# Analytic solution for one dimensional inverse heat conduction problem of semi-infinite bar

## TL;DR

This paper derives an analytical solution for the inverse heat conduction problem in a semi-infinite bar using fractional derivatives, providing a new mathematical approach with an existence theorem.

## Contribution

It introduces an explicit analytical formula and an existence theorem for the inverse heat conduction problem using fractional calculus and function space properties.

## Key findings

- Derived an analytical solution expressed as an infinite series of fractional derivatives.
- Established an existence theorem for the solution within a specific function space.
- Provides a mathematical framework for solving inverse heat conduction problems analytically.

## Abstract

We present analytical formula along with its existence theorem for solution of inverse heat conduction problem of semi-infinite bar, equivalent to a Volterra integral equation of first kind, as an infinite series of fractional derivatives. The mathematical method is based on some properties of function space M[0, T] (proved here) with respect to fractional integration and derivatives.