SOLVING NONLINEAR BOUNDARY VALUE PROBLEMS BY THE APPROXIMATE METHOD - Uteuova, Н.; Shiyapov, К.; Bekbauova, А.; Sharipova, Б. (Bulletin of Abai KazNPU. Series of Physical and Mathematical sciences, 2023)
The solution of nonlinear boundary value problems is usually constructed using various iterative methods based on well-known methods of successive approximations. The choice of the method is ambiguous, it depends on the nature of the boundary value problem itself, the type of differential equations included in it, the degree of nonlinearity, and the capabilities of computers used to solve it. In this regard, various approaches to solving nonlinear boundary value problems have been developed. Among them, projection and variational methods such as Bubnov and Ritz methods, the method of finite differences and the method of finite elements are known. In this paper is proposed a new iterative method for solving the abstract nonlinear equation A(u)=f, which converges for any value of the initial approximation, where the operator A(u) is not necessarily contractive. Thus, an approximate variational method for solving nonlinear boundary value problems has been developed. The convergence of the method is proved. The theoretical presentation of the effectiveness of the method is confirmed by numerical experiments.