  In this paper we obtain approximate bound state solutions of $N$-dimensional
time independent fractional Schr\"{o}dinger equation for generalised
pseudoharmonic potential which has the form
$V(r^{\alpha})=a_1r^{2\alpha}+\frac{a_2}{r^{2\alpha}}+a_3$. Here
$\alpha(0<\alpha<1)$ acts like a fractional parameter for the space variable
$r$. The entire study is composed with the Jumarie type derivative and the
elegance of Laplace transform. As a result we successfully able to express the
approximate bound state solution in terms of Mittag-Leffler function and
fractionally defined confluent hypergeometric function. Our study may be
treated as a generalization of all previous works carried out on this topic
when $\alpha=1$ and $N$ arbitrary. We provide numerical result of energy
eigenvalues and eigenfunctions for a typical diatomic molecule for different
$\alpha$ close to unity. Finally, we try to correlate our work with Cornell
potential model which corresponds to $\alpha=\frac{1}{2}$ with $a_3=0$ and
predict the approximate mass spectra of quarkonia.
