Tomasz Blaszczyk

Numerical solution of fractional oscillator equation

Abstract We focus on a numerical scheme applied for a fractional oscillator equation in a finite time interval. This type of equation includes a complex form of left- and right-sided fractional derivatives. Its analytical solution is represented by a series of left and right fractional integrals and therefore is difficult in practical calculations. Here we elaborated two numerical schemes being dependent on a fractional order of the equation. The results of numerical calculations are compared with analytical solutions. Then we illustrate convergence and stability of our schemes.

NUMERICAL SOLUTION OF FRACTIONAL STURM-LIOUVILLE EQUATION IN INTEGRAL FORM

In this paper a fractional differential equation of the Euler-Lagrange/Sturm-Liouville type is considered. The fractional equation with derivatives of order α ∈ (0, 1] in the finite time interval is transformed to the integral form. Next the numerical scheme is presented. In the final part of this paper examples of numerical solutions of this equation are shown. The convergence of the proposed method on the basis of numerical results is also discussed.

Fractional oscillator equation - Transformation into integral equation and numerical solution

In this paper we propose a numerical solution of a fractional oscillator equation (being a class of the fractional Euler-Lagrange equation). At first, we convert the fractional differential equation of order α 0 to an equivalent integral equation (including boundary conditions). Next, we present a numerical solution of the integral form of the considered equation for two cases: α ? ( 0 , 1 and α ? ( 1 , 2 . We show illustrative examples of solutions for checking the correctness of the proposed solution method of the equation. Also, we determine the convergence order of numerical schemas.

Fractional Euler–Bernoulli beams: Theory, numerical study and experimental validation

Abstract In this paper the classical Euler – Bernoulli beam (CEBB) theory is reformulated utilising fractional calculus. Such generalisation is called fractional Euler–Bernoulli beams (FEBB) and results in non-local spatial description. The parameters of the model are identified based on AFM experiments concerning bending rigidities of micro-beams made of the polymer SU-8. In experiments both force as well as deflection data were recorded revealing significant size effect with respect to outer dimensions of the specimens. Special attention is also focused on the proper numerical solution of obtained fractional differential equation.

Fractional oscillator equation: analytical solution and algorithm for its approximate computation

In this paper the fractional oscillator equation in a finite time interval is considered. The fractional equation with derivatives of order α ∈ ( 0 , 1 ] is transformed into its corresponding integral form, by using the symbolic calculus method, in which the binomial expansion of the inverse integral operator is used. A new fractional integral operator is introduced. A numerical algorithm to approximate the solution of the considered equation is proposed. In the final part of this paper examples of numerical solutions of this equation are given.

Exact and numerical solutions of the fractional Sturm–Liouville problem

Abstract In the paper, we discuss the regular fractional Sturm-Liouville problem in a bounded domain, subjected to the homogeneous mixed boundary conditions. The results on exact and numerical solutions are based on transformation of the differential fractional Sturm-Liouville problem into the integral one. First, we prove the existence of a purely discrete, countable spectrum and the orthogonal system of eigenfunctions by using the tools of Hilbert-Schmidt operators theory. Then, we construct a new variant of the numerical method which produces eigenvalues and approximate eigenfunctions. The convergence of the procedure is controlled by using the experimental rate of convergence approach and the orthogonality of eigenfunctions is preserved at each step of approximation. In the final part, the illustrative examples of calculations and estimation of the experimental rate of convergence are presented.

The fractional SturmLiouville problemNumerical approximation and application in fractional diffusion

The numerical method of solving the fractional eigenvalue problem is derived in the case when the fractional SturmLiouville equation is subjected to the mixed boundary conditions. This non-integer order differential equation is discretized to the scheme with the symmetric matrix representing the action of the numerically expressed composition of the left and the right Caputo derivative. The numerical eigenvalues are thus real, and the eigenvectors associated to distinct eigenvalues are orthogonal in the respective finite-dimensional Hilbert space. The advantage of the proposed method is the formulation which allows us to construct the approximate eigenfunctions which form an orthonormal function system in the infinite-dimensional weighted Lebesgue integrable function space. The developed numerical method of calculation of the eigenvalues and eigenfunctions is then applied in construction of the approximate solution to the 1D space-time fractional diffusion problem in a bounded domain. A new numerical method for solving a fractional eigenvalue problem has been proposed.The approximate eigenfunctions are orthogonal at each step of the proposed method.The method is applied in the approximate solution to fractional diffusion problem.

An Exact Solution of the Second-Order Differential Equation with the Fractional/Generalised Boundary Conditions

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.