A generalized Fourier method of solving a mixed problem for a hyperbolic system (Q1380246)

Consider the problem \[ {\partial u\over \partial t}=a^0 (x){\partial u\over \partial x}+a^1(x,t)u +f(x,t), \quad a<x<b,\;0<t\leq T; \tag{1} \] \[ (2)\quad \alpha^1 {\partial u\over \partial x} + \alpha^0 u|_{x=a} +\beta^1{\partial u\over \partial x}+\beta^0 u|_{x=b} =0; \qquad (3) \quad u(x,0)= h(x), \] where \(a^i\), \(\alpha^i\), and \(\beta^i\) are \((N\times N)\)-matrices; and \(u, f\), and \(h\) are \((N\times 1)\)-matrices, and \(da^0/dx\) and \(a^1\) are continuous with respect to the arguments \(a\leq x\leq b\) and \(0\leq t\leq T\). It is supposed, that system (1) is hyperbolic in the restricted sense, that the boundary conditions are regular, that \(h'(x)\in L^N_2(a,b)\), and \(h(a)= h(b)=0\), \(f_t'\in L^N_2((a,b) \times(0,T))\), and \(h(x)\in L^N_2(a,b)\), and \(f(x,t)\in L_2^N ((a,b) \times(0,T))\). In contrast to previous work, the ``higher'' operator on the right-hand side of (1) and in condition (2) are not self-adjoint. By using a solution in the case \(a^1(x,t) \equiv 0\), we accomplish a simple method of reducing a solution to problem (1)--(3) to a Volterra-type integral equation.