On the Solution of Some Simple Fractional Differential Equations

The dlfferintegration or fractional derivative of complex order , is a generalization of the ordinary concept of derivative of order n, from positive integer v=n to complex values of u, including also, for u-n a negative integer, the ordinary n-th primitive. Substituting, in an ordinary differential equation, derivatives of integer order by derivatives of non-integer order, leads to a fractional differential equation, which is generally'a integro-dlfferential equation. We present simple methods of solution of some classes of fractional differential equations, namely those with constant coefficients (standard I) and those with power type coefficients with exponents equal to the orders of differintegration (standard II). The fractional differential equations of standard I (II), both homogeneuus, and inhomogeneous with exponential (power-type) forcing, can be solved in the 'Liouville' ('Riemann') systems of differlntegration. The standard I (II) is linear with constant (non-constant) coefficients, and some results are also given for a class of non-linear fractional differential equations (standard III).

th present paper.
Two classe of fractional differential equations, which as mentioned in the introduction (I) can be solved by very simple methods akin to those in the theory of O.D.E.s are the linear equation with constant coefficients (standard I), and with power type coefficients with exponent equal to the order of differintegration (standard II). Concerning the former (Part I), we start by considering a simple case, to address the following issue: (3) given that an O.D.E. of order N has N linearly independent particular integrals (L.I.P.I), how many L.I.P.I. has a F.D.E.?; We proceed to solve the general linear F.D.E. of Standard I, in the homogeneous case (4), and in the inhomogeneous case with exponential forcing (5), concluding with an example (8) of a forced oscillator with memory-type damping. Concerning the F.D.E.
of standard II we must use (Part II), the 'Riemann" system of differintegration, instead of the 'Liouville' system used (Part I) for standard I, the two systems being incompatible (Lavoie and Tremblay and Osler [3] Campos [25]). The solution of the standard II is as straightforward as of standard I, both for the simplest (8) and most general (9) homogenous F.D.E., and for the inhomogeneous F.D.E. with power-type forcing ( I0); although all of the preceding F.D.E.s are linear, similar simple methods can be used to solve (II) a restricted class (standard III) of non-linear viz, for v -n a negative integer, (2.2) is the n-times repeated integral from to z; (il) for complex order other than a negative integer (Nishlmoto [5]): Re(v) -I,-2,...: DVF/Dz V {r(l + v)/2i} !z+)exp(i arg z)(-z) -V-I F() d, ( We may seek a solution of (3.1) in the form (3.3a): where the constant a satisfies (3.3b); the latter is an algebraic equation with roots a k given by: , unless the coefficients C k 0 vanish, beyond a certain order k > m; the latter is thus a necessary condition for convergence. If x < O, then exp{ek2Wx} < for all k, and a sufficient condition for the convergence of the series (3.7), is that the series of coefficients E C k converges for k= -m,...,+.   In the case where a k is a root of multiplicity s of the characteristic pseudopolynomial (4.2), and Q(a k) * O; P(a) (a ak )s Q(a),  has characteristic pseudopolynomial (4.12), Hence, a particular integral of (5.11) is (i) given by ( to the particular integral (5.13) or (5.15). 6. OSCILLATION AND RESONANCE WITH NON-INTEGRAL DAMPING EXPONENT.
As a demonstration of the simplicity of the present method of approach to fractional differential equations, we reconsider a free or forced harmonic oscillator with memory-type damping, which was solved elsewhere (Duarte [22]), using Fourier analysis to study the free motion only. When a particle moves along a trajectory x(t), defined by a coordinate x as a function of time t, the viscous damping force is taken as proportional and opposite to the velocity 0 < r < 2; s(t) -V Drx/Dtr, which, on account of (5.23), is a Volterra [21] [25]) for all complex ,v other than B a negative integer. Since the actual calculations are very simple, and somewhat analogous to those in Part I, they are mentioned briefly in the following sections.
Thus the exponent a of (9.2a) is a root of (9.2b), which, for v n a positive integer, is a polynomial equation of degree n, viz.
SOLUTIONS OF SOME SIMPLE FRACTIONAL DIFFERENTIAL EQUATIONS 491 r(! + a)/r(t + a-n) -a(a-I)...(an + I), (9.3) The n roots a k with k 1,...,n specify the n particular integrals of (9.1), which is, in this case, the original Euler equation. If v is non-lntegral, the values of a are roots of the equation (9.2b) involving Gamma functions.
Similarly, the general homogeneous fractional differential equation (7.1) of Standard II, has particular integrals of the form (9.2a), where a is a root of the dlscrlminant equation . z . C k k=a =0 where the C k are constants.
In the case s k when all roots are simple, (9.8) o simplifies to (9.5) with C k =-C k.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos). We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable:

Manuscript Due
December 1, 2008 First Round of Reviews March 1, 2009