Dirichlet Averages , Fractional Integral Operators and Solution of Euler-Darboux Equation on Hölder Spaces

In the present paper, we discuss the solution of Euler-Darboux equation in terms of Dirichlet averages of boundary conditions on Hölder space and weighted Hölder spaces of continuous functions using Riemann-Liouville fractional integral operators. Moreover, the results are interpreted in alternative form.


Introduction
The subject of Dirichlet averages has received momentum in the last decade of 20th century with reference to the solution of certain partial differential equations.Not much work has been registered in this area of Applied Mathematics except some papers devoted to evaluation of Dirichlet averages of elementary functions as well as higher treanscendental functions interpreting the results in more general special functions.The present paper is ventured to give the interpretation of solution of a typical partial differential equation and prove its inclusion properties with respect to Hölder spaces.The Euler-Darboux equation (ED-equation) is a certain kind of degenerate hyperbolic partial differential equation of the type (see Nahušev [1]), ( ) Saigo [2]- [4] considered and studied the ED-equation given by The solution of ED-Equation ( 2), due to Saigo [22], is given by where x and y are restricted in the domain Srivastava and Saigo [5] evaluated the results on multiplication of fractional integral operators and the solution of ED-equation.Deora and Banerji [6] represented the solution of Equation ( 2) in terms of Dirichlet averages of boundary condition functions given in (3) as follows or simply H λ as well as on weighted Hölder Space of continuous functions.In the present paper we discuss the Dirichlet averages on Hölder Space via right-sided Riemann-Liouville fractional integral operators and prove the solution of Equation (2) to be justified on such spaces.In what follows are the preliminaries and definitions related to fractional integral operators, Dirichlet averages, and Hölder spaces of continuous functions.

Hölder Spaces
, i.e., ( ) f x is m-times differentiable function and its m-th derivative is continuous and satisfies the inequality where 0 be the space of Hölder continuous functions and Then we denote by [ ] ( ) , where ( ) 0 x ρ ≥ .
Dirichlet Measure: The complex measure b µ , defind on E by ( ) ( ) ( ) for ( ) , is called the Dirichlet measure.Particularly, for , we write by using (3), the following: Dirichlet Average: and .u z denotes a convex linear combination of 1 , , k z z  .Then the Dirichlet average of a holomorphic function f is defined by (See Carlson [22]) where  denotes the parameters.The convex combination is given by Particularly,when 2 k = , the Dirichlet average, so extracted out of (5), is called the single Dirichlet average of f over the line segment from 0 to 1.It is expressed as where The equation analogous to (11), is expressed as

Fractional Integral Operators
Fractional calculus is the generalization of ordinary n-times iterated integrals and th n derivatives of continuous functions to that of any arbitrary order real or complex.The most commonly used definition of fractional integral operators of order α is due to Riemann-Liouville.A detailed account of fractional calculus is given in Samko et al. [23] and the applications of it are elaborated in Hilfer [24] and Podlubney [25].Vyas [26] interpreted the angle of collision occurring in the study of transport properties of Noble gases at low density configuration in terms of Fractional Integral Operators.

Theorem
where ( ) is the right-sided Riemann-Liouville fractional integral operator defined in (17).
where ( )( ) , 0 denotes the new fractional operator defined by Owing to the proposition 1 to proposition 4 we conclude the proof of theorem

)
Beta Function of k-variables: Let k C > denotes the th k cartesian product of open right half plane and n E E = is the standard simplex in n R .The beta function of k-variables can be expressed as

∈
, then without the loss of characteristics of such spaces, the Dirichlet average of ( ) Dirichlet average of ( ) z ϕ in two variables x and y in [ ] 0,1 .
Invoking the proposition 1 and using the result (32), we find that the fractional integral representation of single Dirichlet average of ( ) rise to the result (35).This justifies that the Dirichlet av- erages, so evaluated, belong to the Hölderian class.associatd with the ED-Equation (2) and if ( ) , u x y denotes the solution of ED-equation, given by(5), in terms of Using Equation (5), Theorem 1 and the Corollary 1, theorem 2 can be proved easily under the proposition 4.