Modification of the Parametrization Method for Solving a Boundary Value Problem for Loaded Differential Epcag

Abstract

The functional differential equation plays important role in mathematical modeling of biological problems. In the present research work, we investigate a boundary value problem (BVP) for a functional differential equation. This equation includes loaded terms and a term with generalized piecewise constant argument. We apply a modified version of the Dzhumabaev parameterization method. The method's goal is to lead the original problem into an equivalent multi-point BVP for ordinary differential equations with parameters, which is composed of a problem with initial and additional conditions. The multi-point BVP is leaded to a system of linear algebraic equations in parameters, which are introduced as the values of the desired solution at the dividing points. The found parameters are plugged into auxiliary Cauchy problems on the partition subintervals, whose solutions are the restrictions of the solution to the original problem. The obtained results are verified by a numerical example. Numerical analysis showed high efficiency of the constructed modified version of the Dzhumabaev parameterization method.

A novel numerical implementation for solving problem for loaded DEPCAG

“…, is called a solution to problem ( 9)-( 18) if the functions 𝑤𝑤𝑤𝑤 𝑟𝑟𝑟𝑟 * (𝑡𝑡𝑡𝑡), 𝑟𝑟𝑟𝑟 = 1,4 ���� , are continuously differentiable on [𝑡𝑡𝑡𝑡 𝑟𝑟𝑟𝑟−1 , 𝑡𝑡𝑡𝑡 𝑟𝑟𝑟𝑟 ) and satisfy equations ( 9), ( 11), ( 13), (15) with respective initial conditions and additional conditions ( 17), (18) with 𝜇𝜇𝜇𝜇 𝑗𝑗𝑗𝑗 = 𝜇𝜇𝜇𝜇 𝑗𝑗𝑗𝑗 * , 𝑗𝑗𝑗𝑗 = 1,4 ���� .…”

“…and 𝜇𝜇𝜇𝜇 * = (𝑥𝑥𝑥𝑥 * (𝑡𝑡𝑡𝑡 0 ), 𝑥𝑥𝑥𝑥 * (𝑡𝑡𝑡𝑡 1 ), 𝑥𝑥𝑥𝑥 * (𝑡𝑡𝑡𝑡 2 ), 𝑥𝑥𝑥𝑥 * (𝑡𝑡𝑡𝑡 3 )), is a solution of problem ( 9)- (15). Conversely, if a pair 3)-( 6) and solve them as ordinary differential equations subject to the initial conditions 𝑥𝑥𝑥𝑥 * (𝑡𝑡𝑡𝑡 𝑟𝑟𝑟𝑟−1 ) = 𝜇𝜇𝜇𝜇 𝑟𝑟𝑟𝑟 * .…”

“…Differential equations with piecewise constant argument of generalized type, introduced by M. Akhmet [10], arise in modeling of diverse phenomena and are widely used in applications such as neural networks, hybrid systems, dynamic systems with discontinuities, etc. The theory of such equations has been extensively developed; see, for instance, [10][11][12][13][14][15]. However, there still remain open questions regarding boundary-value problems for such equations on a finite interval.…”