We consider an inverse problem for partial differential equation with nonlinear conductivity term in one-dimensional space within a finite interval. In the considered problem, a temperature history is unknown in a boundary of domain. The homotopy perturbation technique is used. Moreover, we have presented a numerical example.

Inverse heat conduction problems (IHCPs) rely on temperature heat flux measurements for estimating unknown quantities in the analysis of physical problems in thermal engineering. As an example, inverse problems dealing with heat conduction have been generally associated with estimating an unknown boundary heat flux by using temperature measurements taken below the boundary surfaces. Therefore, while in the classical direct heat conduction problem the cause (boundary heat flux) is given and the effect (temperature field in the body) is determined, the inverse problem involves the estimation of the cause from the knowledge of the effect. An advantage of IHCP is that it enables a much closer collaboration between experimental and theoretical researchers in order to obtain the maximum of information regarding the physical problem under study.

Difficulties encountered in the solution of IHCPs should be recognized. IHCPs are mathematically classified as ill-posed in a general sense because their solutions may become unstable, as a result of the errors inherent to the measurements used in the analysis. Inverse problems were initially taken as not of physical interest due to their ill-posedness.

In recent years, some new methods for estimating
surface heat flux in IHCPs developed in the theory and practice. Consequently,
some methods for approximate solution of these problems have developed. To
obtain stable results, special numerical techniques should be used. Examples
include iterative gradient methods, optimization algorithms, regularization
methods, function specification methods, space-marching method, conjugate
gradient method, Levenberg-Marquardt method, iterative techniques, variational
iteration, finite elements, finite volumes, boundary elements methods, and
so on. In some works, the numerical methods have been used for IHCPs [

The homotopy perturbation method (HPM) was first
proposed by the Chinese mathematician He [

In this paper, we consider a one-dimensional nonlinear inverse heat conduction problem with nonlinear diffusivity term that temperature history is unknown in a boundary. Then, using finite difference method and discrete time variable, the partial differential equation converts to a system of nonlinear ordinary differential equations. Consequently, by applying the homotopy perturbation technique and estimate solution, the temperature distribution in domain at discrete times will be found. Finally, a numerical experiment is given.

We begin with
the following definition which is presented in [

Let

To illustrate homotopy perturbation method, we
consider the following nonlinear equation:

In [

Noticing that

Let

Now, consider the following nonlinear differential
equation:

Suppose that

Now, we can write (

For simplicity, define

By twice integration of (

By homotopy perturbation method, we may choose a
convex homotopy such that [

Let

Obviously, the above assumptions satisfy consideration
of conditions. The exact solution is

Exact solution, approximate solution, and relative
error for the above problem are given in Table

Exact and approximate solution of

Exact solution | Approximate solution | Relative error | |
---|---|---|---|

0.1 | 0.2628402542 | 0.2628399122 | |

0.2 | 0.3013610167 | 0.3013563938 | |

0.3 | 0.3655622875 | 0.3655396925 | |

0.4 | 0.4554440667 | 0.4553735995 | |

0.5 | 0.5710063542 | 0.5708355086 | |

0.6 | 0.7122491501 | 0.7118965132 | |

0.7 | 0.8791724543 | 0.8785215163 | |

0.8 | 1.071776267 | 1.070669376 | |

0.9 | 1.290060588 | 1.288293071 | |

1 | 1.534025417 | 1.531339891 |

Exact and approximate solution of

Exact solution | Approximate solution | Relative error | |
---|---|---|---|

0.1 | 0.5164872127 | 0.5164865028 | |

0.2 | 0.5659488508 | 0.5659409374 | |

0.3 | 0.6483849144 | 0.6483481508 | |

0.4 | 0.7637954034 | 0.7636830954 | |

0.5 | 0.9121803178 | 0.9119111460 | |

0.6 | 1.093539658 | 1.092988531 | |

0.7 | 1.307873423 | 1.306862885 | |

0.8 | 1.555181613 | 1.553473905 | |

0.9 | 1.835464230 | 1.832754112 | |

1 | 2.148721271 | 2.144629711 |

Exact and approximate solution of

Exact solution | Approximate solution | Relative error | |
---|---|---|---|

0.1 | 0.7711700002 | 0.7711686828 | |

0.2 | 0.8346800007 | 0.8346667665 | |

0.3 | 0.9405300015 | 0.9404704769 | |

0.4 | 1.088720003 | 1.088540598 | |

0.5 | 1.279250004 | 1.278823053 | |

0.6 | 1.512120006 | 1.511249735 | |

0.7 | 1.787330008 | 1.785739520 | |

0.8 | 2.104880011 | 2.102199488 | |

0.9 | 2.464770014 | 2.460526294 | |

1 | 2.867000017 | 2.860607703 |

Exact and approximate solution of

Exact solution | Approximate solution | Relative error | |
---|---|---|---|

0.1 | 1.027182818 | 1.027180602 | |

0.2 | 1.108731273 | 1.108709841 | |

0.3 | 1.244645364 | 1.244550232 | |

0.4 | 1.434925092 | 1.434640074 | |

0.5 | 1.679570457 | 1.678894579 | |

0.6 | 1.978581458 | 1.977207417 | |

0.7 | 2.331958096 | 2.329452661 | |

0.8 | 2.739700370 | 2.735487036 | |

0.9 | 3.201808281 | 3.195152502 | |

1 | 3.718281828 | 3.708279035 |

In the obtained results of problem, we see that the
approximate solutions for small increment