (8) is given. Then the reduced problem reads as follows:  
\[\beta(t)=(F^{\prime}(t)-\
u_{m}(B_{x}(X,t)-B_{x}(0,t)))/g(t),\ \ \ g(t)=u_{r}(t)-u_{l}(t), \tag{10}\]4 \[\beta(t)=(F^{\prime}(t)-\
u_{m}(B_{x}(X,t)-B_{x}(0,t)))/g(t),\ \ \ g(t)=u_{r}(t)-u_{l}(t), \tag{10}\]5 \[\beta(t)=(F^{\prime}(t)-\
u_{m}(B_{x}(X,t)-B_{x}(0,t)))/g(t),\ \ \ g(t)=u_{r}(t)-u_{l}(t), \tag{10}\]6 \[\beta(t)=(F^{\prime}(t)-\
u_{m}(B_{x}(X,t)-B_{x}(0,t)))/g(t),\ \ \ g(t)=u_{r}(t)-u_{l}(t), \tag{10}\]7 \[\beta(t)=(F^{\prime}(t)-\
u_{m}(B_{x}(X,t)-B_{x}(0,t)))/g(t),\ \ \ g(t)=u_{r}(t)-u_{l}(t), \tag{10}\]8 \[\beta(t)=(F^{\prime}(t)-\
u_{m}(B_{x}(X,t)-B_{x}(0,t)))/g(t),\ \ \ g(t)=u_{r}(t)-u_{l}(t), \tag{10}\]9 \[U(x,t) = u(x,t)-\Big{(}(1-\frac{x}{X})u_{l}(t)+\frac{x}{X}u_{r}(t)\Big{)}, \tag{11}\]0  
with initial conditions (15), (16) and zero Dirichlet boundary conditions. In all cases, there appear terms \(U_{x}(0,t)\), \(U_{x}(X,t)\) or \(b_{x}(0,t)\), \(b_{x}(X,t)\) in the differential equations.