An Integral Method for Solving Differential Equations in the Modeling of Objects with Distributed Parameters

• Видавець: Кам'янець-Подільський національний університет імені Івана Огієнка
• Дата: 2018
• Автори: Іванюк, Віталій Анатолійович, Костьян, Наталія Леонідівна
• Формат: Стаття
• Мова: Ukrainian
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2018
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/159386

The article deals with the method of obtaining one-dimensional integral dynamic models of systems with distributed parameters in the integral form on the basis of applying the differential equations with fractional derivatives obtained by transformations of irrational transfer functions. Such transfer functions occur in the description of the problems of heat conductivity, diffusion, oscillatory processes and other problems, which are described by differential equations with partial derivatives of a parabolic and hyperbolic type. Transfer functions that describe the semi-integral or semi-inertial links in which the Laplace variable is under the root may be the typical examples. The received Cauchy problem for an ordinary differential equation with fractional derivatives is given in the form of the Volterra integral equation of the second kind of convolution type. The applying of this approach is considered in solving various differential equations: the ordinary differential equation of the order 0 < a <1, the ordinary differential equation of the fractional order a > 1, the differential equation of the n-th order with fractional derivatives. Solving of the last equation is based on the compilation of the characteristic equation, which leads to solving of ordinary differential equations of order a. An important example is the researched approach to solving a system of differential equations with fractional derivatives. Due to the transition to equivalent integral equations, the researched problem can be solved by various approximate methods, which are based on quadrature methods with applying the approximation models of polynomial form, in particular, Chebyshev polynomials. The suggested approach makes it possible to construct one-dimensional integral dynamic models of systems of interconnected objects with distributed parameters that can provide high accuracy and stability of solving.