Solution of the First Boundary-Value Problem for a System of Autonomous Second-Order Linear Partial Differential Equations of Parabolic Type with a Single Delay

Josef Diblík,Zdeněk Svoboda,Denis Khusainov,Oleksandra Kukharenko

doi:10.1155/2012/219040

Abstract

The first boundary-value problem for an autonomous second-order system of linear partial differential equations of parabolic type with a single delay is considered. Assuming that a decomposition of the given system into a system of independent scalar second-order linear partial differential equations of parabolic type with a single delay is possible, an analytical solution to the problem is given in the form of formal series and the character of their convergence is discussed. A delayed exponential function is used in order to analytically solve auxiliary initial problems (arising when Fourier method is applied) for ordinary linear differential equations of the first order with a single delay.

Highlights

In this paper, we deal with an autonomous second-order system of linear partial differential equations of the parabolic type with a single delay ∂u x, t ∂t a11∂2u x, t − ∂x2 τ a12∂2v x, t − ∂x2 τ b11∂2 u x, ∂x2 t b12∂2 v x, ∂x2 t a21 a22 b21 b22, where the matrices of coefficients A a11 a21
A delayed exponential function defined in part 2 together with the description of its main properties is used in order to analytically solve auxiliary initial problems arising when Fourier method is applied for ordinary linear differential equations of the first-order with a single delay
Due to delayed arguments, complications arise in solving analytically auxiliary initial Cauchy problems for first-order linear differential equations with a single delay. We overcome this circumstance by using a special function called a delayed exponential, which is a particular case of the delayed matrix exponential as defined, e.g., in 7–10

Summary

Introduction

We deal with an autonomous second-order system of linear partial differential equations of the parabolic type with a single delay. Assuming that a decomposition of system 1.1 into a system of independent scalar second-order linear partial differential equations of parabolic type with a single delay is possible, an analytical solution to the problem 1.9 – 1.14 is given in the form of formal series in part 3. A delayed exponential function defined in part 2 together with the description of its main properties is used in order to analytically solve auxiliary initial problems arising when Fourier method is applied for ordinary linear differential equations of the first-order with a single delay. Due to delayed arguments, complications arise in solving analytically auxiliary initial Cauchy problems for first-order linear differential equations with a single delay We overcome this circumstance by using a special function called a delayed exponential, which is a particular case of the delayed matrix exponential as defined, e.g., in 7–10.

First-Order Homogeneous Linear Differential Equations with a Single Delay

First-Order Nonhomogeneous Linear Differential Equations with a Single Delay

Partial Differential Systems with Delay

Formal Solution of the Boundary Value Problem

Convergence of Formal Series