Mariusz Ciesielski

A Numerical Method for Solution of Ordinary Differential Equations of Fractional Order

In this paper we propose an algorithm for the numerical solution of arbitrary differential equations of fractional order. The algorithm is obtained by using the following decomposition of the differential equation into a system of differential equation of integer order connected with inverse forms of Abel-integral equations. The algorithm is used for solution of the linear and non-linear equations.

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 treatment of an initial-boundary value problem for fractional partial differential equations

This paper deals with numerical solutions to a partial differential equation of fractional order. Generally this type of equation describes a transition from anomalous diffusion to transport processes. From a phenomenological point of view, the equation includes at least two fractional derivatives: spatial and temporal. In this paper we proposed a new numerical scheme for the spatial derivative, the so-called Riesz-Feller operator. Moreover, using the finite difference method, we show how to employ this scheme in the numerical solution of fractional partial differential equations. In other words, we considered an initial-boundary value problem in one-dimensional space. In the final part of this paper some numerical results and plots of simulations are shown as examples.

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 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.

Application of Control Volume Method Using the Voronoi Tessellation in Numerical Modelling of Solidification Process

The paper presents the method to analyse the thermal processes occurring in the cast composite solidification. The cast is formed by a bundle of parallel fibres randomly immersed in a host metal matrix. The heat is transferred from the metal matrix and absorbed by the fibres. The objective of this paper is to evaluate the volumetric fraction of the fibres for which the solidification of the metal matrix occurs only due to the presence of fibres playing a role of internal chills. Our method is to compute Voronoi diagrams with Voronoi regions representing the geometric location of the fibres in the metal matrix and to use these regions as control volumes within a variant of the Control Volume Method.

Numerical Model of Thermal Processes in Domain of Thin Film Subjected to a Cyclic External Heat Flux

Thermal processes in a thin metal film subjected to a cyclic external short-pulse heating are considered (axially-symmetrical 3D problem). The heat transfer proceeding in domain analyzed is here described by the dual phase lag model (DPLM). According to the newest opinions the DPLM constitutes a very good description of real heat transfer processes proceeding in the micro-scale domains subjected to the strong external heat flux. The base of DPLM formulation is a generalized form of Fourier law (GFL) in which two times τq, τT appear (the relaxation time and thermalization one, respectively). The acceptation of GFL leads to DPLM equation [1, 2]. Thermal processes proceeding in a thin metal film subjected to a cyclic external short-pulse heating are considered (axially symmetrical 3D problem). In the paper the thermal interactions between cyclic external heat source qb and cylindrical micro-domain are analyzed. The external heat source is the function dependent on spatial co-ordinates and time. On the remaining parts of the boundary the no-flux conditions are assumed. It should be pointed out that the DPL model requires the adequate transformation of boundary conditions which appear in the typical macro heat conduction models. The initial conditions are also known (initial temperature of domain and initial heating rate). Numerical model of the process discussed bases on a certain variant of finite differences method and in the final part of the paper the examples of computations are shown.

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.