Formula for the fundamental solution of the heat equation on the sphere (Q5950598)

Asymptotic formula for the fundamental solution of the heat equation on the sphere [see \textit{K. Yosida}, Ann. Math. Stat. 20, 292-297 (1949; Zbl 0033.38503)] \(\frac{\partial f}{\partial t}=\frac 12\Delta f\), \(f_{|t=0}=\delta\), where \(\delta\) is a Dirac \(\delta\)-function at the north pole of the unit sphere is obtained. Using spherical coordinates \(\varphi,\theta\) and rotational symmetry, the considered equation is represented in the form NEWLINE\[NEWLINE \frac{\partial v}{\partial t}=\frac 12\left(\frac{\partial^2 v} {\partial\theta^2}-\cot\theta\frac{\partial v}{\partial\theta}+ \frac 1{\sin^2\theta}v\right),\quad v_{|t=0}=\delta, NEWLINE\]NEWLINE where \(v(\theta,t)=2\pi f(\theta,t)\sin\theta\). The function \(v(\theta,t)\) is taken in the form NEWLINE\[NEWLINE\frac{\sin\theta}t\exp(-\theta^2/2t) \sum_{j=0}^\infty w_j(\theta)t^j,NEWLINE\]NEWLINE where for the functions \(w_j(\theta)\) the recurrence relation is obtained. The leading term \(f_0(\theta,t)\) and three next approximations of \(f(\theta,t)\) are found. The Maple software was used.