Volume 9, no. 2Pages 117 - 123
Numerical Research of the Barenblatt - Zheltov - Kochina Stochastic ModelS.I. Kadchenko, Е.А. Soldatova, S.А. Zagrebina
At present, investigations of Sobolev-type models are actively developing. In the solution of applied problems the results allowing to get their numerical solutions are very significant. In the article the algorithm for numerical solving of the initial boundary value problem is developed. The problem describes the pressure distribution of the homogeneous fluid in the horizontal layer in the circle. The layer is opened by a vertical well of a small radius. In our research we suppose that random disturbing loads have an influence on the fluid. The problem was solved under two assumptions. Firstly, we suppose that an unstable fluid flow is axially symmetric, and secondly, that in initial moment the pressure in the layer is constant. After the process of the discretization we modify the original model to the Cauchy problem for the system of ordinary differential equations. For the numerical solution we use algorithms based on explicit one-step formulas of the Runge - Kutta type with the seventh-order accuracy and with the selection of the integration step. We also use the scheme of the eighth-order accuracy to evaluate the calculation accuracy on each steps of time. According to the results of this control, we choose the time-step. A lot of numerical experiments have shown high numerical efficiency of the algorithm that we use to solve the investigated initial-boundary problem. Full text
- stochastic Sobolev type equation; numerical solution; Barenblatt - Zheltova - Kochina model; Cauchy problem.
- 1. Barenblatt G.I., Zheltov Yu.P., Kochina I.N. Basic Concepts in the Theory of Seepage of Homogeneous Fluids in Fissurized Rocks. Journal of Applied Mathematics and Mechanics, 1960, vol. 24, no. 5, pp. 1286-1303.
2. Zagrebina S.A., Soldatova Е.А. Linear Sobolev Type Equations with Relatively p-Bounded Operators and Additive White Noise. Bulletin of the Irkutsk State University. Series: Mathematics, 2013, vol. 6, no. 1, pp. 20-34. (in Russian)
3. Hallaire M. On a Theory of Moisture-Transfer. Inst. Rech. Agronom., 1964, no. 3, pp. 60-72.
4. Chen P.J., Gurtin M.E. On a Theory of Heat Conduction Involving Two Temperatures. Z. Angew. Math. Phys., 1968, vol. 19, pp. 614-627.
5. Sviridyuk G.A. Couchy Problem for the Linear Singular Sobolev Type Equation. Differential Equations, 1987, vol. 23, no. 12, pp. 2169-2171.
6. Sviridyuk G.A., Manakova N.A. The Dynamical Models of Sobolev Type with Showalter - Sidorov Condition and Additive 'Noise'. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 1, pp. 90-103. (in Russian) DOI: 10.14529/mmp140108.
7. Novikov Е.А., Shornikov Yu.V. Control of the Stability for the Fulberg Method with Seventh-Order Accuracy. Computational Technologies, 2006, vol. 11, no. 4, pp. 65-72.
8. Novikov Е.А. Algorithms of Variable Structure Development for the Solving of the Stiff Problems. Krasnoyarsk, 2014. 123 p. (in Russian)