The solving of boundary value problems by numerical integration of stochastic equations

The probability approach to numerical solution of nonlinear parabolic equations

“…Here we apply it to constructing some layer methods. To show this let us consider the Cauchy problem 13) which is connected with the system (2.7)-(2.9).…”

“…Boundary value problems for nonlinear parabolic equations will be considered in a separate work. The probability approach to linear boundary value problems is treated in [13] and [14].…”

Improvement of the Euler scheme for the solution of a stationary diffusion equation by extrapolation

“…Therefore, it is advisable to use more complex higher-order schemes (see, e.g. [1,9]). In this paper, we propose a new method for increasing the order of convergence of the discrete Euler scheme, which is associated with the property of infinite divisibility of normal random variables.…”

“…Another method for increasing the order of convergence is to use random walk by small spheres [9] r n+1 = r n + v(r n )τ + σ (r n )ν n √ kτ. (2.4) Here k is the space dimension, {ν n } is the sequence of independent random vectors uniformly distributed over the surface of a unit sphere.…”

“…In addition, we introduce the ε-neighbourhood of the domain boundary, reaching which the chain terminates. Under the conditions of the regularity of the problem specified in [9], this scheme ensures the order of the error q = 1 by increasing the computational cost.…”