Another way of solving a free boundary problem related to DCIS model

Abstract

In this paper, we consider a free boundary problem related to ductal carcinoma in situ (DCIS) model. Existence and uniqueness theorem is proved by using Banach fixed point theorem. For computation part, the decomposition method of Adomian is first implemented to get an approximate solution of intermediate function $v(x,t)$, which satisfies a nonlinear equation with initial boundary value conditions, and then based on relationships between $v(x,t)$ and $s\left(t\right)$, numerical solutions $\tilde{s}\left(t\right)$ of free boundary $s\left(t\right)$ are obtained. The approximate solution $\tilde{u}(x,t)$ can be straightly derived by applying Adomian decomposition method to the linear equation involving $u(x,t)$. Finally, a numerical example is presented to show the validity and applicability of our method.

Keywords:

• Free boundary problem
• Adomian decomposition method (ADM)
• parabolic equation
• ductal carcinoma in situ (DCIS) model
• approximation solution

2010 Mathematics Subject Classifications:

• 35K05
• 35K20
• 35Q68
• 35R30
• 35R35

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The present investigation was supported in part by the National Natural Science Foundation of the People's Republic of China [grant number 11201070] and the Science Research Fund of Department of Guangdong Province of the People's Republic of China [grant number Yq2013161].