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The fundamental solutions of diffusion equation for the local-equilibrium andnonlocal models are considered as the limiting cases of the solution of aproblem related to consideration of the Brownian particles random walks. Thedifferences between fundamental solutions were studied. It was shown that onthe period of observation time and distances exceeding the time and spaceassociated with one step of random walk the fundamental solutions of diffusionand telegraph equations are very close to each other. In particular, thesefundamental solutions are very close to the values of the probability densitydistribution of diffusive particles that is obtained from the accurate solutionof the random walk problem. The new modified non-local diffusion equation issuggested. It contains only microparameters of the random walk problem.

About Relationship Between Flux and Concentration Gradient of Particles at Description of Diffusion with the Usage of Random Walk Model