  In this paper we find fractional Riemann-Liouville derivatives for the
Takagi-Landsberg functions. Moreover, we introduce their generalizations called
weighted Takagi-Landsberg functions which have arbitrary bounded coefficients
in the expansion under Schauder basis. The class of the weighted Takagi-
Landsberg functions of order $H>0$ on $[0,1]$ coincides with the $H$-H\"{o}lder
continuous functions on $[0,1]$. Based on computed fractional integrals and
derivatives of the Haar and Schauder functions, we get a new series
representation of the fractional derivatives of a H\"{o}lder continuous
function. This result allows to get the new formula of a Riemann-Stieltjes
integral. The application of such series representation is the new method of
numerical solution of the Volterra and linear integral equations driven by a
H\"{o}lder continuous function.
