On Nonlinear Equations of $p$-adic Strings for Scalar Tachyon Fields

We consider boundary value problems for open and closed $p$-adic strings for scalar tachyon fields. Estimates for solutions to these problems and possible ways of constructing these solutions are obtained by reducing the problems to linear parabolic equations with nonlinear boundary conditions. We give an application of Gauss-type quadrature formulas to the numerical solution of the boundary value problems, and discuss the possibility of using these methods in multidimensional problems ($d=2$).