Geothermics, mathematical problems in

Mathematical problems involved in the study of thermal processes taking place in the Earth. In geothermics one distinguishes between surface phenomena connected with temperature fluctuations in the upper layers of the Earth produced by solar radiation, and deep phenomena related to the temperature distribution produced by radioactive heat sources.

Mathematical problems in geothermics are mainly involved in solving quasi-linear parabolic equations the coefficients of which vary with the depth of the immersion and depend on the temperature. In the study of freezing in the upper layers of the Earth or the thawing processes in its deeper layers allowance is made for phase transitions, i.e. changes in the physical state of the medium. This gives rise to the so-called Stefan problem, or the phase-transition problem. The most effective numerical method for solving these problems is the calculus of finite differences, which is extensively employed in practice.

A number of problems in geothermics involve studies of the interaction between the field of temperatures and other physical phenomena. In the analysis of problems involving the freezing of soil with allowance for the inflow of water, the equations of thermal conductance and the filtration equations are solved simultaneously. Studies of temperature distribution in bodies of water necessitate a simultaneous solution of the thermal conductance and thermal convection equations. Thermoelastic stresses in the Earth and the related effects of dilatation and deformation of the Earth are analyzed by simultaneously solving the equation of thermal conductance and the equation of elastic equilibrium in a gravity field.

There exist several specific problems in this field. Thus, the problem of determining the historical climate of the Earth posed the inverse mathematical problem of the thermal conductance equation, where the temperature is to be determined at the moment of time $t<t_0$ from the given temperature distribution over a depth at the moment of time $t=t_0$. The solution of the thermal conductance equation $u(x,t)$ in the domain $x>0$, $-\infty<t<t_0$ is uniquely determined by the given values $u(x,t_0)=\phi(x)$ if at least one derivative of the solution with respect to $x$ is uniformly bounded, $|\partial^nu/\partial x^n|<M$ [1].

The study of the effect of solar radiation on the temperature conditions prevailing on Earth and on other celestial bodies lead to the problem of the equation of thermal conductance when the boundary conditions are non-linear, in particular for radiation obeying the Stefan–Boltzmann law. These problems may be reduced to non-linear integral equations of Volterra type [1].

References

[1]	A.N. Tikhonov, Dokl. Akad. Nauk SSSR , 1 : 5 (1935) pp. 294–300
[2]	A.N. Tikhonov, Izv. Akad. Nauk SSSR Otd. Mat. Estestv. Nauk. Ser. Geogr. , 3 (1937) pp. 461–479
[3]	A.N. Tikhonov, "On boundary conditions containing derivatives of an order higher than the order of the equation" Mat. Sb. , 26 (68) : 1 (1950) pp. 35–56 (In Russian)
[4]	E.A. Lyubimova, "Thermics of the Earth and the moon" , Moscow (1968) (In Russian)

References

[a1]	H.S. Carslaw, J.C. Jaeger, "Conduction of heat in solids" , Clarendon Press (1959)