Article · Bulletin of the Tomsk Polytechnic University Geo Assets Engineering · April 29, 2026

Existence of solutions for direct problem of soil moisture movement in landslide massif with instant and cord sources

D. Zakirova, A. Satybaev, Z. Malikova

Abstract

Relevance. One of the main causes of landslide processes is associated with the infiltration of moisture from snow, rain, and hail into the landslide mass, indicating a significant dependence of landslides on moisture transfer processes in the landslide-prone rock mass. This article examines the direct problem of soil moisture movement in the landslide mass as one of the primary causes of landslide occurrences. The direct problem is analyzed considering instant and cord sources, which are the most appropriate initial and boundary conditions for modeling this natural phenomenon. When studying direct problems, primary attention is given to the correctness of their solutions, including issues of existence, uniqueness, and stability of solutions. Methods. Methods of characteristic straightening, method of feature extraction, and finite difference method. Aim. To prove the existence of solutions to the direct problem under specific conditions defined by the input parameters of the equation and the problem conditions. Results and conclusions. Using the methods of summation by parts and differentiation of products, all terms of the final equation were transformed, which enabled a step-by-step analysis of the mathematical relationships and confirmed the validity of the proposed approach. It was demonstrated that a specific mathematical equality is valid under the given conditions, confirming both the theoretical soundness and the applicability of the developed algorithm. While proving the theorems, it was shown that piecewise-continuous functions exhibit boundedness and linearity. All these functions converge to the corresponding functions, and it was also noted that they converge weakly to other functions. Considering the behavior of these functions and their limits, it was proven that the mathematical model allows for the derivation of a generalized solution to the direct problem. For citation: Zakirova D.A., Satybaev A.Dzh., Malikova Z.T. Existence of solutions for direct problem of soil moisture movement in landslide massif with instant and cord sources. Bulletin of the Tomsk Polytechnic University. Geo Assets Engineering, 2026, vol. 337, no. 4, pp. 220-233.

Differential Equations and Boundary ProblemsLandslides and related hazardsNonlinear Differential Equations Analysissoil water movementforward problemexistence of solutions

Existence of solutions for direct problem of soil moisture movement in landslide massif with instant and cord sources

Abstract

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