Numerical solution of a nonlinear integral equation for the determination of the unknown time-dependent diffusivity of radioactive materials

The thermal diffusion coefficient in radioactive materials is not a constant value, and this makes the heat transfer problem different in such materials and materials that are not homogeneous and have been destroyed or are disintegrating in some way. In the problem under discussion, the heat diffusion coefficient is time-dependent and satisfies a nonlinear integral equation. The existence and uniqueness of the solution to the integral equation in question are discussed in detail in Chapter ۱۳ of the Book "Encyclopedia of the One-Dimensional Heat Equation" by Cannon, J. R. The integral equation in question is not a standard Volterra integral equation and therefore has not been studied much from a numerical perspective. For example, if we apply the fixed point method, which is a powerful tool in the analysis of existence and uniqueness, discussed in Chapter ۱۳ of the aforementioned Book, to a numerical solution, we cannot go even one step forward with this method. Since the unknown function is located at the kernel of a nested integral, applying canonical methods becomes difficult. Therefore, in this paper, we have discussed a hybrid method of numerical integration and iterative methods that solves the problem with sufficient accuracy. In Section ۵, we have extracted several sample problems using the properties of the heat equation in the case where the thermal diffusivity is Time-dependent. The numerical solution of these sample problems in the Section ۶ demonstrates the efficiency and accuracy of the proposed method.

کلیدواژه ها:

Nonlinear integral equation ، Heat equation ، Numerical solution ، Time-dependent thermal diffusivity