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The method of the fundamental solutions (MFS) is used to construct anapproximate solution for a partial differential equation in a bounded domain.It is demonstrated by combining the fundamental solutions shifted to the pointsoutside the domain and determining the coefficients of the linear sum tosatisfy the boundary condition on the finite points of the boundary. In thispaper, the existence of the approximate solution by the MFS for the Neumannproblems of the modified Helmholtz equation in disk domains is rigorouslydemonstrated. We reveal the sufficient condition of the existence of theapproximate solution. Applying Green's theorem to the Neumann problem of themodified Helmholtz equation, we bound the error between the approximatesolution and exact solution into the difference of the function of the boundarycondition and the normal derivative of the approximate solution by boundaryintegrations. Using this estimate of the error, we show the convergence of theapproximate solution by the MFS to the exact solution with exponential order,that is, $N^2a^N$ order, where $a$ is a positive constant less than one and $N$is the number of collocation points. Furthermore, it is demonstrated that theerror tends to $0$ in exponential order in the numerical simulations withincreasing number of collocation points $N$.

Method of fundamental solutions for Neumann problems of the modified Helmholtz equation in disk domains