_{1}

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

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 [

Saigo [

which implies the Equation (1) for

by characteristic coordinates. The boundary conditions used for the solution of Equation (2) are

The solution of ED-Equation (2), due to Saigo [

where x and y are restricted in the domain

Srivastava and Saigo [

where

Kilbas et al. [

For

if

where

Let

Then we denote by

Carlson [

Standard Simplex: Denote the standard simplex in

Beta Function of k-variables: Let

Dirichlet Measure: The complex measure

for

Dirichlet Average: Let

where

Particularly,when

where

If we consider the continuous function

and for

where

Fractional calculus is the generalization of ordinary n-times iterated integrals and

Let

Proposition 1: Let

Proposition 2: Let

Generalization of fractional integral operators is due to Saigo [

Proposition 3: Let

Proposition 4: Let

By setting

Theorem 1: Let

where

Proof: Using Equation (16), we write

Using the transformation

which upon using (17), can be expressed as

which, for

where

Owing to the proposition 1 to proposition 4 we conclude the proof of theorem 1.

Corollary 1: If

Proof: Invoking the proposition 1 and using the result (32), we find that the fractional integral representation of single Dirichlet average of

Theorem 2: Let

Proof: Using Equation (5), Theorem 1 and the Corollary 1, theorem 2 can be proved easily under the proposition 4.

The author is indebted to P. K. Banerji, Jodhpur, India for fruitful discussions during the preparation of this paper. Financial support under Technical Education Quality Improvement Programme (TEQIP)-II, a programme of Ministry of Human Resource Development, Government of India is highly acknowledged. Author is also thankful to worthy refree for his/her valuable suggestions upon improvement.

D. N. Vyas, (2016) Dirichlet Averages, Fractional Integral Operators and Solution of Euler-Darboux Equation on Hölder Spaces. Applied Mathematics,07,1498-1503. doi: 10.4236/am.2016.714129