Boundary-value problems for a nonlinear hyperbolic equation with divergent part and Lévy Laplacian

We propose an algorithm for the solution of the boundary-value problem

$$ U\left( {0,x} \right) = {u_0},\,\,\,\,U\left( {t,0} \right) = {u_1} $$

and the external boundary-value problem

$$ U\left( {0,x} \right) = {\upsilon_0},\,\,\,\,{\left. {U\left( {t,x} \right)} \right|_{\Gamma }} = {\upsilon_1},\,\,\,\,\mathop{{\lim }}\limits_{{{{\left\| x \right\|}_{{H \to \infty }}}}} U\left( {t,x} \right) = {\upsilon_2} $$

for the nonlinear hyperbolic equation

$$ \frac{\partial }{{\partial t}}\left[ {k\left( {U\left( {t,x} \right)} \right)\frac{{\partial U\left( {t,x} \right)}}{{\partial t}}} \right] = {\Delta_L}U\left( {t,x} \right) $$

with divergent part and infinite-dimensional Lévy Laplacian ∆ _L .