Point of Minimum Temperature

A homogeneous convex centrosymmetric body with constant thermal properties is initially at temperature zero and its boundary is maintained at a temperature Tb > 0. Prove or disprove that at any time t > 0, the point of minimum temperature is the center. Also, prove or disprove that the isothermal surfaces are convex and centrosymmetric. Note that the convexity of the isothermal surfaces will imply that the center is the point of minimum temperature. SOLUTIONS Two Integrals Arising from a Cloud Model Problem 90-12, by JOHANNES VERLINDE (Colorado State University). Determine a closed-form solution or a good approximation to the following integrals, where d is a rational number: 00 (1.1) We can get a general form for the closed form of this integral for the special case d = 1 from Prudnikov, Brychkov, and Marichev [1, form. 2.10.3.2]: (1.3) Ir (d = 1) =-(ao + v)) F,(v,acx+v; v+ 1;-b +rw (a er(a + v) / (1.4) I(d= 1)= Cb(a+) 2Fi(v,a+v;v+l;-) This problem arises in the mean collection growth equation as used in large mesoscale numerical cloud models. For simplicity, it is assumed that all parameters a, v, b, c, d are real and positive, to ensure convergence of the integrals. Since oo a1 ebtd Fr(v) F(a) (1) IF+I=F(V)I ta t le dt=r()() it is sufficient to consider the integral L, only. By inversion of the Mellin transform