Ritz–Galerkin method for solving an inverse problem of parabolic equation with moving boundaries and integral condition

ABSTRACT

In this paper, we consider an inverse problem for parabolic equation with moving boundaries and integral condition. Existence and uniqueness are proved using tool from double-layer heat potential and Volterra integral equations of the second kind. For computation part, the Ritz–Galerkin method is first implemented to get an approximate solution of intermediate function H(x, t), and then based on relationships between H(x, t) and $\lambda (t),u(x,t)$, respectively, numerical solutions of $\lambda (t)$ and u(x, t) are obtained. Finally, numerical examples are presented to show the validity and applicability of our method.

KEYWORDS:

• Ritz–Galerkin method
• inverse problem
• parabolic equation
• moving boundaries
• initial boundary value problem
• approximation solution

AMS Subject Classifications:

• 65L09
• 65M32
• 35K05
• 35K15
• 35K20
• 35Q68

Notes

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]; the Science Research Fund of Department of Guangdong Province of the People’s Republic of China [grant number Yq2013161].