# Non-local initial problem for second order time-fractional and space-singular equation

## TL;DR

This paper develops an explicit solution for a second order time-fractional PDE with space singularities, using Fourier-Bessel series and special functions, addressing non-local boundary conditions.

## Contribution

It introduces a novel explicit solution method for a class of fractional PDEs with non-local conditions, involving Fourier-Bessel series and special functions.

## Key findings

- Explicit solution expressed via Fourier-Bessel series.
- Solution involves multinomial Mittag-Leffler and Bessel functions.
- Addresses non-local boundary conditions in fractional PDEs.

## Abstract

In this work, we consider an initial problem for second order partial differential equations with Caputo fractional derivatives in the time-variable and Bessel operator in the space-variable. For non-local boundary conditions, we present a solution of this problem in an explicit form representing it by the Fourier-Bessel series. The obtained solution is written in terms of multinomial Mittag-Leffler functions and first kind Bessel functions.