@tomleslie Thank you. i am satisfied with the answer you gave, however, i am able to plot the figure you did plot, my problem comes when i have to plot eq1. so i can assign r=2,sigma=0.5, alpha=0.5 as well and you are right i have to plot u(x,t). how can you go about this one?

@J F Ogilvie are there any other methods or softwares that can help in solving this kind of problem? If the equation did not have a gamma function and t^(1-alpha) then i would apply laplace transform. So am not quite sure which way to take unless the problem is not solvable

@Thomas Richard indeed for alpha is equal to 1 the solution is trivial, i tried substituting values of alpha which i considered as fractions and i got an error "Error, invalid input: diff expects 2 or more arguments, but received 1
NULL;"

so how do we go about solving fractional orders?

@Thomas Richard Does this mean i can use dsolve for any order of alpha in the interval [0 1)? is a way to find a solution without taking a  specific real number within the interval?

what i mean is , is irt possible to find a general solution of order alpha where alpha is in [0 1)?

 thank you