## Abstract

We describe the formulation of forward and inverse problems in heat transfer and illustrate several techniques applicable to their solution.

### Keywords

- heat transfer
- finite differences
- regularization
- Fourier transforms
- interpolation

## 1. Introduction

Heat flow is described by the heat (or diffusion) partial differential equation, which takes the form

### 1.1 Derivation of homogeneous and non-homogeneous equations

We now discuss two simple scenarios which can be used to derive homogeneous and non-homogeneous forms of the heat Equation [2]. Heat is the flow of energy from a warmer to a cooler location. There are several types of heat transfer including conduction (flow of heat through stationary material), convection (flow of heat through fluids), and radiation (flow of heat through electromagnetic waves) [2]. In this chapter we are concerned mainly with heat conduction for which the differential heat model applies. We take

Next, using Taylor’s approximation

For the exothermic slab case exhibited in Figure 2 on the right, we obtain an inhomogeneous equation. We assume the slab is filled with a material which emits heat at a rate

### 1.2 Numerical schemes

Discretization of the derivative operators yields the finite difference formulation. A number of different schemes are possible based on the difference schemes used to replace the derivatives in the equation resulting in either an explicit method (where the update for *forward* or *inverse*, in the sense that they either predict the temperature profile at later time from the knowledge of the initial temperature distribution at time

## 2. Forward problem

The forward problem is concerned by the prediction of heat profile in future time from the initial condition. We first consider the heat equation initial boundary value problem in 1D:

with constant

A quick calculation shows that

which results in the explicit finite difference scheme:

This can be re-written as

If we use the backward time finite difference approximation for

Again using

Note that this takes the form of the linear system

In 2D, the standard explicit scheme arises from the discretization:

Like in the 1D scheme, suitable finite difference approximations yield the implicit method.

A practically useful direction to consider is the use of non-standard finite difference (nsfd) schemes [5]. Based on (5), we define the discretized operators:

and proceed to evaluate

Setting

Hence, the nsfd scheme follows if we make the following replacement in (6):

When

By this it follows that for small

### 2.1 Numerical experiments

Below, we present some sample results for the case of constant

An interesting case is that of heat transfer with an exothermic material, which can be modeled with a heat source distributed throughout [3]. Considering an infinitesimal segment as in Figure 1 yields the energy balance equation:

## 3. Inverse problem

The inverse problem consists of predicting the initial condition from the observation of the heat profile at some later time

In this case, separation of variables can be used and after a well-known sequence of steps, the following series solution can be derived [3]:

It follows that

A more general approach for inversion is to utilize the inverse problem formulations of the matrix heat propagation formulation. For example, for the standard implicit or Crank-Nicholson scheme, we have the system

with

This means that

This can be accomplished by replacing the naive solution *predictor-corrector* scheme to iteratively improve the solution. In this approach, the approximation to

### 3.1 Numerical experiments

As an example, consider the 1D cosine based initial heat profile with

fuT=**fft** (u (:, n)); *% temperature profile at time T*

R=**exp** (k.*k ′*T);

**filter** (R);

init_cond_approx=**real** (**ifft** (fuT.*R));

The term

## 4. Conclusions

This article discusses numerical approaches for the heat equation, both for the forward and inverse problems with examples for homogeneous and non-homogeneous cases. For the forward heat propagation modeling problem, we have presented a simple non-standard finite difference formulation, which satisfies the differential model and appears to result in improved performance over the standard finite difference scheme with similar magnitude time and spatial step sizes. For the inverse problem, we have described approaches based on Fourier filtering and Tikhonov regularization.