On the solvability and approximate solution of integro-differential Wiener-Hopf equations on one fourth of the plane (Q2761533)

This paper deals with integro-differential equation NEWLINE\[NEWLINE\sum_{\nu=0}^{s}\sum_{\mu=0}^{m}a_{\nu\mu}{\partial^{\nu+\mu}\phi(x,y)\over\partial x^{\nu}\partial y^{\mu}}+{1\over 2\pi}\sum_{l=0}^{p}\sum_{j=0}^{q}\int_{0}^{\infty}\int_{0}^{\infty} k_{lj}(t-x,\tau-y){\partial^{l+j}\phi(t,\tau)\over\partial t^{l}\partial \tau^{j}} dt d\tau=h(x,y),NEWLINE\]NEWLINE \(x>0\), \(y>0\) where \(a_{\nu\mu}\) are known constants; \(k_{lj}(x,y)\in L\), \(h(x,y)\in L_2\) are known functions. The authors study the problem of solvability of the considered equation and, in particular, they prove the following result.NEWLINENEWLINENEWLINELet us denote NEWLINE\[NEWLINEA(x,y)=(x+i)^{-a}(y+i)^{-b} \sum_{\nu=0}^{s}\sum_{\mu=0}^{m}(-ix)^{\nu}(-iy)^{\mu}a_{\nu\mu}+ \sum_{l=0}^{p}\sum_{j=0}^{q}(-ix)^{l}(-iy)^{j}K_{lj }(x,y),NEWLINE\]NEWLINE where \(a=\max(s,p), b=\max(m,q)\), \(K_{lj}(x,y)\) is the Fourier transformation of the function \(k_{lj}(x,y)\). Let \(A(x,y)\in H_{\alpha\beta}\), \(0<\alpha,\beta<1\), then the considered equation is normally solvable if and only if \(A(x,y)\neq 0, (x,y)\in R\times R\). A method of approximate solving of integro-differential Wiener-Hopf equation is proposed.