Application of a finite-difference integral transform to Ionkin-Samarskij problems (Q5935946)

For the Ionkin-Samarskij problem \[ u_t=a^2u_{xx}+F(x,t); \quad 0<x<1, \;0<t<T< \infty, \tag{1} \] \[ u(x,0)=f(x) \quad \text{for} \;x\in (a_i,a_{i+1}), \quad i=0,\dots ,n-1,\tag{2} \] \[ u(0,t)=\nu (t), \quad\int_0^1K(x)u(x,t) dx = \mu(t), \tag{3} \] one says that the boundary conditions (3) are proper if there exist \(M>0\) and \(m\in \mathbb{R}\) such that \[ |\Delta (\lambda)|\geq \frac{M}{\lambda^m}, \quad \lambda \in R_\delta = \{\lambda \in C;\;|\lambda|\geq R,\;|\arg\lambda|\leq \tfrac{\pi}{4} + \delta\}, \tag{\(*\)} \] where \(\Delta\) is the denominator in the Green function of the associated parametric problem, \(R\) a sufficiently large positive number, and \(\delta\) an arbitrary given number such that \(0<2\delta<\frac{\pi}{4}-|\arg a|\). Supposing \((*)\) and the hypotheses \(1^\circ\) \(\text{Re }a^2 >0\); \(2^\circ\) \(K\in L_2([0,1])\); \(3^\circ\) \(F\in C(]0,1]\times [0,T])\); \(\mu,\nu\in C((0,T])\cap L([0,T])\); \(4^\circ\) \(f\in C([a_i,a_{i+1}])\), \(i=0,\dots,n-1\), an integral representation for the solution \(u(x,t)\) of the problem (1)--(3) is given, if the problem is solvable. In addition if \((*)\), \(1^\circ\), \(2^\circ\), \(4^\circ\) and \(F(x,t)\equiv 0\), \(\nu(t)\equiv 0\), \(\mu(t)\equiv 0\), then the problem (1)--(3) has a unique solution for which one has the integral representation and \(u(a_i,0)=\frac{1}{2}[f(a_i-0)+f(a_i+0)]\); \(i= 1, \dots,n-1.\)