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Paper IPM / M / 7375  


Abstract:  
In this paper a twodimensional heat equation is considered. The
problem has both Numann and Dirichlet boundary conditions and one
nonlocal condition in which an integral of the unknown solution
u occurs. The Dirichlet boundary condition contains an
additional unknown function μ(t). In this paper the numerical
solution of this equation is treated. Due to the structure of the
boundary conditions a reduced onedimensional heat equation for
the new unknown v(y,t)=∫u(x,y,t)dx can be formulated. The
resulting problem has a nonlocal boundary condition. This
onedimensional heat equation is solved by Saulyev's formula. From
the solution of this onedimensional problem an approximation of
the function μ(t) is obtained. Once this approximation is
known, the given twodimensional problem reduces to a standard
heat equation with the usual Neumann's boundary conditions. This
equation is solved by an extension of the Saulyev's techniques.
Results of numerical experiments are presented.
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