Laplace transform for solving some families of fractional differential equations and its applications

Advances in Difference Equations, Sep 2018

In many recent works, many authors have demonstrated the usefulness of fractional calculus in the derivation of particular solutions of a significantly large number of linear ordinary and partial differential equations of the second and higher orders. The main objective of the present paper is to show how this simple fractional calculus method to the solutions of some families of fractional differential equations would lead naturally to several interesting consequences, which include (for example) a generalization of the classical Frobenius method. The methodology presented here is based chiefly upon some general theorems on (explicit) particular solutions of some families of fractional differential equations with the Laplace transform and the expansion coefficients of binomial series. MSC:26A33, 33C10, 34A05.

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Laplace transform for solving some families of fractional differential equations and its applications

Shy-Der Lin Chia-Hung Lu In many recent works, many authors have demonstrated the usefulness of fractional calculus in the derivation of particular solutions of a significantly large number of linear ordinary and partial differential equations of the second and higher orders. The main objective of the present paper is to show how this simple fractional calculus method to the solutions of some families of fractional differential equations would lead naturally to several interesting consequences, which include (for example) a generalization of the classical Frobenius method. The methodology presented here is based chiefly upon some general theorems on (explicit) particular solutions of some families of fractional differential equations with the Laplace transform and the expansion coefficients of binomial series. MSC: 26A33; 33C10; 34A05 1 Introduction, definitions and preliminaries In the past two decades, the widely investigated subject of fractional calculus has remarkably gained importance and popularity due to its demonstrated applications in numerous - special functions, etc. (see, for example, []). ential equations. where the Euler gamma function () is defined by (z) = tzet dt L f (t) (s) = F(s) = The Mittag-Leffler function (cf. [, ]) is defined by z, , C, R() > . The simplest Wright function (cf. [, ]) is defined by (ai, i),p z := p q(z) = p q (bj, j),q where z, ai, bj C, i, j R, i = , , . . . , p and j = , , . . . , q. The Riemann-Liouville fractional derivatives Da+y and Dby of order C ( () ) are defined by n = n = respectively, where [ ()] means the integral part of (). The Pochhammer symbol (or the shifted factorial, since ()n = n! for n N = {, , , . . .}) (cf. []) given by The binomial coefficients are defined by n = !(! n)! = = , r= r! (ai, i),p t L p q (bj, j),q (, ), (ai, i),p (bj, j),q s est Df (t) dt t f (n)( ) (n ) (t )n+ d dt es(u+ )un du d esuun du d es f (n)( ) d = snL f (n)(t) (s) = sn snL f (t) snf () snf () f (n)() The interchange of the order of integration in the above derivation can be justified by applying Fubinis theorem. 2 Solutions of the fractional differential equations Throughout this section, we let y(t) be such that for some value of the parameter s, the Laplace transform L[y] converges. y (t) + ay()(t) + by(t) = y(t) = c (r + k + )(at)r [( )r + k + ]r! (b)ktk+ (b)ktk+ k= r= (b)ktk+ (r + k + )(at)r [( )r + k + ]r! (r + k + )(at)r [( )r + k + ]r! (r + k + )(at)r [( )r + k + ]r! . Equation (.) yields L[y] = cs + c + acs + acs s + as + b = c Thus, from Equation (.), we derive the following solution by the inverse Laplace transform to Equation (.): (b)k sk y(t) = c = c (k + r)!(a)r [( )r + k + ] (k + r)!(a)r [( )r + k + ] (k + r)!(a)r [( )r + k + ] (k + r)!(a)r [( )r + k + ] (r + k + )(at )r [( )r + k + ]r! (b)k tk+ (b)k tk+ k= r= (b)k tk+ (r + k + )(at )r [( )r + k + ]r! (r + k + )(at )r [( )r + k + ]r! (r + k + )(at )r [( )r + k + ]r! . Example . The fractional differential equation of a generalized viscoelastic free damping oscillation (cf. []) y (t) + ay( )(t) + by(t) = y(t) = c (r + k + )(at )r [ r + k + ]r! (b)k tk+ (b)k tk+ (b)k tk+ (r + k + )(at )r [ r + k + ]r! (r + k + )(at )r [ r + k + ]r! (r + k + )(at )r [( )r + k + ]r! . y(t) = c + + ()k tk+ ()k tk+ ()k tk+ (r + k + )(t )r [ r + k + ]r! (r + k + )(t )r [ r + k + ]r! (r + k + )(t )r r= [ r + k + ]r! (r + k + )(t )r r= [ r + k + ]r! y()(t) + ay (t) + by(t) = y(t) = c (r + k + )(a)rt()r+k [( )r + k + ]r! (r + k + )(a)rt()r+k+ [( )r + k + ]r! (r + k + )(a)rt()r+k+ [( )r + k + ]r! That is, Equation (.) yields L[y] = = c Thus, from Equation (.), we derive the following solution by the inverse Laplace transform to Equation (.): y(t) = c (r + k + )(a)r [( )r + k + ] (r + k + )(a)r [( )r + k + ] (r + k + )(a)r [( )r + k + ] This solution can be expressed by the Wright function as y(t) = c y( )(t) y (t) y(t) = has a solution y(t) = c k k k (r + k + )t r + k+ (r + k + )t r + k+ y(t) = c s L[y] cs bL[y] = , we have L[y] = y(t) = c = bs = cs y(t) = c y(t) = cE, w t Competing interests The authors declare that they have no competing interests.


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Shy-Der Lin, Chia-Hung Lu. Laplace transform for solving some families of fractional differential equations and its applications, Advances in Difference Equations, 2013, 137, DOI: 10.1186/1687-1847-2013-137