Boundary value conditions for linear differential equations with power degenerations

Aigerim Kalybay

doi:10.1186/s13661-020-01412-6

Abstract

On the interval [0,1] we consider the nth order linear differential equation, the coefficient of the highest derivative of which is equivalent to the power function t^{mu } when trightarrow 0. The main aim of the paper is to pose “generalized” Cauchy conditions for the given equation at the point of singularity t=0, which would be correct for any mu >0.

Highlights

Let us consider the following nth order linear differential equation:n ai(t)y(i)(t) = f (t), t ∈ [0, 1], (1)i=0 where the coefficients ai(·), i = 0, 1, . . . , n, and the right-hand side f (·) are continuous functions on [0, 1]
Conditions (4) are the required “generalized” Cauchy conditions, and in the paper we prove that problem (1) and (4) with (3) has a unique solution
Let us note that the operator Dnα gives the basis for a space Wpn,α = Wpn,α(I) of functions y : I → R with the finite semi-norm y Wpn,α(I) = Dnα y p, where 1 < p < ∞, I = (0, 1) or I = (1, ∞)

Summary

Introduction

Using this set of numbers α for y(t) ∈ Cn(0, 1], we construct the following operations: D0αy(t) = tα0 y(t), Diα y(t) tαi d dt tαi–1 d dt tα1 d dt tα0 y(t), i = 1, 2, . At the point of singularity t = 0 of equation (1) we pose the boundary conditions Let us note that the operator Dnα gives the basis for a space Wpn,α = Wpn,α(I) of functions y : I → R with the finite semi-norm y Wpn,α(I) = Dnα y p, where 1 < p < ∞, I = (0, 1) or I = (1, ∞).

Objectives

Results

Conclusion