It is usually stated that to solve any mathematical problem described by a differential equation, we must have the number of boundary conditions equal to the order of the PDE that is to be solved; in addition, if the problem is transient in nature, an initial condition must also be prescribed. Further, the number of unknowns being sought must be equal to the number of available equations. This is discussed in general in Section 3.3.6. For integral equations, the boundary conditions are usually implicit in the limits of the integrals. Suppose, however, that these conditions are not met; for example, more than the required number of boundary conditions are prescribed on some surface or surfaces, and it is desired to find the missing boundary conditions on other surfaces. In some cases of this type, the number of equations may not match the number of unknowns. These problems fall under the class of inverse problems, because the requirements for standard solution are not met. Problems of this type may be very sensitive to the imposed values of the given boundary conditions, and solutions may vary wildly. This type of problem is referred to as ill-posed. Inverse solutions in heat transfer may be required when experimental observations of temperature or heat flux are not available at the physical location where they are needed, or radiative property distributions in a participating medium must be obtained from remote measurements. For example on the re-entry heat shield of a spacecraft (Fig. 10.20), it may be impossible to measure temperature Ts(t) or heat flux Failed to parse (lexing error): {{q"}_s}(t)

```on the ablating surface of the shield, but transient temperatures T(t) and heat fluxes Failed to parse (lexing error): q"(t)
can be measured on the interior surface. Can these remote measurements be used to determine the unknown temperature and heat flux on the heat shield surface? Intuitively, it would seem so: It isn't so easy!
```

The governing equations for inverse problems tend to be mathematically ill-posed, and predicting conditions on the remote boundary can result in multiple solutions, physically unrealistic solutions, or solutions that oscillate greatly in space and time. In this Section, we examine why these difficulties arise, and show one solution method for handling them. Various methods may be applied for overcoming the ill-posed nature of the governing equations. Tikhonov (1963) and Phillips (1962) are often credited with developing the first systematic treatment for these types of inverse problems. For problems dominated by conduction, there are texts and monographs available that demonstrate many of these methods (Tikhonov, 1963; Alifanov, 1994; Alifanov et al. 1995; Beck et al., 1995; Özişik and Orlande, 2000).

Figure 10.20 Determining the heat shield hot surface temperature Ts(t) and heat flux Failed to parse (lexing error): {{q"}_s}(t)
```from measurement of remote temperature and heat flux (T(t) and Failed to parse (lexing error): q"(t)
```
)

Alifanov, O.M., 1994, Inverse Heat Transfer Problems, Springer, Berlin.

Alifanov, O.M., Artyukhin, E.A., and Rumyantsev, S.V., 1995, Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems, Begell House, New York, NY.

Beck, J.V., Blackwell, B., and St. Clair, Jr., C.R., 1995, Inverse Heat Conduction: Ill-Posed Problems, Wiley-Interscience, New York, NY.

Daun, K.J., Erturk, H., Gamba, M. Hosseini Sarvari, M. and Howell, J.R., 2003, “The Use of Inverse Methods for the Design and Control of Radiant Sources,” JSME International Journal, Series B, 46, no. 4, 470-478, 2003.

França, F.H.R., Howell, J.R., Ezekoye, O.A., and Morales, J.C., 2002, “Inverse Design of Thermal Systems,” Chap. 1, Advances in Heat Transfer, J.P. Hartnett and T.F. Irvine, eds., Elsevier, Vol. 36, 1-110.

Hansen, P.C., 1998, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM, Philadelphia, PA.

Özişik, M.N. and Orlande, H.R.B., 2000, Inverse Heat Transfer: Fundamentals and Applications, Taylor and Francis, New York, NY.

Phillips, D.L., 1962, “A Treatment for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Computing Machinery, Vol. 9, pp. 84-97.

Tikhonov, A.N., Solution of Incorrectly Formulated Problems and the Regularization Method, Soviet Math. Dokl., 4, 1035-1038, 1963,. [Engl. trans. Dokl. Akad. Nauk. SSSR, 151, 501-504].

Wing, G.W., 1991, A Primer on Integral Equations of the First Kind, SIAM, Philadelphia, PA.