^{1}

^{2}

^{1}

^{2}

We analysed the initial/boundary value problem for the second-order homogeneous differential equation with constant coefficients in this paper. The second-order differential equation with respect to the fractional/generalised boundary conditions is studied. We presented particular solutions to the considered problem. Finally, a few illustrative examples are shown.

The second-order differential equations provide an important mathematical tool for modelling the phenomena occurring in dynamical systems. Examples of linear or nonlinear equations appear in almost all of the natural and engineering sciences and arise in many fields of physics.

Many scientists have studied various aspects of these problems, such as physical systems described by the Duffing equation [

Classical differential equations are defined by using the integer order derivatives. In recent years, the class of differential equations containing fractional derivatives (known as fractional differential equations) have become an important topic. There are two approaches to obtain these types of equations. The first one is to replace the integer order derivative in classical differential equations with a fractional derivative (see, e.g., [

The second approach is a generalisation of a method known in classical and quantum mechanics, where the differential equations are obtained from conservative Lagrangian or Hamiltonian functions. These equations are known in the literature as fractional Euler-Lagrange equations, and they contain both the left and right fractional derivatives. New mechanics models for nonconservative systems, in terms of fractional derivatives, were developed by Riewe in [

In contrast to the above-mentioned references, where the authors analysed the integer order differential equations with classical boundary/initial conditions or fractional differential equations with the Dirichlet or natural boundary conditions, in this paper we consider the second-order differential equation with the fractional/generalised boundary conditions.

In this paper, we solve the second-order differential equation

Let us consider two particular cases of (

Here, we study the following differential equation:

The fractional differentiation of general solution (

Let

Let

We use Taylor’s series expansions of sine and cosine functions [

Note that the infinite series included in formulas (

The second problem has the following form:

Let

Let

Here we apply Taylor’s series expansions of the hyperbolic sine and cosine functions [

In formulas (

The boundary conditions, written in the general form (

Equation (

Equation (

Equation (

One can note that the fractional boundary conditions

On the basis of the proposed method, to find the particular solutions to the considered equations (

The particular solutions of (

The particular solutions of (

The particular solutions of (

The initial/boundary value problem for the second-order homogeneous differential equations with constant coefficients has been considered. The general solutions to these equations are widely known and involve arbitrary constants. Our aim was to find the particular solutions to this problem which satisfy the generalised boundary conditions. Such boundary conditions complement the set of classic boundary conditions (including the Dirichlet, Neumann, and Robin types) by including the fractional ones.

The use of the fractional boundary conditions in the considered initial/boundary value problem required the fractional differentiation of the general solutions. We derived the formulas for the left- and right-sided Riemann-Liouville fractional derivatives of the sine, cosine, hyperbolic sine, and hyperbolic cosine functions that occur in the general solutions. On this basis, the integration constants in these solutions were determined analytically.

On the plots, one can observe that the obtained results for the fractional boundary conditions are located between the solutions to the considered problem with respect to the classical (integer order) boundary conditions. Such behaviour of the particular solutions gives new possibilities in physical phenomena modelling, like the harmonic oscillator modelling, among others. In the future, we plan to apply this approach to seek solutions to other types of the initial/boundary value problems, in particular to the four-order problems.

The authors declare that they have no conflicts of interest.

The research is supported by Faculty of Mechanical Engineering and Computer Science, Czestochowa University of Technology.