Solution of Boundary Value Problems for Certain Nonlinear Differential Equations Using the Bubnov-Galerkin Method

Introduction. This study investigates the possibility of numerically solving a boundary value problem with a nonlinear differential equation, continuous coefficients, and a right-hand side using the modified Bubnov-Galerkin method. In the problem formulation, the partial derivatives of the equationʼs coefficients are continuous functions of all arguments. The order of the nonlinear differential equation n is strictly less than the number of coordinate functions m.

Materials and Methods. To numerically solve the nonlinear boundary value problem, the modifi Petrov-Galerkin method and the uniqueness property of decomposing a smooth function into a system of linearly independent polynomial basis functions on the interval [−1,1] with a unit Chebyshev norm for each function in the system are used. The system of linear algebraic equations includes linearly independent boundary conditions. The matrix elements and the right-hand side of the system depend on the simple iteration index s. The coefficient  vector of the solution decomposition into basis functions also depends on the index s. The inverse matrix of the system was computed using the Msimsl linear algebra library in Fortran.

Results. Sufficient conditions for the existence and uniqueness of the solution to the boundary value problem with a nonlinear differential equation using the simple iteration method have been formulated. When the sufficient conditions are met, the decomposition coefficients decrease absolutely as the basis function index increases.

Discussion and Conclusion. Three boundary value problems with a second-order nonlinear equation and one problem with a third-order equation were solved exactly. The analytical solutions were compared with numerical solutions, with the uniform norm of the difference having an order of 10^‒13, 10^‒11, 10^‒10, 10^‒10, respectively. The modified Bubnov-Galerkin method allows for solving each branch of a multivalued function in boundary value problems with nonlinear differential equations.

1. Bahvalov N.S., Zhidkov N.P., Kobelkov G.M. Numerical methods: a textbook for students of physics and mathematics specialties of higher educational institutions. Binom. lab. Knowledge; 2011. 636 p. (In Russ.)

2. Ershova T.Ya. Boundary value problem for a third-order differential equation with a strong boundary layer. Bulletin of Moscow University. Episode 15: Computational mathematics and cybernetics. 2020;1:30–39. (In Russ.) https://doi.org/10.3103/S0278641920010057

3. Morozova E.A. Solvability of a boundary value problem for a system of ordinary differential equations. Bulletin of Perm University. Mathematics. Mechanics. Computer science. 2010;3(3):46‒50. (In Russ.)

4. Volosova N.K. et al. Modified Bubnov-Galerkin method for solving boundary value problems with a linear ordinary differential equation. Computational Mathematics and Information Technologies. 2024;8(3):23‒33. (In Russ.) https://doi.org/10.23947/2587-8999-2024-8-3-23-33

5. Volosova N.K. et al. Increasing the accuracy of solving boundary value problems with a linear ordinary differential equation using the Bubnov-Galerkin method. Computational Mathematics and Information Technologies. 2024;8(4): 7‒18. (In Russ.) https://doi.org/10.23947/2587-8999-2024-8-4-7-18

6. Bahvalov N.S., Lapin A.V., Chizhonkov E.V. Numerical methods in problems and exercises. M.: BINOM. Knowledge laboratory; 2010. 240 p. (in Russ.)

7. Petrov A.G. High-precision numerical schemes for solving plane boundary value problems for a polyharmonic equation and their application to problems of hydrodynamics. Applied Mathematics and Mechanics. 2023;87(3):343–368 (In Russ.) https://doi.org/10.31857/S0032823523030128

8. SukhinovA.I. et al. A two-dimensional hydrodynamic model of coastal systems, taking into account evaporation. Computation Mathematics and Information Technologies. 2023;7(4):9‒21. (In Russ.) https://doi.org/10.23947/2587-8999-2023-7-4-9-21

9. Galaburdin A.V. The application of neural networks to solve the problems of a conductor for regions of regional form. Computational Mathematics and Information Technologies. 2024;8(2):68–79. (In Russ.) https://doi.org/10.23947/2587-8999-2023-7-4-9-21

10. Sidoryakina V.V., Solomaha D.A. Symmetrized versions of the Seidel and upper relaxation methods for solving two- dimensional difference problems of elliptic. Computational Mathematics and Information Technologies. 2023;7(3):12–19. (In Russ) https://doi.org/10.23947/2587-8999-2023-7-3-12-19

11. Alekseev V.M., Galeev E.M., Tikhomirov V.M. Collection of optimization problems: Theory. Examples. Problems. Moscow: Fizmatlit; 2008. 256 p. (In Russ.)

12. Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis. Textbook. Moscow: Fizatlit, 2012. 572 p. (In Russ.)