ASYMPTOTICS OF THE SOLUTION OF THE BOUNDARY-VALUE PROBLEM WITH THE INTERNAL INITIAL JUMP
Nurgabyl Duisebek Nurgabyluly, Kanatkyzy Zere
Zhetysu State University named after I. Zhansugurov
Abstract. In the paper the authors consider the boundary-value problem for the second-order linear differential equation with the small parameter at the derivative. An algorithm for constructing the asymptotic expansion of the solution of the boundary-value problem is described. Uniform asymptotic approximation of the solution of the singularly perturbed boundary-value problem is constructed accurate within random order as the small parameter tends to zero. On the basis of asymptotics of the solution of the initial problem, existence and uniqueness of the solution of the boundary-value problem are proved. A degenerate problem is formulated. In case of sufficiently small the authors find difference estimate between solutions of perturbed and unperturbed problems. The phenomenon of the internal initial jump is investigated.
Key words and phrases: дифференциальные уравнения, краевые задачи, малый параметр, начальный скачок, асимптотическое разложение, differential equations, boundary-value problems, small parameter, initial jump, asymptotic expansion
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