Investigation of the solutions of the Cauchy problem and boundary-value problems for the ordinary differential equations with continuously changing order of the derivative

TL;DR

This paper explores initial and boundary value problems for linear ordinary differential equations with continuously changing order of derivatives, providing analytic solutions and addressing a less studied area of modern mathematics.

Contribution

It introduces an analytic method for solving differential equations with continuously changing derivative order, a novel approach in this emerging field.

Findings

Derived explicit solutions for these differential equations.

Identified invariant functions that remain unchanged under derivatives of any positive order.

Contributed to the theoretical understanding of differential equations with variable derivative order.

Abstract

As is known, the problems for the differential equations with continuously changing order of the derivatives are not considered completely. In this paper we consider the initial and boundary value problems for this type of linear ordinary differential equations with constant coefficients and obtain analytic representations of solutions of these problems. It should be noted that this area is one of the less studied fields of modern mathematics and there are not effective methods for the study of problems for such differential equations, just as we study the problem for partial differential equations, with both with additive and a multiplicative derivatives. The method used here is based on an invariant for the above mentioned derivative, i.e. on the functions that do not change for the derivative of any real positive order.