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**Approximation of solutions to the boundary value problems for the generalized Boussinesq equation.**
*(English)*
Zbl 1401.35288

Summary: The paper is devoted to one of the Sobolev type mathematical models of fluid filtration in a porous layer. Results that allow to obtain numerical solutions are significant for applied problems. We propose the following algorithm to solve the initial-boundary value problems describing the motion of a free surface filtered in a fluid layer having finite depth. First, the boundary value problems are reduced to the Cauchy problems for integro-differential equations, and then the problems are numerically integrated. However, numerous computational experiments show that the algorithm can be simplified by replacing the integro-differential equations with the corresponding approximating Riccati differential equations, whose solutions can also be found explicitly. In this case, the numerical values of the solution to the integro-differential equation are concluded between successive values of approximating solutions. Therefore, we can pointwise estimate the approximation errors. Examples of results of numerical integration and corresponding approximations are given.

### MSC:

35Q79 | PDEs in connection with classical thermodynamics and heat transfer |

35A35 | Theoretical approximation in context of PDEs |

### Keywords:

Sobolev type equation; boundary value problem; integro-differential equation; free surface; Riccati equation
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\textit{V. Z. Furaev} and \textit{A. I. Antonenko}, Vestn. Yuzhno-Ural. Gos. Univ., Ser. Mat. Model. Program. 10, No. 4, 145--150 (2017; Zbl 1401.35288)

### References:

[1] | [1] Dzektser E. S., Shadrin G. A., “On the Motion of Groundwater with a Free Surface”, Industrial and Civil Engineering, 10 (1971), 22–44 (in Russian) |

[2] | [2] Sviridyuk G. A., Manakova N. A., “The Barenblatt–Zheltov–Kochina Model with Additive White Noise in Quasi-Sobolev Spaces”, Journal of Computational and Engineering Mathematics, 3:1 (2016), 61–67 · Zbl 1359.60087 |

[3] | [3] Sviridyuk G. A., Manakova N. A., “The Phase Space of the Cauchy-Dirichlet Problem for the Oskolkov Equation of Nonlinear Filtration”, Russian Mathematics (Izvestiya VUZ. Matematika), 2003, no. 9, 33–38 · Zbl 1076.35064 |

[4] | [4] Zamyshlyaeva A. A., Bychkov E. V., Tsyplenkova O. N., “Mathematical Models Based on Boussinesq–Love Equation”, Applied Mathematical Sciences, 8:110 (2014), 5477–5483 |

[5] | [5] Sviridyuk G. A., “A Problem for the Generalized Boussinesq Filtration Equation”, Soviet Mathematics (Izvestiya VUZ. Matematika), 33:2 (1989), 62–73 · Zbl 0699.35127 |

[6] | [6] Furaev V.Ż., On the Solvability of Boundary Value Problems and Cauchy Problems for the Generalized Boussinesq Equation in the Theory of Nonstationary Filtration, The dissertation of the candidate physical and mathematical sciences, The Peoples’ Friendship University named Patrice Lumumba, M., 1983 (in Russian) |

[7] | [7] Furaev V.Ż., “Solvability in the Large of the First Boundary Value Problem for the Generalized Boussinesq Equation”, Differential Equations, 19:11 (1983), 2014–2015 (in Russian) · Zbl 0543.76024 |

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