fractional means non-locality be it in space or time. So how can it be local fractional calculus? Anyways follow Mainardi for applications and Podlubny for maths..

THEORY AND APPLICATIONS

OF FRACTIONAL DIFFERENTIAL

EQUATIONS

ANATOLYA. KILBAS, HARI M. SRIVASTAVA,JUAN J. TRUJILLO

ELSEVIER, 20016

This is a reasonably good book.

The above answers related to the traditonal nonlocal and heritary Fractional Calculus. Local fractional Calculus is a different type which is proposed in 1996 by Kolwankar and Gangal. The current materials on LFD are mainly in research papers. Prof. Xiaojun Yang published several books about LFD. Maybe you can refer to.

Thank to all for your valuable suggestion. 

This is good question with many possible answers.

In response to @Vikash Pandey's question about the local nature of fractional calculus, see

https://www.researchgate.net/publication/301257318_Introduction_to_local_fractional_derivative_and_integral_operators

and 

https://www.researchgate.net/publication/301257086_Local_fractional_Laplace_transform_and_applications

Also, see the many related papers available in Hari Mohan Srivastava's RG page at 

https://www.researchgate.net/profile/Hari_Srivastava

Chapter Introduction to local fractional derivative and integral operators

Chapter Local fractional Laplace transform and applications

a link to a book

@Otman Jaber: a link to a book....

What is the link?

hope these links could help.

http://theory.sinp.msu.ru/~tarasov/PDF/IJACM2015.pdf

http://fcag-egypt.com/members/prof_yang_xiao_jun.pdf

https://arxiv.org/pdf/1307.0739.pdf

Dear Vaijanath Chinchane,

I would like to suggest the following books on Local fractional Calculus:

* "Advanced Local Fractional Calculus & Its Applications" by Xiao-Jun Yang

* "Local Fractional Calculus: a Review" by Kiran M. Kolwankar

* "Local Fractional Derivatives of Differentiable Functions are Integer-order Derivatives or Zero" by Vasily E. Tarasov.

Thanks and regards,

Krushna