NUMERICAL-ANALYTICAL METHOD OF SOLUTION OF A NONLINEAR 
UNSTEADY HEAT-CONDUCTION EQUATION

V. D. Belik, B. A. Uryukov,
G. A. Frolov, and G. V. Tkachenko

UDC 536.2.083

A method of solution of a one-dimensional nonlinear unsteady 
heat-conduction equation has been proposed. The use of the method 
of Green's functions made it possible to transform the resulting equation 
to a nonlinear Volterra integral equation of the second kind for temperature, 
which is solved by the quadratic-form method. A system of recurrence 
relations, which is solved numerically, has been obtained. The influence 
of the nonlinearity on the temperature profiles has been analyzed. 
A comparison to the numerical finite-element method has shown that 
the numerical-analytical technique allows a reduction of more than 
103 times in the calculation time.

I. N. Frantsevich Institute of Problems of Materials Science, 
National Academy of Sciences of Ukraine, 3 Krzhizhanovskii 
Str., Kiev, 03142, Ukraine; email: uryukov@meta.ua. 
Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 81, No. 6, 
pp. 1058-1062, November-December, 2008. Original article 
submitted June 12, 2007; revision submitted March 18, 2008.