The Use Of Fractional B Splines Wavelets In Fractional Differential Equations

In this paper，we discuss the existence and uniqueness of the solution of thenonhomogeneous linear differential equations of arbitrary positive real order by using thefractional B-Splines wavelets and the Mittag-Leffler function. The scheme of solving thefractional differential equations and the explicit expression of the solution is given in thispaper. We also show the asymptotic solution of the differential equations of fractionalorder and corresponding truncated error in theory.In section1，we introduce the background of the fractional calculus and fractionaldifferential equations and give the reasons why we choose the fractional B splineswavelet as a basis function to solve the fractional differential equations in this paper.In section2, we recall the definitions of fractional derivative and integral and relatedproperties used in the paper, give the representation of Mittag-Leffler function andgeneralized Mittag-Leffler function, and then introduce the fractional B-splines and somerelated properties of wavelet.In Section3, by applying the technique of the Laplace Transforms, and consideringthe proprieties of the generalized Mittag-Leffler function, we prove the Lemma of thedifferential equations of arbitrary positive real order, which make sure the solutionbelongs to the spaceL2; and because the solution can be expressed as the form ofwavelet series and the basis function is the orthogonal fractional B-splines wavelet whichyields the Riesz basis for the spaceL2, then we prove the uniqueness of the coefficientsof representation of the solution, which gain the uniqueness of the solution of thefractional differential equations and validate the representation of solution. Thus, we havefinished the proof of the theorem.In section4, the asymptotic solution of the differential equations of fractional orderα∈Q+and correspondence truncated error will be discussed.In section5, we extend the fractional differential equations with its initial valuesinto two aspects: in the first aspect, we replace the case of Riemann-Liouville fractionaldifferential operator by Caputo sense; and in the second aspect, we consider the fractionaldifferential equations with its initial values which are non-zeros, then give the proof ofthe existence and uniqueness of the solution of the nonhomogeneous linear fractionalordinary differential equations and the explicit expression of the solution.