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differential equations with fractional derivatives are used for modeling non - markovian processes in various areas of science such as physics , biology and economics [ 1 - 5 ] . the methods for analytical solution of ordinary and partial fractional differential equations can be applied to only special types of equations and initial conditions . in practical applications the solutions of the fractional differential equations are determined by numerical methods . the development of efficient methods for numerical solution of fractional differential equations is an active research field . an important approximation for the caputo fractional derivative is the @xmath0 approximation . it has been successfully used for numerical solution of ordinary and partial fractional differential equations . when the function @xmath1 is a smooth function , the accuracy of the @xmath0 approximation is @xmath2 . in previous work @xcite we determined the second order approximation by modifying the first three coefficients of the @xmath0 approximation with the value of the riemann zeta function at the point @xmath3 , and we computed the numerical solutions of the fractional relaxation and subdiffusion equations with sufficiently differentiable solutions . the solutions of fractional differential equations often have a singularity at the initial point . in this case the numerical solutions may converge to the exact solution but their accuracy may be lower than the expected accuracy from the numerical analysis . in this paper we propose a method for improving the accuracy of the numerical solutions of the fractional relaxation equation @xmath4 where @xmath5 , and the fractional subdiffusion equation @xmath6,t>0 , &   \\ u(x,0)=\sin x,\;u(0,t)=0,\;u(\pi , t)=0 . \end{array }          \right .\ ] ] when the solutions of the relaxation and subdiffusion equations are smooth functions the accuracy of the numerical solutions using the @xmath0 approximation are @xmath2 and @xmath7 and the numerical solutions using the modified @xmath0 approximation have accuracy @xmath8 and @xmath9 . the solutions of the fractional relaxation and subdiffusion equations and have differentiable singularities at the initial point . the accuracy of the numerical solutions and of the fractional relaxation equation is @xmath10 and the numerical solutions and of the fractional subdiffusion equation have accuracy @xmath11 . equations and are linear fractional differential equations . the form of the equations allows us to compute the miller - ross derivatives of the solutions at the initial point . in section 3 and section 4 we use the fractional taylor polynomials to improve the differentiability properties of the solutions and the accuracy of the numerical methods . the caputo derivative of order @xmath12 , where @xmath13 is defined as @xmath14 when the function @xmath15 is defined on the interval @xmath16 $ ] , the lower limit of the integral in the definition of caputo derivative is @xmath17 . the value of the fractional derivative @xmath18 depends on the values of the function on the whole interval of definition . one approach for discretizing the caputo derivative is to divide the interval to subintervals of small length and approximate the derivative of the function @xmath1 using spline interpolation or a lagrange polynomial @xcite . let @xmath19 be a small positive number . denote @xmath20 the @xmath0 approximation for caputo derivative is derived @xcite by approximating the derivative of the function @xmath1 on the interval @xmath21}$ ] with its value @xmath22 at the midpoint of the interval . @xmath23 where @xmath24 and @xmath25 when the function @xmath1 has a continuous second derivative , the accuracy of the @xmath0 approximation is @xmath26 ( @xcite ) . the modified @xmath0 approximation for the caputo derivative @xmath27 has coefficients @xmath28 for @xmath29 , @xmath30 where @xmath31 is the riemann zeta function . the modified @xmath0 approximation has accuracy @xmath32 .    by approximating the caputo derivative at the point @xmath33 with and we obtain the numerical solutions and of the fractional relaxation equation . in table 1 and table 2 we compute the maximum error and the order of the numerical solutions and for the following equations @xmath34 @xmath35 equations and have solutions @xmath36 and @xmath37 . the solution of equation has an unbounded second derivative at @xmath38 . while the numerical solutions of equation converge to the exact solution , their accuracy is smaller than @xmath2 . 99 a. cartea , d. del castillo - 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the accuracy of the numerical solution of a fractional differential equation depends on the differentiability class of the solution . 
 the derivatives of the solutions of fractional differential equations often have a singularity at the initial point , which may result in a lower accuracy of the numerical solutions . 
 we propose a method for improving the accuracy of the numerical solutions of the fractional relaxation and subdiffusion equations based on the fractional taylor polynomials of the solution at the initial point . + * ams subject classification : * 33f05 , 34a08 , 57r10 + * key words : * fractional differential equation , caputo derivative , numerical solution , fractional taylor polynomial .