Closed-form solution to a class of boundary-value problems for the 1+1-dimensional Kardar-Parisi-Zhang equation

• The deterministic KPZ equation is considered on generic bounded intervals • Exact solution in the time domain is derived by means of inverse Laplace transforms • Cases with incompatible initial and boundary conditions are included • The closed-form expression of the solution offers new alternative and perspectives This paper presents a new approach for solving a class of boundary value problems for the 1+1-dimensional Kardar-Parisi-Zhang equation on generic bounded intervals. The Hopf-Cole transformation is used to solve the given problem, based on a closed-form solution for the linear reaction-diffusion equation on finite intervals, published recently. The exact solution to the deterministic equation with fixed Neumann boundary conditions is derived in the time domain using inverse Laplace transforms. Even if analytic inverses cannot be found in transform tables, highly efficient algorithms are available. Numerical inverses in the time domain are always feasible, regardless of the complexity of the Laplace domain expressions. In comparison with solutions derived using series expressions or numerical methods, closed-form solutions, even in the Laplace domain, offer novel insights. Furthermore, the utilization of the numerical inverse Laplace transforms is shown to be a more efficient computational approach, thereby establishing a reference point for numerical and semi-analytical methods.