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We consider an inverse problem of determining the time-dependent lowest ordercoefficient of two-dimensional (2D) heat equation with Ionkin boundary andtotal energy integral overdetermination condition. The well-posedness of theproblem is obtained by generalized Fourier method combined by the Banach fixedpoind theorem. For obtaining a numerical solution of the inverse problem, wepropose the discretization method from a new combination. On the one hand, itis known the traditional method of uniform finite difference combined withnumerical integration on a uniform grid (trapezoidal and Simpson's), on theother hand, we give the method of non-uniform finite difference is combined bya numerical integration on a non-uniform grid (with Gauss-Lobatto nodes).Numerical examples illustrate how to implement the method.

Inverse Problem of Finding the Coefficient of the Lowest Term in Two-dimensional Heat Equation with Ionkin-type Boundary Condition